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b(:)g(:)g(:)i(;)e(n)905 b Fr(\(1)p Fj(:)p Fr(3\))50 1263 y(where)27 b Fj(x)362 1277 y Fu(0)428 1263 y Fk(\021)e Fr(0.)40 b(Of)26 b(course)1035 1195 y Fe(P)1131 1217 y Fi(n)1131 1290 y(i)p Fu(=1)1268 1263 y Fj(a)1316 1277 y Fi(i)1372 1263 y Fr(=)f(1,)j(and)1740 1195 y Fe(Q)1826 1217 y Fi(n)1826 1290 y(i)p Fu(=1)1963 1263 y Fj(\013)2021 1277 y Fi(i)2077 1263 y Fr(=)d(2.)40 b(The)27 b(h)m(yp)s(ercub)s(e)f Fk(S)2959 1277 y Fi(N)3055 1263 y Fr(can)h(then)g(b)s(e)g(con-)50 1409 y(sidered)e(as)h(a)h Fj(n)p Fr(-fold)e(pro)s(duct)g Fk(S)1186 1423 y Fi(N)1280 1409 y Fr(=)1376 1340 y Fe(Q)1462 1363 y Fi(n)1462 1436 y(i)p Fu(=1)1599 1409 y Fk(S)1654 1425 y Fi(N)19 b Fu(ln)13 b Fi(\013)1851 1434 y Fm(i)1881 1425 y Fi(=)f Fu(ln)h(2)2044 1409 y Fr(.)39 b(W)-8 b(e)28 b(will)c(write)h Fj(\033)k Fr(=)24 b Fj(\033)2885 1423 y Fu(1)2926 1409 y Fj(\033)2978 1423 y Fu(2)3034 1409 y Fj(:)15 b(:)g(:)h(\033)3207 1423 y Fi(n)3283 1409 y Fr(where)26 b Fj(\033)3594 1423 y Fi(i)3649 1409 y Fk(2)50 1554 y(S)105 1570 y Fi(N)19 b Fu(ln)13 b Fi(\013)302 1579 y Fm(i)331 1570 y Fi(=)g Fu(ln)g(2)494 1554 y Fr(.)41 b(Usually)29 b(w)m(e)h(will)e(assume)i (that)h Fj(x)1757 1568 y Fu(1)1823 1554 y Fj(>)24 b Fr(0,)31 b Fj(x)2071 1568 y Fi(n)2146 1554 y Fr(=)25 b(1,)31 b(and)e(all)g Fj(a)2693 1568 y Fi(i)2749 1554 y Fj(>)c Fr(0.)41 b(Then)29 b(the)i(Gaussian)50 1699 y(pro)s(cess)23 b Fj(X)435 1713 y Fi(\033)507 1699 y Fr(can)h(b)s(e)f(represen)m(ted)g(in)f(the)i (follo)m(wing)d(w)m(a)m(y)-8 b(.)40 b(T)-8 b(ak)m(e)25 b Fj(\013)2366 1666 y Fi(N)2366 1722 y Fu(1)2442 1699 y Fr(+)6 b(\()p Fj(\013)2612 1713 y Fu(1)2653 1699 y Fj(\013)2711 1713 y Fu(2)2752 1699 y Fr(\))2787 1666 y Fi(N)2862 1699 y Fr(+)g Fk(\001)15 b(\001)g(\001)7 b Fr(+)f(\()p Fj(\013)3221 1713 y Fu(1)3263 1699 y Fj(\013)3321 1713 y Fu(2)3377 1699 y Fk(\001)15 b(\001)g(\001)h Fj(\013)3556 1713 y Fi(n)3606 1699 y Fr(\))3641 1666 y Fi(N)50 1845 y Fr(indep)s(enden)m(t)38 b(standard)i(Gaussian)f(random)h(v)-5 b(ariables)39 b(en)m(umerated)i(as)f(follo)m(ws:)60 b(The)40 b(\014rst)g Fj(\013)3527 1812 y Fi(N)3527 1867 y Fu(1)3637 1845 y Fr(of)50 1990 y(them)33 b Fj(X)360 2004 y Fi(\033)402 2013 y Fn(1)476 1990 y Fr(are)g(en)m(umerated)h(b)m(y)f Fj(\013)1313 1957 y Fi(N)1313 2013 y Fu(1)1416 1990 y Fr(con\014gurations)g(of)g(spins)e Fj(\033)2395 2004 y Fu(1)2466 1990 y Fk(2)f(f\000)p Fr(1)p Fj(;)15 b Fr(+1)p Fk(g)2919 1957 y Fi(N)20 b Fu(ln)13 b Fi(\013)3117 1966 y Fn(1)3153 1957 y Fi(=)g Fu(ln)g(2)3317 1990 y Fr(,)34 b(the)g(next)50 2136 y(\()p Fj(\013)143 2150 y Fu(1)184 2136 y Fj(\013)242 2150 y Fu(2)283 2136 y Fr(\))318 2103 y Fi(N)416 2136 y Fr(of)28 b(them)g Fj(X)822 2150 y Fi(\033)864 2159 y Fn(1)900 2150 y Fi(\033)942 2159 y Fn(2)1011 2136 y Fr(|)g(b)m(y)g(con\014gurations)g(of)g(spins)e Fj(\033)2218 2150 y Fu(1)2259 2136 y Fj(\033)2311 2150 y Fu(2)2377 2136 y Fk(2)f(f\000)p Fr(1)p Fj(;)15 b Fr(+1)p Fk(g)2825 2103 y Fi(N)20 b Fu(ln)q(\()p Fi(\013)3039 2112 y Fn(1)3075 2103 y Fi(\013)3122 2112 y Fn(2)3158 2103 y Fu(\))p Fi(=)13 b Fu(ln)g(2)3350 2136 y Fr(,)29 b(etc)g Fj(:)15 b(:)g(:)32 b Fr(,)50 2281 y(\014nally)i(the)i(last)f(\()p Fj(\013)763 2295 y Fu(1)805 2281 y Fj(\013)863 2295 y Fu(2)919 2281 y Fk(\001)15 b(\001)g(\001)h Fj(\013)1098 2295 y Fi(n)1148 2281 y Fr(\))1183 2248 y Fi(N)1286 2281 y Fr(=)34 b(2)1436 2248 y Fi(N)1542 2281 y Fr(of)i(them)f Fj(X)1963 2295 y Fi(\033)2005 2304 y Fn(1)2041 2295 y Fi(\033)2083 2304 y Fn(2)2119 2295 y Fi(;:::)11 b(;\033)2282 2304 y Fm(n)2366 2281 y Fr(are)37 b(en)m(umerated)f(b)m(y)f(con\014gurations)50 2427 y Fj(\033)102 2441 y Fu(1)143 2427 y Fj(\033)195 2441 y Fu(2)250 2427 y Fj(:)15 b(:)g(:)i(\033)424 2441 y Fi(n)498 2427 y Fk(2)25 b(f\000)p Fr(1)p Fj(;)15 b Fr(+1)p Fk(g)946 2394 y Fi(N)1017 2427 y Fr(.)40 b(Then)30 b(the)g(pro)s(cess)g Fj(X)1868 2441 y Fi(\033)1948 2427 y Fr(has)g(the)h(same)f(distribution)d(as)477 2656 y Fj(X)552 2670 y Fi(\033)626 2656 y Fk(\021)722 2593 y(p)p 798 2593 89 4 v 63 x Fj(a)846 2670 y Fu(1)887 2656 y Fj(X)962 2670 y Fi(\033)1004 2679 y Fn(1)1065 2656 y Fr(+)1155 2593 y Fk(p)p 1231 2593 V 63 x Fj(a)1279 2670 y Fu(2)1320 2656 y Fj(X)1395 2670 y Fi(\033)1437 2679 y Fn(1)1473 2670 y Fi(\033)1515 2679 y Fn(2)1575 2656 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)1883 2593 y Fk(p)p 1959 2593 98 4 v 63 x Fj(a)2007 2670 y Fi(n)2057 2656 y Fj(X)2132 2670 y Fi(\033)2174 2679 y Fn(1)2210 2670 y Fi(\033)2252 2679 y Fn(2)2287 2670 y Fi(:::)o(\033)2394 2679 y Fm(n)2443 2656 y Fj(;)107 b Fr(if)29 b Fj(\033)f Fr(=)d Fj(\033)2886 2670 y Fu(1)2927 2656 y Fj(\033)2979 2670 y Fu(2)3035 2656 y Fj(:)15 b(:)g(:)h(\033)3208 2670 y Fi(n)3258 2656 y Fj(;)240 b Fr(\(1)p Fj(:)p Fr(4\))50 2886 y(The)30 b(partition)f(function)g(of)h(the)h(GREM)g(at)g(the)f(in) m(v)m(erse)h(temp)s(erature)f Fj(\014)35 b Fr(is)30 b(de\014ned)f(as) 158 3095 y Fj(Z)220 3109 y Fi(N)s(;\014)377 3095 y Fr(=)c(2)518 3058 y Fh(\000)p Fi(N)692 3009 y Fe(X)659 3204 y Fi(\033)r Fh(2S)797 3213 y Fm(N)871 3095 y Fj(e)913 3058 y Fi(\014)957 3003 y Fh(p)p 1017 3003 65 4 v 55 x Fi(N)7 b(X)1142 3067 y Fm(\033)377 3354 y Fr(=)25 b(2)518 3316 y Fh(\000)p Fi(N)854 3268 y Fe(X)659 3463 y Fi(\033)701 3472 y Fn(1)737 3463 y Fh(2S)831 3475 y Fm(N)15 b Fn(ln)c Fm(\013)1001 3489 y Fn(1)1036 3475 y Fm(=)h Fn(ln)e(2)1119 3354 y Fj(e)1161 3316 y Fi(\014)1205 3267 y Fh(p)p 1266 3267 140 4 v 1266 3316 a Fi(a)1305 3325 y Fn(1)1340 3316 y Fi(N)d(X)1465 3325 y Fm(\033)1502 3339 y Fn(1)1650 3268 y Fe(X)1455 3463 y Fi(\033)1497 3472 y Fn(2)1533 3463 y Fh(2S)1627 3475 y Fm(N)16 b Fn(ln)10 b Fm(\013)1797 3489 y Fn(2)1832 3475 y Fm(=)i Fn(ln)e(2)1915 3354 y Fj(e)1957 3316 y Fi(\014)2001 3267 y Fh(p)p 2062 3267 V 2062 3316 a Fi(a)2101 3325 y Fn(2)2137 3316 y Fi(N)c(X)2261 3325 y Fm(\033)2298 3339 y Fn(1)2333 3325 y Fm(\033)2370 3339 y Fn(2)2430 3354 y Fj(:)15 b(:)g(:)2649 3268 y Fe(X)2445 3463 y Fi(\033)2487 3472 y Fm(n)2532 3463 y Fh(2S)2626 3475 y Fm(N)g Fn(ln)c Fm(\013)2796 3484 y(n)2840 3475 y(=)h Fn(ln)e(2)2923 3354 y Fj(e)2965 3316 y Fi(\014)3009 3267 y Fh(p)p 3070 3267 149 4 v 3070 3316 a Fi(a)3109 3325 y Fm(n)3154 3316 y Fi(N)c(X)3278 3325 y Fm(\033)3315 3339 y Fn(1)3350 3325 y Fm(\033)3387 3339 y Fn(2)3423 3325 y Fm(:::)o(\033)3522 3334 y(n)3576 3354 y Fj(:)3523 3575 y Fr(\(1)p Fj(:)p Fr(5\))50 3720 y(The)30 b(Gibbs)e(measure)j Fj(\026)910 3734 y Fi(\014)s(;N)1075 3720 y Fr(is)e(de\014ned)g(as)1491 3995 y Fj(\026)1546 4009 y Fi(\014)s(;N)1681 3995 y Fr(\()p Fj(\033)s Fr(\))d Fk(\021)1939 3933 y Fj(e)1981 3900 y Fi(\014)2025 3846 y Fh(p)p 2086 3846 65 4 v 2086 3900 a Fi(N)6 b(X)2210 3909 y Fm(\033)p 1939 3974 319 4 v 2000 4057 a Fj(Z)2062 4071 y Fi(\014)s(;N)3523 3995 y Fr(\(1)p Fj(:)p Fr(6\))159 4317 y(T)-8 b(o)27 b(b)s(e)e(able)h(to)h (state)g(our)f(main)f(results,)h(w)m(e)h(m)m(ust)f(recall)f(some)i (notation)f(and)g(results)f(from)g([BK1].)159 4535 y(Recall)30 b(the)g(function)g Fj(u)998 4549 y Fu(ln)13 b Fi(\013;N)1209 4535 y Fr(\()p Fj(x)p Fr(\),)31 b Fj(x)25 b Fk(2)g Fg(R)39 b Fr(giv)m(en)30 b(b)m(y)698 4779 y Fj(u)750 4793 y Fu(ln)13 b Fi(\013;N)961 4779 y Fr(\()p Fj(x)p Fr(\))26 b(=)1205 4697 y Fk(p)p 1281 4697 293 4 v 82 x Fr(2)15 b(ln)f Fj(\013N)31 b Fr(+)1854 4718 y Fj(x)p 1696 4758 369 4 v 1696 4777 a Fk(p)p 1772 4777 293 4 v 77 x Fr(2)15 b(ln)f Fj(\013N)2096 4779 y Fk(\000)2197 4718 y Fr(ln)g Fj(N)31 b Fr(+)19 b(ln)14 b(ln)g Fj(\013)21 b Fr(+)f(ln)14 b(4)p Fj(\031)p 2197 4758 828 4 v 2404 4854 a Fr(2)2449 4777 y Fk(p)p 2525 4777 293 4 v 77 x Fr(2)h(ln)g Fj(\013N)3036 4779 y(:)462 b Fr(\(1)p Fj(:)p Fr(7\))159 5098 y(The)30 b(main)f(result)g (from)h([BK1])i(that)f(w)m(e)f(will)e(need)i(in)f(the)i(sequal)f(is)50 5316 y Fd(Theorem)44 b(1.1:)64 b Fq(L)-5 b(et)38 b Fj(\013)1005 5330 y Fi(i)1070 5316 y Fk(\025)c Fr(1)p Fq(,)39 b Fr(0)c Fj(<)f(a)1520 5330 y Fi(i)1584 5316 y Fj(<)g Fr(1)p Fq(,)39 b Fj(i)c Fr(=)f(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(n)p Fq(,)2441 5248 y Fe(Q)2527 5271 y Fi(n)2527 5344 y(i)p Fu(=1)2665 5316 y Fj(\013)2723 5330 y Fi(i)2787 5316 y Fr(=)34 b(2)p Fq(,)3004 5248 y Fe(P)3100 5271 y Fi(n)3100 5344 y(i)p Fu(=1)3238 5316 y Fj(a)3286 5330 y Fi(i)3350 5316 y Fr(=)g(1)p Fq(.)57 b(L)-5 b(et)50 5462 y Fj(J)100 5476 y Fu(1)141 5462 y Fj(;)15 b(J)231 5476 y Fu(2)273 5462 y Fj(;)g(:)g(:)g(:)31 b(;)15 b(J)539 5476 y Fi(m)648 5462 y Fk(2)39 b Fg(N)52 b Fq(b)-5 b(e)40 b(the)h(indic)-5 b(es)40 b(such)h(that)g Fr(0)e(=)g Fj(J)2092 5476 y Fu(0)2172 5462 y Fj(<)f(J)2331 5476 y Fu(1)2411 5462 y Fj(<)h(J)2571 5476 y Fu(2)2651 5462 y Fj(<)g Fk(\001)15 b(\001)g(\001)40 b Fj(<)e(J)3065 5476 y Fi(m)3173 5462 y Fr(=)h Fj(n)p Fq(.)64 b(Denote)p eop %%Page: 4 4 4 3 bop 50 295 a Fo(4)3309 b Fl(Se)l(ction)27 b Fo(1)50 513 y Fq(by)36 b Fr(\026)-46 b Fj(a)219 527 y Fi(l)276 513 y Fk(\021)377 445 y Fe(P)473 467 y Fi(J)513 477 y Fm(l)473 540 y Fi(i)p Fu(=)p Fi(J)595 550 y Fm(l)p Ff(\000)p Fn(1)701 540 y Fu(+1)813 513 y Fj(a)861 527 y Fi(i)891 513 y Fq(,)44 b Fr(\026)-54 b Fj(\013)1012 527 y Fi(l)1070 513 y Fk(\021)1170 445 y Fe(Q)1256 467 y Fi(J)1296 477 y Fm(l)1256 540 y Fi(i)p Fu(=)p Fi(J)1378 550 y Fm(l)p Ff(\000)p Fn(1)1484 540 y Fu(+1)1596 513 y Fj(\013)1654 527 y Fi(i)1685 513 y Fq(,)35 b Fj(l)d Fr(=)d(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(m)p Fq(,)36 b(and)g(intr)-5 b(o)g(duc)g(e)36 b(standar)-5 b(d)38 b(Gaussian)50 658 y(r)-5 b(andom)35 b(variables)154 867 y Fr(\026)128 890 y Fj(X)210 826 y Fi(\033)252 835 y Fn(1)288 826 y Fi(:::)n(\033)394 835 y Fm(J)429 851 y(l)p Ff(\000)p Fn(1)203 903 y Fi(\033)245 912 y Fm(J)280 928 y(l)p Ff(\000)p Fn(1)387 912 y(+1)469 903 y Fi(\033)511 912 y Fm(J)546 928 y(l)p Ff(\000)p Fn(1)653 912 y(+2)735 903 y Fh(\001\001\001)o Fi(\033)842 912 y Fm(J)877 928 y(l)938 890 y Fk(\021)1034 789 y Fe(\020)1088 835 y Fk(p)p 1164 835 292 4 v 55 x Fj(a)1212 904 y Fi(J)1252 914 y Fm(l)p Ff(\000)p Fn(1)1359 904 y Fu(+1)1455 890 y Fj(X)1530 904 y Fi(\033)1572 913 y Fm(J)1607 927 y Fn(1)1648 904 y Fi(:::)o(\033)1755 913 y Fm(J)1790 928 y(l)p Ff(\000)p Fn(1)1901 904 y Fi(\033)1943 913 y Fm(J)1978 928 y(l)p Ff(\000)p Fn(1)2085 913 y(+1)2192 890 y Fr(+)2283 835 y Fk(p)p 2358 835 V 2358 890 a Fj(a)2406 904 y Fi(J)2446 914 y Fm(l)p Ff(\000)p Fn(1)2553 904 y Fu(+2)2650 890 y Fj(X)2725 904 y Fi(\033)2767 913 y Fm(J)2802 927 y Fn(1)2842 904 y Fi(:::)o(\033)2949 913 y Fm(J)2984 928 y(l)p Ff(\000)p Fn(1)3095 904 y Fi(\033)3137 913 y Fm(J)3172 928 y(l)p Ff(\000)p Fn(1)3279 913 y(+1)3361 904 y Fi(\033)3403 913 y Fm(J)3438 928 y(l)p Ff(\000)p Fn(1)3545 913 y(+2)933 1090 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)1241 1031 y Fk(p)p 1316 1031 120 4 v 1316 1090 a Fj(a)1364 1104 y Fi(J)1404 1114 y Fm(l)1436 1090 y Fj(X)1511 1104 y Fi(\033)1553 1113 y Fm(J)1588 1127 y Fn(1)1628 1104 y Fi(:::)o(\033)1735 1113 y Fm(J)1770 1128 y(l)p Ff(\000)p Fn(1)1881 1104 y Fi(\033)1923 1113 y Fm(J)1958 1128 y(l)p Ff(\000)p Fn(1)2065 1113 y(+1)2147 1104 y Fi(:::)o(\033)2254 1113 y Fm(J)2289 1128 y(l)2325 989 y Fe(\021)2379 1090 y Fj(=)2424 1020 y Fk(p)p 2500 1020 76 4 v 2501 1090 a Fr(\026)-46 b Fj(a)2548 1104 y Fi(l)2576 1090 y Fj(:)3523 1226 y Fr(\(1)p Fj(:)p Fr(8\))159 1444 y Fq(Assume)47 b(that)h(the)f(p)-5 b(artition)49 b Fj(J)1315 1458 y Fu(1)1356 1444 y Fj(;)15 b(J)1446 1458 y Fu(2)1488 1444 y Fj(;)g(:)g(:)g(:)32 b(;)15 b(J)1755 1458 y Fi(m)1871 1444 y Fq(satis\014es)48 b(the)f(fol)5 b(lowing)48 b(c)-5 b(ondition:)72 b(for)48 b(al)5 b(l)47 b Fj(l)53 b Fr(=)50 1590 y(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(m)33 b Fq(and)g(al)5 b(l)33 b Fj(k)j Fq(such)d(that)h Fj(J)1343 1604 y Fi(l)p Fh(\000)p Fu(1)1484 1590 y Fr(+)20 b(2)26 b Fk(\024)f Fj(k)j Fk(\024)d Fj(J)1963 1604 y Fi(l)566 1853 y Fr(\()p Fj(a)649 1867 y Fi(k)714 1853 y Fr(+)20 b Fj(a)853 1867 y Fi(k)r Fu(+1)1005 1853 y Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)1270 1867 y Fi(J)1310 1877 y Fm(l)1337 1867 y Fh(\000)p Fu(1)1455 1853 y Fr(+)g Fj(a)1594 1867 y Fi(J)1634 1877 y Fm(l)1665 1853 y Fr(\))p Fj(=)q Fr(\026)-46 b Fj(a)1793 1867 y Fi(l)1847 1853 y Fk(\025)25 b Fr(ln)o(\()p Fj(\013)2112 1867 y Fi(k)2157 1853 y Fj(\013)2215 1867 y Fi(k)r Fu(+1)2367 1853 y Fk(\001)15 b(\001)g(\001)h Fj(\013)2546 1867 y Fi(J)2586 1877 y Fm(l)2613 1867 y Fh(\000)p Fu(1)2711 1853 y Fj(\013)2769 1867 y Fi(J)2809 1877 y Fm(l)2840 1853 y Fr(\))p Fj(=)f Fr(ln\()9 b(\026)-54 b Fj(\013)3105 1867 y Fi(l)3133 1853 y Fr(\))p Fj(:)330 b Fr(\(1)p Fj(:)p Fr(9\))50 2116 y Fq(Then)33 b(the)g(p)-5 b(oint)34 b(pr)-5 b(o)g(c)g(ess)159 2356 y Fk(P)229 2309 y Fu(\()p Fi(m)p Fu(\))222 2383 y Fi(N)184 2529 y Fk(\021)299 2442 y Fe(X)235 2633 y Fi(\033)277 2642 y Fn(1)312 2633 y Fi(:::)o(\033)419 2642 y Fm(J)454 2656 y Fn(1)464 2529 y Fj(\016)504 2557 y Fi(u)547 2528 y Ff(\000)p Fn(1)547 2581 y(ln)18 b(\026)-38 b Fm(\013)653 2595 y Fn(1)688 2581 y Fm(;N)767 2557 y Fu(\()816 2541 y(\026)795 2557 y Fi(X)855 2566 y Fm(\033)892 2580 y Fn(1)928 2566 y Fm(:::)o(\033)1027 2582 y(J)1062 2596 y Fn(1)1107 2557 y Fu(\))1235 2442 y Fe(X)1112 2633 y Fi(\033)1154 2642 y Fm(J)1189 2656 y Fn(1)1225 2642 y(+1)1307 2633 y Fi(:::)o(\033)1414 2642 y Fm(J)1449 2656 y Fn(2)1459 2529 y Fj(\016)1499 2568 y Fi(u)1542 2539 y Ff(\000)p Fn(1)1542 2592 y(ln)17 b(\026)-38 b Fm(\013)1647 2606 y Fn(2)1682 2592 y Fm(;N)1762 2568 y Fu(\()1810 2552 y(\026)1790 2568 y Fi(X)1855 2517 y Fm(\033)1892 2531 y Fn(1)1927 2517 y Fm(:::\033)2027 2532 y(J)2062 2546 y Fn(1)1850 2580 y Fm(\033)1887 2596 y(J)1922 2610 y Fn(1)1957 2596 y(+1)2040 2580 y Fm(:::)o(\033)2139 2596 y(J)2174 2610 y Fn(2)2219 2568 y Fu(\))2267 2529 y Fk(\001)15 b(\001)g(\001)2409 2442 y Fe(X)2221 2633 y Fi(\033)2263 2642 y Fm(J)2298 2657 y(m)p Ff(\000)p Fn(1)2439 2642 y(+1)2521 2633 y Fi(:::)o(\033)2628 2642 y Fm(J)2663 2651 y(m)2653 2529 y Fj(\016)2693 2577 y Fi(u)2736 2547 y Ff(\000)p Fn(1)2736 2601 y(ln)i(\026)-37 b Fm(\013)2842 2610 y(m)2901 2601 y(;N)2981 2577 y Fu(\()3029 2561 y(\026)3009 2577 y Fi(X)3074 2517 y Fm(\033)3111 2531 y Fn(1)3146 2517 y Fm(:::\033)3246 2532 y(J)3281 2547 y(m)p Ff(\000)p Fn(1)3069 2589 y Fm(\033)3106 2604 y(J)3141 2619 y(m)p Ff(\000)p Fn(1)3281 2604 y(+1)3364 2589 y Fm(:::)o(\033)3463 2604 y(J)3498 2613 y(m)3568 2577 y Fu(\))50 2882 y Fq(c)-5 b(onver)g(ges)27 b(we)-5 b(akly)27 b(to)g(the)f(pr)-5 b(o)g(c)g(ess)29 b Fk(P)1347 2849 y Fu(\()p Fi(m)p Fu(\))1500 2882 y Fq(on)d Fg(R)1683 2849 y Fi(m)1784 2882 y Fq(which)h(is)f(char)-5 b(acterise)g(d)29 b(by)d(the)h(fol)5 b(lowing)27 b(gener)-5 b(ating)50 3027 y(functions:)245 3293 y Fj(F)303 3307 y Fu(\001)363 3316 y Fn(1)399 3307 y Fh(\002\001\001\001)o(\002)p Fu(\001)638 3317 y Fm(k)681 3293 y Fr(\()p Fj(z)t Fr(\))27 b Fk(\021)e Fg(E)10 b Fj(z)1021 3172 y Fe(P)1122 3267 y Fm(x)1158 3281 y Fn(1)1211 3233 y Fu(1)-23 b(I)1250 3245 y Ff(f)p Fm(x)1319 3259 y Fn(1)1355 3245 y Ff(2)p Fn(\001)1448 3259 y(1)1483 3245 y Ff(g)1521 3233 y Fh(\001\001\001)1599 3172 y Fe(P)1695 3267 y Fm(x)1731 3283 y(k)1787 3233 y Fu(1)g(I)1826 3245 y Ff(f)p Fm(x)1895 3261 y(k)1936 3245 y Ff(2)p Fn(\001)2029 3261 y Fm(k)2068 3245 y Ff(g)824 3466 y Fr(=)25 b Fj(f)965 3480 y Fu(1)p Fi(;)p Fu(\001)1083 3489 y Fn(1)1122 3466 y Fr(\()p Fj(f)1202 3480 y Fu(2)p Fi(;)p Fu(\001)1320 3489 y Fn(2)1359 3466 y Fr(\()p Fj(f)1439 3480 y Fu(3)p Fi(;)p Fu(\001)1557 3489 y Fn(3)1612 3466 y Fk(\001)15 b(\001)g(\001)h Fr(\()p Fj(f)1813 3480 y Fi(k)r Fh(\000)p Fu(1)p Fi(;)p Fu(\001)2029 3490 y Fm(k)q Ff(\000)p Fn(1)2152 3466 y Fr(\()p Fj(f)2232 3480 y Fi(k)r(;)p Fu(\001)2355 3490 y Fm(k)2397 3466 y Fr(\()p Fj(z)t Fr(\)\)\))f Fk(\001)g(\001)g(\001)k Fr(\)\))p Fj(;)201 b Fk(j)p Fj(z)t Fk(j)27 b Fj(<)d Fr(1)3477 3357 y(\(1)p Fj(:)p Fr(10\))50 3716 y Fq(wher)-5 b(e)35 b Fj(f)353 3730 y Fi(i;)p Fu(\001)461 3739 y Fm(i)493 3716 y Fr(\()p Fj(z)t Fr(\))29 b(=)f Fj(e)779 3683 y Fi(K)840 3692 y Fm(i)870 3683 y Fu(\()p Fi(z)s Fh(\000)p Fu(1\)\()p Fi(e)1120 3656 y Ff(\000)p Fm(a)1204 3671 y(i)1237 3683 y Fh(\000)p Fi(e)1329 3656 y Ff(\000)p Fm(b)1407 3671 y(i)1440 3683 y Fu(\))1473 3716 y Fq(,)35 b Fr(\001)1612 3730 y Fi(i)1670 3716 y Fr(=)28 b(\()p Fj(a)1852 3730 y Fi(i)1883 3716 y Fj(;)15 b(b)1962 3730 y Fi(i)1992 3716 y Fr(])35 b Fq(with)g Fj(a)2299 3730 y Fi(i)2329 3716 y Fj(;)15 b(b)2408 3730 y Fi(i)2467 3716 y Fk(2)28 b Fg(R)43 b Fq(or)35 b Fj(b)2814 3730 y Fi(i)2872 3716 y Fr(=)28 b Fk(1)p Fq(,)35 b Fj(i)28 b Fr(=)g(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(k)s Fq(.)50 3862 y(with)33 b(c)-5 b(onstants)35 b Fr(0)26 b Fj(<)f(K)891 3876 y Fu(1)932 3862 y Fj(;)15 b(:)g(:)g(:)31 b(;)15 b(K)1225 3876 y Fi(m)1320 3862 y Fk(\024)25 b Fr(1)p Fq(.)50 4080 y Fs(Remark:)53 b Fr(The)34 b(v)-5 b(alues)33 b(of)h(the)f(constan)m(ts)i Fj(K)1709 4094 y Fi(l)1771 4080 y Fr(are)f(giv)m(en)g(in)e([BK1].)52 b(Here)34 b(w)m(e)g(just)g(note)g(that)g(if)f(all)50 4225 y Fj(J)100 4239 y Fi(l)148 4225 y Fk(\000)20 b Fj(J)289 4239 y Fi(l)p Fh(\000)p Fu(1)431 4225 y Fk(\000)g Fr(1)30 b(inequalities)e(in)h(\(1.9\))j(for)f Fj(k)d Fr(=)d Fj(J)1759 4239 y Fi(l)p Fh(\000)p Fu(1)1901 4225 y Fr(+)20 b(2)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(J)2304 4239 y Fi(l)2363 4225 y Fr(are)31 b(strict,)f Fj(K)2855 4239 y Fi(l)2908 4225 y Fr(=)25 b(1.)50 4444 y Fs(Remark:)45 b Fr(The)28 b(pro)s(cess)g Fk(P)1057 4411 y Fu(\()p Fi(m)p Fu(\))1213 4444 y Fr(is)f(called)h(a)i Fq(Poisson)i(c)-5 b(asc)g(ade)30 b Fr(with)d Fj(m)i Fr(lev)m(els.)40 b(It)28 b(is)g(b)s(est)h(understo)s (o)s(d)50 4589 y(in)24 b(terms)i(of)g(the)g(follo)m(wing)f(iterativ)m (e)h(construction.)39 b(If)25 b Fj(m)g Fr(=)g(1,)j(it)d(is)g(just)h(a)g (P)m(oisson)f(p)s(oin)m(t)g(pro)s(cess)h(on)50 4735 y Fg(R)41 b Fr(with)31 b(in)m(tensit)m(y)h(measure)g Fj(K)1164 4749 y Fu(1)1205 4735 y Fj(e)1247 4702 y Fh(\000)p Fi(x)1350 4735 y Fj(dx)p Fr(.)47 b(T)-8 b(o)33 b(construct)g Fk(P)2129 4702 y Fu(\(2\))2260 4735 y Fr(on)f Fg(R)2447 4702 y Fu(2)2494 4735 y Fr(,)h(w)m(e)g(place)g(the)f(pro)s(cess)g Fk(P)3470 4702 y Fu(\(1\))3601 4735 y Fr(for)50 4880 y Fj(k)g Fr(=)c(1)33 b(on)f(the)h(axis)f(of)h(the)f(\014rst)g(co)s (ordinate)h(and)e(through)h(eac)m(h)i(of)e(its)g(p)s(oin)m(ts)g(dra)m (w)g(a)h(straigh)m(t)f(line)50 5025 y(parallel)20 b(to)j(the)g(axis)f (of)g(the)h(second)f(co)s(ordinate.)38 b(Then)22 b(w)m(e)g(put)g(on)g (eac)m(h)i(of)e(these)h(lines)e(indep)s(enden)m(tly)50 5171 y(a)h(P)m(oisson)f(p)s(oin)m(t)g(pro)s(cess)g(with)f(in)m(tensit)m (y)h(measure)h Fj(K)1957 5185 y Fu(2)1998 5171 y Fj(e)2040 5138 y Fh(\000)p Fi(x)2143 5171 y Fj(dx)p Fr(.)38 b(These)21 b(p)s(oin)m(ts)g(on)g Fg(R)3001 5138 y Fu(2)3069 5171 y Fr(form)g(the)h(pro)s(cess)50 5316 y Fk(P)120 5283 y Fu(\(2\))218 5316 y Fr(.)41 b(This)28 b(pro)s(cedure)g(is)h(no)m(w)h (simply)d(iterated)j Fj(m)g Fr(times.)40 b(Clearly)-8 b(,)29 b(the)h(pro)5 b(jection)30 b(of)g Fk(P)3302 5283 y Fu(\()p Fi(m)p Fu(\))3458 5316 y Fr(on)g(the)50 5462 y(plane)f(of)i(the)f(\014rst)g Fj(k)j Fr(co)s(ordinates)e(is)e (distributed)e(as)k Fk(P)2042 5429 y Fu(\()p Fi(k)r Fu(\))2144 5462 y Fr(,)g Fj(k)d Fr(=)d(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)21 b Fk(\000)f Fr(1.)p eop %%Page: 5 5 5 4 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3127 b Fo(5)50 513 y Fs(Remark:)44 b Fr(Giv)m(en)28 b Fj(a)795 527 y Fi(i)825 513 y Fj(;)15 b(\013)923 527 y Fi(i)954 513 y Fr(,)28 b Fj(i)e Fr(=)f(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(n)p Fr(,)28 b(the)g(partition)e(of)i(indices)d Fj(J)2506 527 y Fu(1)2547 513 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(J)2814 527 y Fi(m)2911 513 y Fr(satisfying)27 b(the)g(condi-)50 658 y(tion)h(\(1.9\))i(is)d(generally)g(not)i (unique.)38 b(Therefore)28 b(this)f(theorem)i(yields)d(a)j Fq(family)g Fr(of)f(con)m(v)m(ergen)m(t)j(p)s(oin)m(t)50 804 y(pro)s(cesses)f(asso)s(ciated)h(with)e(the)h(GREM.)50 1071 y Fs(1.2.)47 b(Con)m(v)m(ergence)36 b(of)f(the)f(partition)h (function.)159 1289 y Fr(W)-8 b(e)28 b(no)m(w)f(turn)e(to)j(the)f(new)f (results)f(of)i(this)f(pap)s(er.)38 b(The)26 b(\014rst)g(and)g(tec)m (hnically)g(the)h(k)m(ey)h(result,)e(will)50 1434 y(relate)35 b(the)h(\015uctuations)e(of)i(the)f(partition)f(function)g(in)g(the)h (thermo)s(dynamic)f(limit)f(to)j(the)f(P)m(oisson)50 1580 y(cascade)h(pro)s(cesses)e(obtained)g(in)f(the)h(preceding)g (theorem.)53 b(This)33 b(is)g(the)i(main)e(step)i(needed)f(to)h(link)50 1725 y(the)30 b(GREM)h(to)g(the)g(mo)s(dels)e(constructed)i(b)m(y)f (Ruelle)f([Ru].)159 1943 y(F)-8 b(or)34 b Fq(any)41 b Fr(sequence)33 b(of)h(indices)d(0)f Fj(<)f(J)1519 1957 y Fu(1)1589 1943 y Fj(<)h Fk(\001)15 b(\001)g(\001)30 b Fj(<)f(J)1975 1957 y Fi(m)2074 1943 y Fr(=)g Fj(n)k Fr(the)g(partition)f(function)f(of)i(the)g(GREM)50 2089 y(can)d(b)s(e)g(written)g(as:)150 2311 y Fj(Z)212 2325 y Fi(\014)s(;N)373 2311 y Fr(=)o Fj(e)485 2197 y Fe(P)582 2219 y Fm(m)582 2292 y(j)s Fn(=1)706 2184 y Fe(\000)747 2258 y Fi(\014)s(N)856 2199 y Fk(p)p 932 2199 277 4 v 59 x Fu(2)r(\026)-38 b Fi(a)1007 2267 y Fm(j)1054 2258 y Fu(ln)21 b(\026)-44 b Fi(\013)1174 2267 y Fm(j)1209 2258 y Fh(\000)p Fi(\014)s(\015)1348 2267 y Fm(j)1382 2258 y Fu([ln\()p Fi(N)20 b Fu(ln)g(\026)-43 b Fi(\013)1689 2267 y Fm(j)1723 2258 y Fu(\)+ln)13 b(4)p Fi(\031)r Fu(])p Fi(=)p Fu(2)p Fh(\000)p Fi(N)19 b Fu(ln)13 b Fi(\013)2309 2267 y Fm(j)2343 2184 y Fe(\001)464 2518 y Fk(\002)619 2431 y Fe(X)555 2622 y Fi(\033)597 2631 y Fn(1)632 2622 y Fi(:::)o(\033)739 2631 y Fm(J)774 2645 y Fn(1)829 2518 y Fj(e)871 2470 y Fi(\014)s(\015)953 2479 y Fn(1)990 2470 y Fi(u)1033 2440 y Ff(\000)p Fn(1)1033 2494 y(ln)k(\026)-38 b Fm(\013)1138 2508 y Fn(1)1173 2494 y Fm(;N)1253 2470 y Fu(\()1301 2454 y(\026)1281 2470 y Fi(X)1341 2479 y Fm(\033)1378 2493 y Fn(1)1413 2479 y Fm(:::\033)1513 2494 y(J)1548 2508 y Fn(1)1592 2470 y Fu(\))1641 2518 y Fk(\001)15 b(\001)g(\001)1950 2431 y Fe(X)1762 2622 y Fi(\033)1804 2631 y Fm(J)1839 2646 y(m)p Ff(\000)p Fn(1)1980 2631 y(+1)2062 2622 y Fi(:::)o(\033)2169 2631 y Fm(J)2204 2640 y(m)2284 2518 y Fj(e)2326 2458 y Fi(\014)s(\015)2408 2467 y Fm(m)2469 2458 y Fi(u)2512 2429 y Ff(\000)p Fn(1)2512 2482 y(ln)j(\026)-38 b Fm(\013)2618 2491 y(m)2678 2482 y(;N)2757 2458 y Fu(\()2805 2442 y(\026)2785 2458 y Fi(X)2850 2399 y Fm(\033)2887 2413 y Fn(1)2922 2399 y Fm(:::\033)3022 2414 y(J)3057 2429 y(m)p Ff(\000)p Fn(1)2845 2470 y Fm(\033)2882 2486 y(J)2917 2501 y(m)p Ff(\000)p Fn(1)3058 2486 y(+1)3140 2470 y Fm(:::)o(\033)3239 2486 y(J)3274 2495 y(m)3344 2458 y Fu(\))3477 2444 y Fr(\(1)p Fj(:)p Fr(11\))50 2844 y(where)39 b(\026)-47 b Fj(a)368 2858 y Fi(l)434 2844 y Fk(\021)543 2776 y Fe(P)639 2798 y Fi(J)679 2808 y Fm(l)639 2871 y Fi(i)p Fu(=)p Fi(J)761 2881 y Fm(l)p Ff(\000)p Fn(1)867 2871 y Fu(+1)979 2844 y Fj(a)1027 2858 y Fi(i)1057 2844 y Fr(,)49 b(\026)-54 b Fj(\013)1180 2858 y Fi(l)1246 2844 y Fk(\021)1354 2776 y Fe(Q)1440 2798 y Fi(J)1480 2808 y Fm(l)1440 2871 y Fi(i)p Fu(=)p Fi(J)1562 2881 y Fm(l)p Ff(\000)p Fn(1)1669 2871 y Fu(+1)1780 2844 y Fj(\013)1838 2858 y Fi(i)1869 2844 y Fr(,)40 b Fj(\015)1981 2858 y Fi(l)2047 2844 y Fk(\021)2155 2779 y(p)p 2231 2779 76 4 v 2232 2844 a Fr(\026)-46 b Fj(a)2279 2858 y Fi(l)2307 2844 y Fj(=)2352 2773 y Fk(p)p 2428 2773 238 4 v 71 x Fr(2)15 b(ln)24 b(\026)-54 b Fj(\013)2638 2858 y Fi(l)2666 2844 y Fr(,)40 b Fj(l)f Fr(=)f(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(m)p Fr(,)40 b(and)e(the)50 2989 y(random)32 b(v)-5 b(ariables)792 2966 y(\026)766 2989 y Fj(X)848 2925 y Fi(\033)890 2934 y Fn(1)926 2925 y Fi(:::)o(\033)1033 2934 y Fm(J)1068 2950 y(l)p Ff(\000)p Fn(1)841 3000 y Fi(\033)883 3009 y Fm(J)918 3025 y(l)p Ff(\000)p Fn(1)1025 3009 y(+1)1107 3000 y Fi(:::)o(\033)1214 3009 y Fm(J)1249 3025 y(l)1318 2989 y Fr(are)33 b(de\014ned)f(in)g (\(1.8\).)51 b(Moreo)m(v)m(er,)36 b(for)d(an)m(y)g(sequence)h Fj(J)3332 3003 y Fu(1)3373 2989 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(J)3640 3003 y Fi(m)50 3135 y Fr(satisfying)30 b(the)h(assumptions)e(of)i(Theorem)g(1.1,)i(the)e(corresp)s(onding)e(p) s(oin)m(t)h(pro)s(cess)h(con)m(v)m(erges.)44 b(But,)50 3280 y(of)30 b(course,)h(at)h(most)e(one)h(of)g(these)g(sequences)g (can)f(pro)m(vide)g(through)g(\(1.11\))j(the)d(righ)m(t)g(scaling)g(of) h(the)50 3426 y(\015uctuations)j(of)i(the)f(partition)f(function.)54 b(As)36 b(in)e(the)h(analysis)f(of)i(the)f(extremal)h(pro)s(cess)e(in)g ([BK1],)50 3571 y(it)i(will)e(b)s(e)i(the)h(pro)s(cess)f(corresp)s (onding)f(to)i(a)h(maximal)d(spacing)h(of)h(the)g(in)m(tegers)g Fj(J)3105 3585 y Fi(i)3172 3571 y Fr(satisfying)f(the)50 3717 y(conditions)25 b(\(1.9\))j(that)g(giv)m(es)f(the)f(relev)-5 b(an)m(t)27 b(con)m(tribution.)38 b(Recall)27 b(that)g(this)e (partition)h(is)f(ac)m(hiev)m(ed)j(b)m(y)50 3862 y(setting)i Fj(J)398 3876 y Fu(0)465 3862 y Fk(\021)25 b Fr(0,)31 b(and)812 4090 y Fj(J)862 4104 y Fi(l)915 4090 y Fk(\021)25 b Fr(min)o Fk(f)p Fj(J)35 b(>)25 b(J)1439 4104 y Fi(l)p Fh(\000)p Fu(1)1585 4090 y Fr(:)h Fj(A)1704 4104 y Fi(J)1744 4114 y Fm(l)p Ff(\000)p Fn(1)1851 4104 y Fu(+1)p Fi(;J)2040 4090 y Fj(>)f(A)2204 4104 y Fi(J)6 b Fu(+1)p Fi(;k)2439 4090 y Fk(8)p Fj(k)28 b Fk(\025)d Fj(J)30 b Fr(+)20 b(1)p Fk(g)p Fj(:)530 b Fr(\(1)p Fj(:)p Fr(12\))50 4319 y(with)28 b Fj(A)324 4333 y Fi(j;k)446 4319 y Fk(\021)542 4251 y Fe(P)637 4273 y Fi(k)637 4346 y(i)p Fu(=)p Fi(j)772 4319 y Fj(a)820 4333 y Fi(i)851 4319 y Fj(=)p Fr(\(2)15 b(ln\()1103 4251 y Fe(Q)1189 4273 y Fi(k)1189 4346 y(i)p Fu(=)p Fi(j)1324 4319 y Fj(\013)1382 4333 y Fi(i)1413 4319 y Fr(\)\).)41 b(The)29 b(sequence)g Fj(J)2160 4333 y Fu(0)2202 4319 y Fj(;)15 b(J)2292 4333 y Fu(1)2333 4319 y Fj(;)g(:)g(:)g(:)32 b(;)15 b(J)2600 4333 y Fi(m)2699 4319 y Fr(of)30 b(\(1.12\))i(satis\014es)d(\(1.9\))i(b)m(y)50 4464 y(Prop)s(osition)18 b(1.4)k(from)e([BK1].)39 b(Ho)m(w)m(ev)m(er,) 24 b(unlik)m(e)19 b(in)g(the)i(case)h(of)e(the)h(extremal)g(pro)s (cess,)h(it)e(m)m(ust)h(b)s(e)f(cut)50 4610 y(at)33 b(some)g(temp)s (erature)g(dep)s(enden)m(t)f(lev)m(el,)h Fj(J)1641 4626 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1771 4610 y Fr(.)47 b(Clearly)-8 b(,)33 b Fj(A)2248 4624 y Fu(1)p Fi(;J)2346 4633 y Fn(1)2415 4610 y Fj(>)c(A)2583 4624 y Fi(J)2623 4633 y Fn(1)2659 4624 y Fu(+1)p Fi(;J)2813 4633 y Fn(2)2882 4610 y Fj(>)g Fk(\001)15 b(\001)g(\001)30 b Fj(>)f(A)3285 4624 y Fi(J)3325 4633 y Fm(m)p Ff(\000)p Fn(1)3466 4624 y Fu(+1)p Fi(;J)3620 4633 y Fm(m)3685 4610 y Fr(.)50 4755 y(Let)i(us)e(put)1144 4901 y Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))26 b Fk(\021)f Fr(max)p Fk(f)p Fj(l)j Fk(\025)d Fr(1)g(:)h Fj(\014)1963 4863 y Fu(2)2004 4901 y Fj(A)2072 4915 y Fi(J)2112 4925 y Fm(l)p Ff(\000)p Fn(1)2219 4915 y Fu(+1)p Fi(;J)2373 4925 y Fm(l)2429 4901 y Fj(>)f Fr(1)p Fk(g)862 b Fr(\(1)p Fj(:)p Fr(13\))50 5098 y(and)29 b Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))26 b Fk(\021)f Fr(0)30 b(if)f Fj(\014)717 5065 y Fu(2)758 5098 y Fj(A)826 5112 y Fu(1)p Fi(;J)924 5121 y Fn(1)989 5098 y Fk(\024)c Fr(1.)41 b(According)30 b(to)g(the)g(notations)g(of)g(\(1.11\))i Fj(A)2726 5112 y Fi(J)2766 5122 y Fm(l)p Ff(\000)p Fn(1)2873 5112 y Fu(+1)p Fi(;J)3027 5122 y Fm(l)3083 5098 y Fr(=)25 b Fj(\015)3231 5065 y Fu(2)3226 5124 y Fi(l)3301 5098 y Fr(and)k(\(1.13\))50 5244 y(implies)e(that)k Fj(\014)5 b(\015)658 5260 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)905 5244 y Fk(\024)25 b Fr(1.)159 5462 y(In)30 b([CCP])g(the)g(limit)e(of)j (the)f(free)h(energy)g(w)m(as)f(found)f(in)g(terms)i(of)f(\(1.12\))j (and)d(\(1.13\):)p eop %%Page: 6 6 6 5 bop 50 295 a Fo(6)3309 b Fl(Se)l(ction)27 b Fo(1)50 513 y Fd(Theorem)37 b(1.2:)51 b Fr([CCP])30 b Fq(With)j(the)g(notation) h(intr)-5 b(o)g(duc)g(e)g(d)35 b(ab)-5 b(ove,)346 688 y Fr(lim)304 747 y Fi(N)7 b Fh(!1)529 688 y Fj(N)612 651 y Fh(\000)p Fu(1)724 688 y Fr(ln)14 b Fj(Z)877 702 y Fi(N)s(;\014)1034 688 y Fk(\021)25 b(\000)56 b Fr(lim)1216 747 y Fi(N)7 b Fh(!1)1440 688 y Fj(F)1498 702 y Fi(\014)s(;N)218 943 y Fr(=)p Fj(\014)345 869 y Fe(\000)387 858 y(p)p 478 858 340 4 v 85 x Fr(2)q(\026)-46 b Fj(a)571 957 y Fu(1)627 943 y Fr(ln)23 b(\026)-54 b Fj(\013)776 957 y Fu(1)837 943 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)1145 841 y Fe(q)p 1236 841 517 4 v 102 x Fr(2)q(\026)-46 b Fj(a)1329 959 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1474 943 y Fr(ln)23 b(\026)-54 b Fj(\013)1623 959 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1752 869 y Fe(\001)1814 943 y Fk(\000)20 b Fr(ln)o(\()9 b(\026)-54 b Fj(\013)2074 957 y Fu(1)2130 943 y Fk(\001)15 b(\001)g(\001)26 b Fr(\026)-54 b Fj(\013)2310 959 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))2439 943 y Fr(\))21 b(+)2666 829 y Fi(n)2623 856 y Fe(X)2526 1051 y Fi(i)p Fu(=)p Fi(J)2648 1063 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2760 1051 y Fu(+1)2868 943 y Fj(\014)2924 905 y Fu(2)2965 943 y Fj(a)3013 957 y Fi(i)3043 943 y Fj(=)p Fr(2)p Fj(;)109 b Fq(a.s.)3477 1161 y Fr(\(1)p Fj(:)p Fr(14\))50 1307 y Fq(and)33 b(also)h(in)f Fj(L)586 1274 y Fi(p)628 1307 y Fq(,)f Fr(1)26 b Fk(\024)f Fj(p)g(<)g Fk(1)p Fq(.)159 1525 y Fr(The)41 b(follo)m(wing)f(theorem)h(yields)f (the)h(\015uctuations)g(of)g(the)h(partition)e(function)g(and)h(of)g (the)h(free)50 1670 y(energy)-8 b(.)50 1888 y Fd(Theorem)44 b(1.3:)64 b Fq(L)-5 b(et)38 b Fj(\013)1005 1902 y Fi(i)1070 1888 y Fk(\025)c Fr(1)p Fq(,)39 b Fr(0)c Fj(<)f(a)1520 1902 y Fi(i)1584 1888 y Fj(<)g Fr(1)p Fq(,)39 b Fj(i)c Fr(=)f(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(n)p Fq(,)2441 1820 y Fe(Q)2527 1843 y Fi(n)2527 1916 y(i)p Fu(=1)2665 1888 y Fj(\013)2723 1902 y Fi(i)2787 1888 y Fr(=)34 b(2)p Fq(,)3004 1820 y Fe(P)3100 1843 y Fi(n)3100 1916 y(i)p Fu(=1)3238 1888 y Fj(a)3286 1902 y Fi(i)3350 1888 y Fr(=)g(1)p Fq(.)57 b(L)-5 b(et)50 2034 y Fj(J)100 2048 y Fu(1)141 2034 y Fj(;)15 b(J)231 2048 y Fu(2)273 2034 y Fj(;)g(:)g(:)g(:)31 b(;)15 b(J)539 2048 y Fi(m)641 2034 y Fk(2)32 b Fg(N)7 b Fq(,)43 b(b)-5 b(e)36 b(the)h(se)-5 b(quenc)g(e)36 b(of)h(indic)-5 b(es)37 b(de\014ne)-5 b(d)37 b(by)f(\(1.12\))i(and)f Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))38 b Fq(de\014ne)-5 b(d)37 b(by)f(\(1.13\).)50 2179 y(Denote)c(by)i Fr(\026)-46 b Fj(a)529 2193 y Fi(l)582 2179 y Fk(\021)678 2111 y Fe(P)774 2134 y Fi(J)814 2144 y Fm(l)774 2207 y Fi(i)p Fu(=)p Fi(J)896 2217 y Fm(l)p Ff(\000)p Fn(1)1002 2207 y Fu(+1)1114 2179 y Fj(a)1162 2193 y Fi(i)1192 2179 y Fq(,)42 b Fr(\026)-54 b Fj(\013)1311 2193 y Fi(l)1364 2179 y Fk(\021)1460 2111 y Fe(Q)1546 2134 y Fi(J)1586 2144 y Fm(l)1546 2207 y Fi(i)p Fu(=)p Fi(J)1668 2217 y Fm(l)p Ff(\000)p Fn(1)1774 2207 y Fu(+1)1886 2179 y Fj(\013)1944 2193 y Fi(i)1974 2179 y Fq(,)33 b Fj(\015)2082 2193 y Fi(l)2135 2179 y Fr(=)2231 2114 y Fk(p)p 2307 2114 76 4 v 2308 2179 a Fr(\026)-46 b Fj(a)2355 2193 y Fi(l)2383 2179 y Fj(=)2428 2109 y Fk(p)p 2504 2109 238 4 v 70 x Fr(2)15 b(ln)23 b(\026)-54 b Fj(\013)2713 2193 y Fi(l)2741 2179 y Fq(,)33 b Fj(l)27 b Fr(=)e(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(m)p Fq(.)50 2398 y(Then)30 b(the)g(pr)-5 b(op)g(erly)32 b(normalise)-5 b(d)32 b(p)-5 b(artition)32 b(function)e(of)f(the)h(GREM)g(c)-5 b(onver)g(ges)30 b(in)f(law)i(to)f(the)g(fol)5 b(low-)50 2543 y(ing)32 b(inte)-5 b(gr)g(al)34 b(on)f Fg(R)717 2510 y Fi(m)792 2543 y Fq(:)281 2786 y Fj(e)323 2651 y Fe(P)419 2674 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))419 2747 y Fm(j)s Fn(=1)545 2639 y Fe(\000)587 2712 y Fh(\000)p Fi(\014)s(N)753 2653 y Fk(p)p 828 2653 277 4 v 828 2712 a Fu(2)r(\026)-38 b Fi(a)903 2721 y Fm(j)950 2712 y Fu(ln)21 b(\026)-44 b Fi(\013)1070 2721 y Fm(j)1105 2712 y Fu(+)p Fi(\014)s(\015)1243 2721 y Fm(j)1277 2712 y Fu([ln\()p 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Fm(i)p Fn(:)p Fm(J)1261 4110 y(l)p Fn(\()p Fm(\014)r Fn(\))1375 4092 y(+1)p Ff(\024)p Fm(i)p Ff(\024)p Fm(J)1610 4110 y(l)p Fn(\()p Fm(\014)r Fn(\)+1)608 4167 y(\()p Fm(a)668 4182 y(J)703 4200 y(l)p Fn(\()p Fm(\014)r Fn(\))817 4182 y(+1)899 4167 y(+)p Ff(\001\001\001)o Fn(+)p Fm(a)1090 4182 y(i)1119 4167 y Fn(\))p Fm(=)r Fn(\026)-33 b Fm(a)1212 4182 y(J)1247 4200 y(l)p Fn(\()p Fm(\014)r Fn(\)+1)1443 4167 y(=ln\()p Fm(\013)1610 4182 y(J)1645 4200 y(l)p Fn(\()p Fm(\014)r Fn(\))1758 4182 y(+1)1840 4167 y Ff(\001\001\001)o Fm(\013)1944 4182 y(i)1973 4167 y Fn(\))p Fm(=)11 b Fn(ln)17 b(\026)-37 b Fm(\013)2148 4182 y(J)2183 4200 y(l)p Fn(\()p Fm(\014)r Fn(\)+1)2054 3994 y Fr(\()2089 3934 y Fe(p)p 2180 3934 299 4 v 60 x Fj(a)2228 4008 y Fi(J)2268 4020 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2382 4008 y Fu(+1)2479 3994 y Fj(Z)2541 4008 y Fi(J)2581 4020 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2695 4008 y Fu(+1)2812 3994 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)3120 3931 y Fk(p)p 3196 3931 79 4 v 63 x 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Fi(\014)s(\015)343 5292 y Fn(1)379 5283 y Fi(x)420 5292 y Fn(1)456 5283 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(\014)s(\015)715 5292 y Fm(m)775 5283 y Fi(x)816 5292 y Fm(m)882 5316 y Fk(P)952 5283 y Fu(\()p Fi(m)p Fu(\))1078 5316 y Fr(\()p Fj(dx)1212 5330 y Fu(1)1269 5316 y Fj(:)15 b(:)g(:)h(dx)1489 5330 y Fi(m)1558 5316 y Fr(\))40 b(o)m(v)m(er)g(the)f(pro)s(cess)g Fk(P)2397 5283 y Fu(\()p Fi(m)p Fu(\))2562 5316 y Fr(on)g Fg(R)2757 5283 y Fi(m)2832 5316 y Fr(.)67 b(The)38 b(k)m(ey)i(facts,)i (that)50 5462 y(w)m(ere)31 b(pro)m(v)m(en)f(in)f(Prop)s(ositions)g (3.1,)i(3.2)h(of)37 b([BK1])32 b(are)f(en)m(umerated)f(in)p eop %%Page: 7 7 7 6 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3127 b Fo(7)50 513 y Fd(Prop)s(osition)49 b(1.4:)57 b Fq(Assume)41 b(that)h(the)f (numb)-5 b(ers)41 b Fj(\015)2032 527 y Fu(1)2073 513 y Fj(;)15 b(:)g(:)g(:)31 b(;)15 b(\015)2336 527 y Fi(m)2446 513 y Fq(ar)-5 b(e)42 b(such)f(that)g Fj(\015)3064 527 y Fu(1)3145 513 y Fj(>)e(\015)3302 527 y Fu(2)3383 513 y Fj(>)g Fk(\001)15 b(\001)g(\001)41 b Fj(>)50 658 y(\015)97 672 y Fi(m)191 658 y Fj(>)25 b Fr(0)33 b Fq(and)h Fj(\014)5 b(\015)645 672 y Fi(m)739 658 y Fj(>)25 b Fr(1)p Fq(.)42 b(Then)50 876 y(\(i\))31 b(F)-7 b(or)31 b(any)g Fj(a)26 b Fk(2)e Fj(R)32 b Fq(the)f(pr)-5 b(o)g(c)g(ess)32 b Fk(P)1306 843 y Fu(\()p Fi(m)p Fu(\))1463 876 y Fq(c)-5 b(ontains)32 b(a.s.)42 b(a)31 b(\014nite)g(numb)-5 b(er)31 b(of)f(p)-5 b(oints)32 b Fr(\()p Fj(x)3090 890 y Fu(1)3132 876 y Fj(;)15 b(:)g(:)g(:)31 b(;)15 b(x)3400 890 y Fi(m)3470 876 y Fr(\))31 b Fq(such)50 1022 y(that)j Fj(\015)283 1036 y Fu(1)324 1022 y Fj(x)376 1036 y Fu(1)436 1022 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)22 b Fr(+)e Fj(\015)792 1036 y Fi(m)861 1022 y Fj(x)913 1036 y Fi(m)1007 1022 y Fj(>)25 b(a)p Fq(.)50 1240 y(\(ii\))32 b(The)h(limit)497 1503 y Fr(lim)439 1557 y Fi(x)p Fh(!\0001)697 1379 y Fe(Z)1092 1552 y Fn(\()p Fm(x)1153 1566 y Fn(1)1188 1552 y Fm(;:::)11 b(;x)1340 1561 y(m)1400 1552 y Fn(\))p Ff(2)p Fb(R)1497 1539 y Fm(m)1561 1552 y(;)759 1611 y Ff(9)p Fm(i;)p Fn(1)p Ff(\024)p Fm(i)p Ff(\024)p Fm(m)p Fn(:)p Fm(\015)1102 1625 y Fn(1)1138 1611 y Fm(x)1174 1625 y Fn(1)1209 1611 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)1399 1626 y(i)1428 1611 y(x)1464 1626 y(i)1493 1611 y(>)p Fn(\()p Fm(\015)1601 1625 y Fn(1)1636 1611 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)1826 1626 y(i)1855 1611 y Fn(\))p Fm(x)1946 1503 y Fj(e)1988 1465 y Fi(\014)s(\015)2070 1474 y Fn(1)2106 1465 y Fi(x)2147 1474 y Fn(1)2183 1465 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(\014)s(\015)2441 1474 y Fm(m)2502 1465 y Fi(x)2543 1474 y Fm(m)2608 1503 y Fk(P)2678 1465 y Fu(\()p Fi(m)p Fu(\))2805 1503 y Fr(\()p Fj(dx)2939 1517 y Fu(1)2996 1503 y Fj(:)k(:)g(:)h(dx)3216 1517 y Fi(m)3285 1503 y Fr(\))50 1805 y Fq(is)32 b(\014nite)h(a.s.)159 2023 y Fr(F)-8 b(or)31 b Fj(m)25 b Fr(=)g(1)30 b(w)m(e)h(recall)e(the)h (de\014nition)e(of)37 b([BKL]:)31 b(for)e(all)g Fj(x)d Fk(2)e Fg(R)2465 1950 y Fe(R)2526 1972 y Fh(1)2508 2055 y Fi(x)2618 2023 y Fj(e)2660 1990 y Fi(\014)s(\015)2742 1999 y Fn(1)2778 1990 y Fi(x)2819 1999 y Fn(1)2859 2023 y Fk(P)2929 1990 y Fu(\(1\))3028 2023 y Fr(\()p Fj(dx)p Fr(\))31 b(b)s(eing)d(a)j(sum)50 2168 y(o)m(v)m(er)38 b(a)e(\014nite)g(n)m(um)m(b)s(er)f(of)i(p)s(oin)m(ts)e(a.s.,)1492 2095 y Fe(R)1552 2118 y Fh(1)1535 2201 y(\0001)1684 2168 y Fj(e)1726 2135 y Fi(\014)s(\015)1808 2144 y Fn(1)1844 2135 y Fi(x)1885 2144 y Fn(1)1925 2168 y Fk(P)1995 2135 y Fu(\(1\))2094 2168 y Fr(\()p Fj(dx)p Fr(\))h Fk(\021)f Fr(lim)2531 2182 y Fi(x)p Fh(!\0001)2794 2095 y Fe(R)2855 2118 y Fh(1)2837 2201 y Fi(x)2947 2168 y Fj(e)2989 2135 y Fi(\014)s(\015)3071 2144 y Fn(1)3107 2135 y Fi(x)3148 2144 y Fn(1)3189 2168 y Fk(P)3259 2135 y Fu(\(1\))3357 2168 y Fr(\()p Fj(dx)p Fr(\))i(and)50 2314 y(this)29 b(limit)f(is)i(\014nite)f(a.s.)41 b(If)30 b(the)h(in)m(tegral)f(in)f (question)g(is)h(de\014ned)f(up)g(to)i Fj(m)20 b Fk(\000)g Fr(1,)31 b(w)m(e)g(ma)m(y)g(consider)283 2448 y Fe(Z)678 2621 y Fn(\()p Fm(x)739 2635 y Fn(1)774 2621 y Fm(;:::)10 b(;x)925 2630 y(m)986 2621 y Fn(\))p Ff(2)p Fb(R)1083 2608 y Fm(m)1146 2621 y(;)344 2680 y Ff(9)p Fm(i;)p Fn(1)p Ff(\024)p Fm(i)p Ff(\024)p Fm(m)p Fn(:)p Fm(\015)687 2694 y Fn(1)723 2680 y Fm(x)759 2694 y Fn(1)795 2680 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)985 2695 y(i)1013 2680 y(x)1049 2695 y(i)1078 2680 y(>)p Fn(\()p Fm(\015)1186 2694 y Fn(1)1222 2680 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)1412 2695 y(i)1440 2680 y Fn(\))p Fm(x)1531 2571 y Fj(e)1573 2534 y Fi(\014)s(\015)1655 2543 y Fn(1)1692 2534 y Fi(x)1733 2543 y Fn(1)1769 2534 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(\014)s(\015)2027 2543 y Fm(m)2088 2534 y Fi(x)2129 2543 y Fm(m)2194 2571 y Fk(P)2264 2534 y Fu(\()p Fi(m)p Fu(\))2391 2571 y Fr(\()p Fj(dx)2525 2585 y Fu(1)2581 2571 y Fj(:)15 b(:)g(:)i(dx)2802 2585 y Fi(m)2871 2571 y Fr(\))293 2881 y(=)423 2767 y Fi(m)389 2794 y Fe(X)389 2990 y Fi(k)r Fu(=1)536 2757 y Fe(Z)981 2886 y Fn(\()p Fm(x)1042 2900 y Fn(1)1077 2886 y Fm(;:::)11 b(;x)1229 2895 y(m)1289 2886 y Fn(\))p Ff(2)p Fb(R)1386 2873 y Fm(m)1450 2886 y(;)609 2945 y Ff(8)p Fm(i)p Fn(=1)p Fm(;:::)f(;k)q Ff(\000)p Fn(1:)p Fm(\015)1030 2959 y Fn(1)1066 2945 y Fm(x)1102 2959 y Fn(1)1137 2945 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)1327 2960 y(i)1356 2945 y(x)1392 2960 y(i)1421 2945 y Ff(\024)p Fn(\()p Fm(\015)1529 2959 y Fn(1)1564 2945 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)1754 2960 y(i)1783 2945 y Fn(\))p Fm(x)786 3004 y(\015)820 3018 y Fn(1)855 3004 y Fm(x)891 3018 y Fn(1)927 3004 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)1117 3020 y(k)1157 3004 y(x)1193 3020 y(k)1232 3004 y(>)p Fn(\()p Fm(\015)1340 3018 y Fn(1)1376 3004 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)1566 3020 y(k)1605 3004 y Fn(\))p Fm(x)1885 2881 y Fj(e)1927 2843 y Fi(\014)s(\015)2009 2852 y Fn(1)2045 2843 y Fi(x)2086 2852 y Fn(1)2122 2843 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(\014)s(\015)2380 2852 y Fm(m)2441 2843 y Fi(x)2482 2852 y Fm(m)2547 2881 y Fk(P)2617 2843 y Fu(\()p Fi(m)p Fu(\))2744 2881 y Fr(\()p Fj(dx)2878 2895 y Fu(1)2935 2881 y Fj(:)15 b(:)g(:)h(dx)3155 2895 y Fi(m)3224 2881 y Fr(\))3477 2757 y(\(1)p Fj(:)p Fr(18\))50 3200 y(where)34 b(eac)m(h)h(term)f(is)f(a.s.)53 b(a)35 b(\014nite)e(linear)g(com)m (bination)g(of)h(already)g(de\014ned)f(in)m(tegrals.)52 b(In)33 b(fact,)k(let)50 3345 y(us)31 b(determine)h(the)g Fj(k)s Fr(th)g(term.)47 b(Giv)m(en)32 b(an)m(y)g(realisation)f(of)i Fk(P)2236 3312 y Fu(\()p Fi(m)p Fu(\))2395 3345 y Fr(in)e Fg(R)2562 3312 y Fi(m)2637 3345 y Fr(,)i(consider)e(its)h(pro)5 b(jection)32 b(on)50 3490 y(the)27 b(plane)f(of)i(the)f(\014rst)f Fj(k)31 b Fr(co)s(ordinates)c(distributed)d(as)j Fk(P)2078 3457 y Fu(\()p Fi(k)r Fu(\))2180 3490 y Fr(.)40 b(By)27 b(Prop)s(osition)f(1.4)i(\(i\))f(this)f(con)m(tains)h(a)50 3636 y(\014nite)e(n)m(um)m(b)s(er)g(of)h(p)s(oin)m(ts)f(\()p Fj(x)1060 3650 y Fu(1)1101 3636 y Fj(;)15 b(:)g(:)g(:)31 b(;)15 b(x)1369 3650 y Fi(k)1415 3636 y Fr(\))26 b(suc)m(h)g(that)g Fj(\015)1916 3650 y Fu(1)1957 3636 y Fj(x)2009 3650 y Fu(1)2061 3636 y Fr(+)12 b Fk(\001)j(\001)g(\001)d Fr(+)g Fj(\015)2391 3650 y Fi(k)2435 3636 y Fj(x)2487 3650 y Fi(k)2557 3636 y Fj(>)25 b Fr(\()p Fj(\015)2735 3650 y Fu(1)2788 3636 y Fr(+)12 b Fk(\001)j(\001)g(\001)d Fr(+)g Fj(\015)3118 3650 y Fi(k)3162 3636 y Fr(\))p Fj(x)p Fr(.)39 b(Whenev)m(er)50 3781 y(the)32 b(\014rst)g Fj(k)j Fr(co)s(ordinates)d(of)g(a)h(p)s(oin)m(t)e(of)h Fk(P)1562 3748 y Fu(\()p Fi(m)p Fu(\))1721 3781 y Fr(in)f Fg(R)1889 3748 y Fi(m)1996 3781 y Fr(are)h(\014xed,)h(the)f(remaining)e Fj(m)22 b Fk(\000)f Fj(k)35 b Fr(co)s(ordinates)50 3927 y(are)40 b(distributed)c(as)j Fk(P)875 3894 y Fu(\()p Fi(m)p Fh(\000)p Fi(k)r Fu(\))1138 3927 y Fr(in)f Fg(R)1313 3894 y Fi(m)p Fh(\000)p Fi(k)1485 3927 y Fr(.)67 b(Then)39 b(the)g(in)m(tegral)g(of)h(the)f(function)f Fj(e)3014 3894 y Fi(\015)3052 3904 y Fm(k)q Fn(+1)3170 3894 y Fi(x)3211 3904 y Fm(k)q Fn(+1)3329 3894 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(\015)3543 3903 y Fm(m)3604 3894 y Fi(x)3645 3903 y Fm(m)50 4072 y Fr(o)m(v)m(er)33 b(these)g(co)s(ordinates)f(is)g (de\014ned)f(b)m(y)h(induction)e(and)i(is)f(\014nite)g(a.s.)47 b(Th)m(us)32 b(the)g Fj(k)s Fr(th)h(term)f(of)g(\(1.18\))50 4218 y(is)27 b(the)h(sum)g(of)g(these)h(in)m(tegrals)f(tak)m(en)h(with) e(the)h(w)m(eigh)m(ts)h Fj(e)2149 4185 y Fi(\015)2187 4194 y Fn(1)2223 4185 y Fi(x)2264 4194 y Fn(1)2300 4185 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(\015)2514 4195 y Fm(k)2554 4185 y Fi(x)2595 4195 y Fm(k)2668 4218 y Fr(o)m(v)m(er)g(p)s (oin)m(ts)e(\()p Fj(x)3218 4232 y Fu(1)3259 4218 y Fj(;)15 b Fk(\001)g(\001)g(\001)32 b Fj(;)15 b(x)3528 4232 y Fi(k)3573 4218 y Fr(\))29 b(of)50 4363 y Fk(P)120 4330 y Fu(\()p Fi(k)r Fu(\))254 4363 y Fr(suc)m(h)i(that)i Fk(8)p Fj(i)27 b Fr(=)h(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(k)25 b Fk(\000)c Fr(1)28 b(:)g Fj(\015)1466 4377 y Fu(1)1507 4363 y Fj(x)1559 4377 y Fu(1)1621 4363 y Fr(+)21 b Fk(\001)15 b(\001)g(\001)22 b Fr(+)f Fj(\015)1979 4377 y Fi(i)2009 4363 y Fj(x)2061 4377 y Fi(i)2119 4363 y Fk(\024)28 b Fr(\()p Fj(\015)2300 4377 y Fu(1)2362 4363 y Fr(+)21 b Fk(\001)15 b(\001)g(\001)22 b Fr(+)f Fj(\015)2720 4377 y Fi(i)2750 4363 y Fr(\))p Fj(x;)48 b(\015)2957 4377 y Fu(1)2998 4363 y Fj(x)3050 4377 y Fu(1)3112 4363 y Fr(+)21 b Fk(\001)15 b(\001)g(\001)22 b Fr(+)f Fj(\015)3470 4377 y Fi(k)3515 4363 y Fj(x)3567 4377 y Fi(k)3639 4363 y Fj(>)50 4509 y Fr(\()p Fj(\015)132 4523 y Fu(1)192 4509 y Fr(+)e Fk(\001)c(\001)g(\001)21 b Fr(+)e Fj(\015)545 4523 y Fi(k)589 4509 y Fr(\))p Fj(x)p Fr(.)41 b(As)30 b(noticed)g(ab)s(o)m(v)m(e,)h(this)e(linear)f(com)m(bination)i(has)f(a) i(\014nite)d(n)m(um)m(b)s(er)h(of)h(terms)g(a.s.)50 4654 y(Finally)e(w)m(e)j(put)394 4788 y Fe(Z)444 4995 y Fc(R)494 4976 y Fm(m)570 4912 y Fj(e)612 4874 y Fi(\014)s(\015)694 4883 y Fn(1)731 4874 y Fi(x)772 4883 y Fn(1)807 4874 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(\014)s(\015)1066 4883 y Fm(m)1127 4874 y Fi(x)1168 4883 y Fm(m)1233 4912 y Fk(P)1303 4874 y Fu(\()p Fi(m)p Fu(\))1430 4912 y Fr(\()p Fj(dx)1564 4926 y Fu(1)1620 4912 y Fj(:)15 b(:)g(:)h(dx)1840 4926 y Fi(m)1910 4912 y Fr(\))404 5155 y Fk(\021)83 b Fr(lim)500 5209 y Fi(x)p Fh(!\0001)758 5031 y Fe(Z)1153 5204 y Fn(\()p Fm(x)1214 5218 y Fn(1)1249 5204 y Fm(;:::)10 b(;x)1400 5213 y(m)1461 5204 y Fn(\))p Ff(2)p Fb(R)1558 5191 y Fm(m)1621 5204 y(;)819 5263 y Ff(9)p Fm(i;)p Fn(1)p Ff(\024)p Fm(i)p Ff(\024)p Fm(m)p Fn(:)p Fm(\015)1162 5277 y Fn(1)1198 5263 y Fm(x)1234 5277 y Fn(1)1270 5263 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)1460 5278 y(i)1488 5263 y(x)1524 5278 y(i)1553 5263 y(>)p Fn(\()p Fm(\015)1661 5277 y Fn(1)1696 5263 y(+)p Ff(\001\001\001)p Fn(+)p Fm(\015)1887 5278 y(i)1915 5263 y Fn(\))p Fm(x)2006 5155 y Fj(e)2048 5117 y Fi(\014)s(\015)2130 5126 y Fn(1)2167 5117 y Fi(x)2208 5126 y Fn(1)2244 5117 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(\014)s(\015)2502 5126 y Fm(m)2563 5117 y Fi(x)2604 5126 y Fm(m)2669 5155 y Fk(P)2739 5117 y Fu(\()p Fi(m)p Fu(\))2866 5155 y Fr(\()p Fj(dx)3000 5169 y Fu(1)3056 5155 y Fj(:)15 b(:)g(:)h(dx)3276 5169 y Fi(m)3346 5155 y Fr(\))50 5462 y(whic)m(h)29 b(exists)h(a.s.)41 b(b)m(y)30 b(Prop)s(osition)f(1.4)i(\(ii\).)p eop %%Page: 8 8 8 7 bop 50 295 a Fo(8)3309 b Fl(Se)l(ction)27 b Fo(1)50 513 y Fs(1.3.)47 b(The)34 b(Gibbs)i(measures.)159 731 y Fr(W)-8 b(e)33 b(will)d(no)m(w)i(turn)f(to)h(an)g(appropriate)g (asymptotic)g(description)e(of)i(the)g(Gibbs)f(measures)h(in)f(the)50 876 y(GREM.)j(In)e(the)i(random)f(energy)g(mo)s(del,)g(w)m(e)h(ha)m(v)m (e)h(considered)d(v)-5 b(arious)32 b(ob)5 b(jects)35 b(to)f(ac)m(hiev)m(e)g(suc)m(h)f(a)50 1022 y(description.)k(The)26 b(simplest)e(one)i(w)m(as)g(Ruelle's)f(pro)s(cess)g(of)i(the)f(Gibbs)e (masses,)j(obtained)f(as)g(the)g(limit)50 1167 y(of)f(the)f(pro)s(cess) h Fk(W)700 1181 y Fi(N)794 1167 y Fk(\021)890 1099 y Fe(P)986 1195 y Fi(\033)1050 1167 y Fj(\016)1090 1184 y Fi(\026)1134 1194 y Fm(\014)r(;N)1252 1184 y Fu(\()p Fi(\033)r Fu(\))1358 1167 y Fr(.)38 b(Let)25 b(us)f(p)s(oin)m(t)g(to)h (the)g(fact)h(that)f(together)h(with)e(the)g(information)50 1313 y(that)g(the)f(replica)f(o)m(v)m(erlap)h(in)f(the)h(REM)h(can)f (tak)m(e)i(on)e(only)f(the)h(v)-5 b(alues)23 b(0)g(and)g(1,)i(this)d (pro)s(cess)g(describ)s(es)50 1458 y(in)i(a)i(w)m(a)m(y)h(fully)c(the)j (structure)f(of)h(the)g(Gibbs)e(measure:)38 b(namely)-8 b(,)27 b(if)d(one)i(is)f(in)m(terested)h(in)e(capturing)h(an)50 1604 y(arbitrary)20 b(large)i(fraction)g(1)s Fk(\000)s Fj(p)f Fr(of)h(the)g(total)h(mass,)g(then)f(it)f(su\016ces)h(to)g (consider)f(for)h(some)g Fj(\017)j Fr(=)g Fj(\017)p Fr(\()p Fj(p)p Fr(\))g Fj(>)g Fr(0)50 1749 y(the)32 b(atoms)g(of)g(the)g(pro)s (cess)g Fk(W)39 b Fr(with)30 b(mass)i(larger)f(than)h Fj(\017)p Fr(,)g(and)f(to)h(place)g(them)g(at)g(random)f(on)h(a)g(set) 50 1895 y(of)38 b(orthogonal)g(v)m(ectors)h(on)f(the)g (in\014nite-dimensional)33 b(unit)j(sphere.)62 b(This)36 b(yields,)i(in)f(a)h(reasonable)50 2040 y(sense,)30 b(an)h(adequate)g (description)d(of)j(the)g(Gibbs)d(mass.)159 2258 y(Ho)m(w)m(ev)m(er,)41 b(in)36 b(the)i(GREM,)f(this)g(picture)f(is)g(insu\016cien)m(t)g(to)h (describ)s(e)f(the)i(Gibbs)d(measure)i(fully)-8 b(.)50 2404 y(The)27 b(reason)h(for)g(this)e(is)h(that)i(the)f(o)m(v)m(erlap)g (distribution)c(ma)m(y)29 b(no)m(w)e(tak)m(e)j(on)e(v)-5 b(alues)27 b(that)h(are)g(di\013eren)m(t)50 2549 y(from)23 b(0)g(and)g(1.)39 b(Th)m(us)22 b(the)i(p)s(oin)m(ts)e(carrying)g(the)i (masses)f(describ)s(ed)f(b)m(y)h(the)h(pro)s(cess)e Fk(W)31 b Fr(are)24 b(distributed)50 2695 y(in)g(a)h(more)g(complicated)g(w)m (a)m(y)h(in)e(space,)j(and)d(m)m(ust)h(retain)g(more)g(information)f (to)h(capture)h(this)e(spatial)50 2840 y(picture.)55 b(Ruelle)34 b(to)s(ok)j(this)d(fact)j(in)m(to)e(accoun)m(t)i(when)e (de\014ning)e(the)j(\\probabilit)m(y)e(cascades")j(in)d(his)50 2985 y(v)m(ersion)28 b(of)h(the)g(GREM.)h(W)-8 b(e)30 b(will)c(describ)s(e)i(these)h(ob)5 b(jects)30 b(in)d(the)i(con)m(text) i(of)e(the)g(GREM)h(and)e(pro)m(v)m(e)50 3131 y(their)h(con)m(v)m (ergence)k(to)e(Ruelle's)e(cascades.)50 3276 y Fs(The)34 b(o)m(v)m(erlap)i(distribution.)159 3495 y Fr(A)28 b(k)m(ey)i(ob)5 b(ject)29 b(considered)e(in)g(the)i(ph)m(ysical)e(literature)g(on)h (spin)f(glasses)h(is)g(the)g(distribution)d(of)k(the)50 3640 y(o)m(v)m(erlap,)i(i.e.)41 b(the)30 b(random)g(probabilit)m(y)e (distribution)1281 3888 y(~)1262 3912 y Fj(f)1307 3926 y Fi(\014)s(;N)1441 3912 y Fr(\()p Fj(q)s Fr(\))e Fk(\021)f Fj(\026)1732 3874 y Fh(\012)p Fu(2)1732 3940 y Fi(\014)s(;N)1882 3912 y Fr(\()p Fj(R)1986 3926 y Fi(N)2055 3912 y Fr(\()p Fj(\033)n(;)15 b(\033)2235 3875 y Fh(0)2262 3912 y Fr(\))26 b Fj(<)f(q)s Fr(\))979 b(\(1)p Fj(:)p Fr(19\))50 4185 y(where)35 b Fj(R)387 4199 y Fi(N)456 4185 y Fr(\()p Fj(\033)n(;)15 b(\033)636 4152 y Fh(0)663 4185 y Fr(\))34 b Fk(\021)862 4149 y Fu(1)p 847 4164 65 4 v 847 4216 a Fi(N)923 4185 y Fr(\()p Fj(\033)n(;)15 b(\033)1103 4152 y Fh(0)1129 4185 y Fr(\).)57 b(In)35 b(the)h(con)m(text)h(of)f (the)g(GREM,)g(it)f(is)g(more)h(natural)e(to)j(in)m(tro)s(duce)50 4330 y(the)30 b(distribution)d(of)k(the)f(hierarc)m(hical)f(distance,)h (i.e.)1273 4603 y Fj(f)1318 4617 y Fi(\014)s(;N)1452 4603 y Fr(\()p Fj(q)s Fr(\))c Fk(\021)f Fj(\026)1743 4564 y Fh(\012)p Fu(2)1743 4630 y Fi(\014)s(;N)1893 4603 y Fr(\()p Fj(d)1975 4617 y Fi(N)2045 4603 y Fr(\()p Fj(\033)n(;)15 b(\033)2225 4565 y Fh(0)2251 4603 y Fr(\))26 b Fj(<)f(q)s Fr(\))990 b(\(1)p Fj(:)p Fr(20\))50 4875 y(As)30 b(w)m(e)h(shall)d(see) j(later,)g(these)g(t)m(w)m(o)g(notions)f(turn)f(out)i(to)g(b)s(e)f (asymptotically)f(equiv)-5 b(alen)m(t)30 b(in)f(a)i(strong)50 5021 y(sense.)54 b(With)34 b(the)h(notation)g(in)m(tro)s(duced)e(in)g (Theorem)i(1.3)h(according)e(to)i(the)f(partition)e Fj(J)3332 5035 y Fu(0)3373 5021 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(J)3640 5035 y Fi(m)50 5166 y Fr(de\014ned)29 b(in)g(\(1.12\),)k(let)d (us)g(set)1599 5383 y Fj(q)1640 5397 y Fi(l)1692 5383 y Fk(\021)1845 5269 y Fi(l)1791 5297 y Fe(X)1788 5489 y Fi(n)p Fu(=1)1951 5321 y Fr(ln)23 b(\026)-54 b Fj(\013)2100 5335 y Fi(n)p 1951 5362 199 4 v 1983 5445 a Fr(ln)13 b(2)3477 5383 y(\(1)p Fj(:)p Fr(21\))p eop %%Page: 9 9 9 8 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3127 b Fo(9)50 513 y Fr(and)1462 700 y Fj(q)1503 714 y Fr(max)1677 700 y(\()p Fj(\014)5 b Fr(\))26 b Fk(\021)1931 579 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1928 614 y Fe(X)1925 806 y Fi(n)p Fu(=1)2088 639 y Fr(ln)d(\026)-54 b Fj(\013)2237 653 y Fi(n)p 2088 679 199 4 v 2119 762 a Fr(ln)14 b(2)3477 700 y(\(1)p Fj(:)p Fr(22\))50 942 y(W)-8 b(e)32 b(will)c(see)k(that)g (the)f(measure)g Fj(f)1282 956 y Fi(\014)s(;N)1447 942 y Fr(con)m(v)m(erges)i(to)f(a)f(limiting)d(random)i(measure)h Fj(f)3116 956 y Fi(\014)3196 942 y Fr(with)e(supp)s(ort)50 1088 y(on)h(the)h(set)g Fk(f)p Fr(0)p Fj(;)15 b(q)646 1102 y Fu(1)687 1088 y Fj(;)g(:)g(:)g(:)i(;)e(q)930 1104 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1059 1088 y Fk(g)p Fr(.)50 1233 y Fs(P)m(oin)m(t)35 b(pro)s(cesses)i(of)e(masses.)159 1451 y Fr(W)-8 b(e)37 b(will)d(in)m(tro)s(duce)h(a)i(n)m(um)m(b)s(er)e (of)h(p)s(oin)m(t)f(pro)s(cesses)h(that)h(app)s(ear)f(to)h(b)s(e)e(go)s (o)s(d)h(candidates)g(for)g(a)50 1597 y(more)30 b(detailed)g (description)e(of)j(the)f(Gibbs)f(measure.)159 1815 y(Let)i(us)e(in)m (tro)s(duce)h(the)g(sets)802 2050 y Fj(B)871 2064 y Fi(l)899 2050 y Fr(\()p Fj(\033)s Fr(\))c Fk(\021)f(f)p Fj(\033)1246 2012 y Fh(0)1298 2050 y Fk(2)g(S)1439 2064 y Fi(N)1533 2050 y Fr(:)g Fj(d)1630 2064 y Fi(N)1700 2050 y Fr(\()p Fj(\033)n(;)15 b(\033)1880 2012 y Fh(0)1906 2050 y Fr(\))26 b Fk(\025)f Fj(q)2104 2064 y Fi(l)2131 2050 y Fk(g)p Fj(;)77 b(l)27 b Fr(=)e(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\))p Fj(:)521 b Fr(\(1)p Fj(:)p Fr(23\))50 2285 y(W)-8 b(e)31 b(de\014ne)f(p)s(oin)m(t)f(pro)s (cesses)i Fk(W)1197 2237 y Fu(\()p Fi(m)p Fu(\))1190 2312 y Fi(\014)s(;N)1355 2285 y Fr(on)f(\(0)p Fj(;)15 b Fr(1])1671 2252 y Fi(m)1772 2285 y Fr(giv)m(en)31 b(b)m(y)825 2552 y Fk(W)922 2505 y Fu(\()p Fi(m)p Fu(\))915 2580 y Fi(\014)s(;N)1075 2552 y Fk(\021)1171 2466 y Fe(X)1215 2657 y Fi(\033)1318 2552 y Fj(\016)1358 2569 y Fu(\()q Fi(\026)1431 2579 y Fm(\014)r(;N)1548 2569 y Fu(\()q Fi(B)1632 2578 y Fn(1)1667 2569 y Fu(\()p Fi(\033)r Fu(\)\))r Fi(;:::)n(;\026)1949 2579 y Fm(\014)r(;N)2067 2569 y Fu(\()p Fi(B)2150 2578 y Fm(m)2210 2569 y Fu(\()p Fi(\033)r Fu(\)\))r(\))2496 2491 y Fj(\026)2551 2505 y Fi(\014)s(;N)2685 2491 y Fr(\()p Fj(\033)s Fr(\))p 2384 2531 540 4 v 2384 2615 a Fj(\026)2439 2629 y Fi(\014)s(;N)2589 2615 y Fr(\()p Fj(B)2693 2629 y Fi(m)2762 2615 y Fr(\()p Fj(\033)s Fr(\)\))3477 2552 y(\(1)p Fj(:)p Fr(24\))50 2827 y(as)f(w)m(ell)g(as)g(their)g(pro)5 b(jection)30 b(on)g(the)h(last)f(co)s(ordinate,)1119 3073 y Fk(R)1196 3026 y Fu(\()p Fi(m)p Fu(\))1196 3101 y Fi(\014)s(;N)1357 3073 y Fk(\021)1453 2987 y Fe(X)1496 3178 y Fi(\033)1599 3073 y Fj(\016)1639 3090 y Fi(\026)1683 3100 y Fm(\014)r(;N)1801 3090 y Fu(\()q Fi(B)1885 3099 y Fm(m)1945 3090 y Fu(\()p Fi(\033)r Fu(\)\))2202 3012 y Fj(\026)2257 3026 y Fi(\014)s(;N)2392 3012 y Fr(\()p Fj(\033)s Fr(\))p 2090 3052 V 2090 3136 a Fj(\026)2145 3150 y Fi(\014)s(;N)2295 3136 y Fr(\()p Fj(B)2399 3150 y Fi(m)2468 3136 y Fr(\()p Fj(\033)s Fr(\)\))3477 3073 y(\(1)p Fj(:)p Fr(25\))50 3380 y(It)g(is)g(easy)h(to)g(see)g(that)g (the)f(pro)s(cesses)h Fk(W)1542 3333 y Fu(\()p Fi(m)p Fu(\))1535 3408 y Fi(\014)s(;N)1700 3380 y Fr(satisfy)615 3661 y Fk(W)712 3613 y Fu(\()p Fi(m)p Fu(\))705 3688 y Fi(\014)s(;N)840 3661 y Fr(\()p Fj(dw)987 3675 y Fu(1)1029 3661 y Fj(;)15 b(:)g(:)g(:)h(;)f(dw)1342 3675 y Fi(m)1412 3661 y Fr(\))26 b(=)1569 3537 y Fe(Z)1660 3559 y Fu(1)1619 3743 y(0)1716 3661 y Fj(W)1815 3622 y Fi(m)p Fu(+1)1802 3688 y Fi(\014)s(;N)1975 3661 y Fr(\()p Fj(dw)2122 3675 y Fu(1)2164 3661 y Fj(;)15 b(:)g(:)g(:)h(;)f(dw)2477 3675 y Fi(m)2547 3661 y Fj(;)g(dw)2699 3675 y Fi(m)p Fu(+1)2861 3661 y Fr(\))2907 3599 y Fj(w)2972 3613 y Fi(m)p Fu(+1)p 2907 3640 227 4 v 2953 3723 a Fj(w)3018 3737 y Fi(m)3477 3661 y Fr(\(1)p Fj(:)p Fr(26\))50 3905 y(where)32 b(the)g(in)m(tegration)g(is)f(of)i(course)f(o)m(v)m(er)i (the)e(last)g(co)s(ordinate)g Fj(w)2459 3919 y Fi(m)p Fu(+1)2621 3905 y Fr(.)46 b(Note)33 b(that)g(these)g(pro)s(cesses)50 4050 y(will)e(in)h(general)h(not)h(all)e(con)m(v)m(erge,)k(but)d(will)e (do)i(so)g(only)g(when)f(for)h(some)h Fj(\033)s Fr(,)h Fj(\026)2918 4064 y Fi(\014)2966 4050 y Fr(\()p Fj(B)3070 4064 y Fi(m)3140 4050 y Fr(\()p Fj(\033)s Fr(\)\))f(is)f(strictly)50 4196 y(p)s(ositiv)m(e.)47 b(F)-8 b(rom)34 b(our)e(exp)s(erience)h(with) e(the)j(Gibbs)d(measure,)j(it)e(is)g(clear)h(that)h(this)e(will)e(b)s (e)j(the)g(case)50 4341 y(precisely)c(when)g Fj(m)c Fk(\024)g Fj(l)r Fr(\()p Fj(\014)5 b Fr(\).)42 b(In)29 b(fact,)j(w)m(e)f(pro)m(v) m(e)g(the)f(follo)m(wing:)50 4559 y Fd(Theorem)47 b(1.5:)72 b Fq(If)40 b Fj(m)f Fk(\024)f Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))p Fq(,)43 b(the)e(p)-5 b(oint)41 b(pr)-5 b(o)g(c)g(ess)43 b Fk(W)2162 4512 y Fu(\()p Fi(m)p Fu(\))2155 4587 y Fi(\014)s(;N)2330 4559 y Fq(on)d Fr(\(0)p Fj(;)15 b Fr(1])2657 4526 y Fi(m)2768 4559 y Fq(c)-5 b(onver)g(ges)41 b(we)-5 b(akly)42 b(to)e(the)50 4705 y(p)-5 b(oint)38 b(pr)-5 b(o)g(c)g(ess)38 b Fk(W)692 4658 y Fu(\()p Fi(m)p Fu(\))685 4732 y Fi(\014)855 4705 y Fq(whose)f(atoms)h Fj(w)r Fr(\()p Fj(i)p Fr(\))g Fq(ar)-5 b(e)37 b(given)f(in)g(terms)h(of)g(the)g(atoms)h Fr(\()p Fj(x)2996 4719 y Fu(1)3037 4705 y Fr(\()p Fj(i)p Fr(\))p Fj(;)15 b(:)g(:)g(:)i(;)e(x)3392 4719 y Fi(m)3462 4705 y Fr(\()p Fj(i)p Fr(\)\))37 b Fq(of)50 4850 y(the)c(p)-5 b(oint)34 b(pr)-5 b(o)g(c)g(ess)35 b Fk(P)809 4817 y Fu(\()p Fi(m)p Fu(\))968 4850 y Fq(by)80 5044 y Fr(\()p Fj(w)180 5058 y Fu(1)221 5044 y Fr(\()p Fj(i)p Fr(\))p Fj(;)15 b(:)g(:)g(:)j(;)d(w)590 5058 y Fi(m)659 5044 y Fr(\()p Fj(i)p Fr(\)\))105 5271 y(=)201 5115 y Fe( )284 5134 y(R)360 5208 y Fk(P)430 5175 y Fu(\()p Fi(m)p Fu(\))556 5208 y Fr(\()p Fj(dy)s Fr(\))p Fj(\016)s Fr(\()p Fj(y)844 5222 y Fu(1)907 5208 y Fk(\000)20 b Fj(x)1050 5222 y Fu(1)1091 5208 y Fr(\()p Fj(i)p Fr(\)\))p Fj(e)1269 5175 y Fi(\014)s Fu(\()p Fi(\015)t(;y)r Fu(\))p 284 5250 1196 4 v 537 5262 a Fe(R)613 5335 y Fk(P)683 5309 y Fu(\()p Fi(m)p Fu(\))809 5335 y Fr(\()p Fj(dy)s Fr(\))p Fj(e)1016 5309 y Fi(\014)s Fu(\()p Fi(\015)t(;y)r Fu(\))1490 5271 y Fj(;)15 b(:)g(:)g(:)i(;)1703 5134 y Fe(R)1779 5208 y Fk(P)1849 5175 y Fu(\()p Fi(m)p Fu(\))1975 5208 y Fr(\()p Fj(dy)s Fr(\))p Fj(\016)s Fr(\()p Fj(y)2263 5222 y Fu(1)2326 5208 y Fk(\000)j Fj(x)2469 5222 y Fu(1)2510 5208 y Fr(\()p Fj(i)p Fr(\)\))15 b Fj(:)g(:)g(:)i(\016)s Fr(\()p Fj(y)2906 5222 y Fi(m)2996 5208 y Fk(\000)j Fj(x)3139 5222 y Fi(m)3208 5208 y Fr(\()p Fj(i)p Fr(\)\))p Fj(e)3386 5175 y Fi(\014)s Fu(\()p Fi(\015)t(;y)r Fu(\))p 1703 5250 1894 4 v 2305 5262 a Fe(R)2381 5335 y Fk(P)2451 5309 y Fu(\()p Fi(m)p Fu(\))2578 5335 y Fr(\()p Fj(dy)s Fr(\))p Fj(e)2785 5309 y Fi(\014)s Fu(\()p Fi(\015)t(;y)r Fu(\))3608 5115 y Fe(!)3477 5462 y Fr(\(1)p Fj(:)p Fr(27\))p eop %%Page: 10 10 10 9 bop 50 295 a Fo(10)3270 b Fl(Se)l(ction)27 b Fo(1)50 517 y Fq(and)40 b(the)f(p)-5 b(oint)40 b(pr)-5 b(o)g(c)g(esses)41 b Fk(R)1096 470 y Fu(\()p Fi(m)p Fu(\))1096 544 y Fi(\014)s(;N)1270 517 y Fq(c)-5 b(onver)g(ge)40 b(to)f(the)h(p)-5 b(oint)40 b(pr)-5 b(o)g(c)g(ess)41 b Fk(R)2545 470 y Fu(\()p Fi(m)p Fu(\))2545 544 y Fi(\014)2710 517 y Fq(whose)f(atoms)h(ar)-5 b(e)39 b(the)h(last)50 662 y(c)-5 b(omp)g(onents)35 b(of)e(the)g(atoms) h(in)f(\(1.27\).)159 880 y Fr(Of)e(course)h(the)g(most)h(complete)f(ob) 5 b(ject)33 b(w)m(e)f(can)g(reasonably)g(study)f(is)f(the)j(pro)s(cess) 3204 857 y Fe(c)3193 880 y Fk(W)3283 894 y Fi(\014)3360 880 y Fk(\021)27 b(W)3555 833 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))3548 908 y Fi(\014)3685 880 y Fr(.)50 1026 y(Analogously)-8 b(,)30 b(w)m(e)h(will)d(set)1051 1003 y Fe(b)1031 1026 y Fk(R)1108 1040 y Fi(\014)1182 1026 y Fk(\021)d(R)1355 979 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1355 1053 y Fi(\014)1484 1026 y Fr(.)159 1244 y(The)38 b(p)s(oin)m(t)f(pro)s(cesses)1011 1221 y Fe(c)1000 1244 y Fk(W)1090 1258 y Fi(\014)1177 1244 y Fr(tak)m(es)j(v)-5 b(alues)38 b(on)g(v)m(ectors)i(whose)e(comp)s (onen)m(ts)g(form)g(decreasing)g(se-)50 1389 y(quences)29 b(in)g(\(0)p Fj(;)15 b Fr(1].)42 b(Moreo)m(v)m(er,)32 b(these)e(atoms)h(are)f(naturally)e(clustered)h(in)f(a)i(hierarc)m (hical)e(w)m(a)m(y)-8 b(.)42 b(These)50 1535 y(pro)s(cesses)30 b(w)m(ere)h(in)m(tro)s(duced)e(b)m(y)h(Ruelle)f([Ru])h(and)g(called)g Fq(pr)-5 b(ob)g(ability)34 b(c)-5 b(asc)g(ades)p Fr(.)159 1753 y(There)34 b(is)f(an)h(in)m(timate)g(relation)f(b)s(et)m(w)m(een)i (the)g(distance)e(distributions)e Fj(f)2807 1767 y Fi(\014)2889 1753 y Fr(and)j(these)h(p)s(oin)m(t)e(pro-)50 1899 y(cesses.)41 b(The)30 b(\014rst)g(result)f(w)m(e)i(shall)e(pro)m(v)m(e)i(is)e(the)i (follo)m(wing)e(theorem.)50 2117 y Fd(Theorem)37 b(1.6:)51 b Fq(With)33 b(the)g(notation)h(intr)-5 b(o)g(duc)g(e)35 b(ab)-5 b(ove,)33 b(we)g(have)g(that)10 2335 y(\(i\))47 b(The)31 b(r)-5 b(andom)34 b(distribution)e(functions)f Fj(f)1596 2349 y Fi(\014)s(;N)1761 2335 y Fq(and)1956 2311 y Fr(~)1936 2335 y Fj(f)1981 2349 y Fi(\014)s(;N)2147 2335 y Fq(c)-5 b(onver)g(ge)31 b(in)g(distribution)h(and)g(in)f(me)-5 b(an)32 b(to)159 2480 y(the)h(same)g(r)-5 b(andom)35 b(distribution)f(function)f Fj(f)1763 2494 y Fi(\014)1811 2480 y Fq(.)-18 2699 y(\(ii\))47 b Fj(f)204 2713 y Fi(\014)285 2699 y Fq(is)33 b(a)g(step)g(function)f(with)i(jumps)f(at)g(the)g (values)g Fk(f)p Fr(0)p Fj(;)15 b(q)2175 2713 y Fu(1)2217 2699 y Fj(;)g(:)g(:)g(:)h(;)f(q)2459 2715 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))2589 2699 y Fk(g)p Fq(.)42 b(F)-7 b(or)33 b(any)h Fj(q)28 b Fk(2)d Fr(\()p Fj(q)3279 2713 y Fi(i)p Fh(\000)p Fu(1)3402 2699 y Fj(;)15 b(q)3483 2713 y Fi(i)3513 2699 y Fr(])704 2996 y Fj(f)749 3010 y Fi(\014)798 2996 y Fr(\()p Fj(q)s Fr(\))26 b(=)1033 2872 y Fe(Z)1150 2973 y(c)1140 2996 y Fk(W)1230 3010 y Fi(\014)1278 2996 y Fr(\()p Fj(dw)1425 3010 y Fu(1)1467 2996 y Fj(;)15 b(:)g(:)g(:)i(;)e (dw)1781 3012 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1911 2996 y Fr(\))p Fj(w)2011 3012 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))2140 2996 y Fr(\(1)21 b Fk(\000)f Fj(w)2397 3010 y Fi(i)2427 2996 y Fr(\))p Fj(;)81 b(i)26 b Fr(=)f(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\);)314 b(\(1)p Fj(:)p Fr(28\))50 3283 y Fj(f)95 3297 y Fi(\014)143 3283 y Fr(\()p Fj(q)s Fr(\))26 b(=)f(1)33 b Fq(for)g Fj(q)28 b Fk(2)d Fr(\()p Fj(q)833 3297 y Fi(l)861 3283 y Fr(\()p Fj(\014)5 b Fr(\))p Fj(;)15 b Fr(1])p Fq(.)159 3501 y Fr(W)-8 b(e)36 b(will)d(pro)m(v)m(e)j(Theorems)f(1.5)h(in)e(Section)i(3)f(directly)f (as)i(a)f(consequence)h(of)g(Theorem)f(1.3.)57 b(W)-8 b(e)50 3647 y(will)29 b(also)j(pro)m(v)m(e)h(a)f(more)h(general)f (Theorem)f(3.1)j(that)e(imply)e(in)h(particular)f(Theorem)i(1.6.)46 b(Its)32 b(pro)s(of)50 3792 y(is)d(somewhat)i(tedious)f(but)f(in)g (principle)e(straigh)m(tforw)m(ard.)50 4007 y Fs(1.4.)47 b(Ghirlanda-Guerra)35 b(iden)m(tities.)159 4225 y Fr(W)-8 b(e)36 b(no)m(w)e(turn)g(to)h(a)g(di\013eren)m(t)f(approac)m(h)h(to)m (w)m(ards)h(the)e(construction)h(of)f(the)h(pro)s(cesses)3381 4202 y Fe(c)3370 4225 y Fk(W)3460 4239 y Fi(\014)3543 4225 y Fr(that)50 4371 y(will)20 b(completely)j(a)m(v)m(oid)h(the)g (use)f(of)g(the)h(P)m(oisson)f(cascades.)39 b(In)23 b(fact,)j(w)m(e)d (will)e(see)j(that)g(the)f(con)m(v)m(ergence)50 4516 y(of)30 b(the)h(pro)s(cesses)714 4493 y Fe(c)703 4516 y Fk(W)793 4530 y Fi(\014)s(;N)958 4516 y Fr(follo)m(ws)f(from)f (general)i(principles)c(once)k(w)m(e)f(can)h(compute)g(the)f(limiting)e (free)50 4662 y(energy)g(as)h(a)f(function)f(of)i(the)f(function)f Fj(A)h Fr(describing)e(the)j(co)m(v)-5 b(ariance)29 b(of)g(the)f (Gaussian)g(pro)s(cess)f Fj(X)3635 4676 y Fi(\033)3685 4662 y Fr(.)50 4807 y(The)d(k)m(ey)h(role)f(in)f(this)g(approac)m(h)i (is)e(pla)m(y)m(ed)i(b)m(y)f(the)h(so-called)f Fq(Ghirlanda-Guerr)-5 b(a)28 b Fr(iden)m(tities)23 b([GG,A)m(C].)50 4953 y(The)31 b(existence)g(of)h(this)e(approac)m(h)h(will)e(b)s(ear)i(its)f(full)f (fruits)h(in)g(the)h(follo)m(w-up)f(pap)s(er)g([BK3])i(where)f(it)50 5098 y(will)d(allo)m(w)h(us)h(to)h(treat)h(the)e(GREM)h(with)e(con)m (tin)m(uous)h(hierarc)m(hies.)159 5316 y(The)20 b(\014rst)g(result)f (is)h(an)g(explicit)f(expression)g(for)h(the)h(mean)g(of)f(the)h(t)m(w) m(o-replica)g(distribution)c(function)50 5462 y Fj(f)95 5476 y Fi(\014)143 5462 y Fr(.)p eop %%Page: 11 11 11 10 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(11)50 513 y Fd(Prop)s(osition)38 b(1.7:)p Fq(The)33 b(me)-5 b(an)33 b(of)g Fj(f)1426 527 y Fi(\014)1507 513 y Fq(is)g(given)f(by) 828 776 y Fg(E)11 b Fj(f)928 790 y Fi(\014)983 776 y Fr(\()p Fj(q)s Fr(\))25 b(=)1218 621 y Fe(\()1322 712 y Fj(\014)1378 679 y Fh(\000)p Fu(1)1476 609 y Fe(q)p 1567 609 225 4 v 1578 666 a Fu(2)12 b(ln)21 b(\026)-44 b Fi(\013)1746 675 y Fm(j)p 1577 691 204 4 v 1644 743 a Fu(\026)-38 b Fi(a)1681 752 y Fm(j)1791 712 y Fj(;)94 b Fq(if)15 b Fj(q)28 b Fk(2)c Fr(\()p Fj(q)2211 726 y Fi(j)t Fh(\000)p Fu(1)2343 712 y Fj(;)15 b(q)2424 726 y Fi(j)2462 712 y Fr(])p Fj(;)48 b(j)31 b Fk(\024)25 b Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))p Fj(;)1322 873 y Fr(1)p Fj(;)518 b Fq(if)15 b Fj(q)28 b(>)d(q)2187 887 y Fi(max)2336 873 y Fr(\()p Fj(\014)5 b Fr(\))3477 776 y(\(1)p Fj(:)p Fr(29\))159 1203 y(Let)42 b(us)g(denote)g(b)m(y)g Fj(\026)959 1164 y Fh(\012)p Fi(k)959 1230 y(\014)s(;N)1136 1203 y Fr(the)h(distribution)38 b(of)k Fj(k)j Fr(indep)s(enden)m(t)40 b(copies)i(of)g Fj(\033)k Fr(under)40 b(the)i(Gibbs)50 1348 y(measure.)68 b(The)38 b(follo)m(wing)g(prop)s(osition)f(corresp)s (onds)h(to)j(the)e(Ghirlanda-Guerra)f(relations)g(in)g(our)50 1494 y(mo)s(dels.)50 1712 y Fd(Prop)s(osition)33 b(1.8:)p Fq(Assume)c(that)h(the)g(p)-5 b(ar)g(ameters)32 b Fj(\013)2027 1726 y Fi(i)2086 1712 y Fq(and)e Fj(a)2307 1726 y Fi(i)2366 1712 y Fq(ar)-5 b(e)30 b(such)f(that)i(none)e(of)g(the)h(ine)-5 b(quali-)50 1857 y(ties)26 b(\(1.9\))i(holds)g(with)f(e)-5 b(quality.)40 b(Then,)28 b(for)f(any)g(b)-5 b(ounde)g(d)28 b(function)f Fj(h)e Fr(:)h Fk(S)2668 1824 y Fi(n)2661 1882 y(N)2755 1857 y Fk(!)f Fg(R)s Fq(,)33 b(for)27 b(al)5 b(l)27 b Fj(i)f Fr(=)f(1)p Fj(;)15 b(:)g(:)g(:)i(;)e(n)178 2113 y Fr(lim)154 2173 y Fi(N)7 b Fh("1)328 1954 y Fe(\014)328 2009 y(\014)328 2063 y(\014)328 2118 y(\014)328 2172 y(\014)358 2113 y Fg(E)k Fj(\026)468 2074 y Fh(\012)p Fi(n)p Fu(+1)468 2141 y Fi(\014)s(;N)687 2012 y Fe(\020)742 2113 y Fj(h)p Fr(\()p Fj(\033)884 2076 y Fu(1)925 2113 y Fj(;)k(:)g(:)g(:)i(;)e(\033)1182 2076 y Fi(n)1232 2113 y Fr(\)1)-23 b(I)1322 2142 y Fi(\033)1366 2120 y Fm(k)1364 2164 y Fn(1)1407 2142 y Fi(:::)n(\033)1515 2120 y Fm(k)1513 2165 y(i)1556 2142 y Fu(=)p Fi(\033)1656 2112 y Fm(n)p Fn(+1)1654 2164 y(1)1778 2142 y Fi(:::)o(\033)1887 2112 y Fm(n)p Fn(+1)1885 2165 y Fm(i)2014 2012 y Fe(\021)348 2450 y Fk(\000)455 2388 y Fr(1)p 450 2429 55 4 v 450 2512 a Fj(n)516 2450 y Fg(E)10 b Fj(\026)625 2411 y Fh(\012)p Fi(n)p Fu(+1)625 2477 y Fi(\014)t(;N)845 2267 y Fe(0)845 2431 y(@)924 2450 y Fj(h)p Fr(\()p Fj(\033)1066 2412 y Fu(1)1108 2450 y Fj(;)15 b(:)g(:)g(:)h(;)f(\033)1364 2412 y Fi(n)1415 2450 y Fr(\))1465 2267 y Fe(0)1465 2431 y(@)1588 2336 y Fi(n)1545 2363 y Fe(X)1551 2559 y Fi(l)p Fh(6)p Fu(=)p Fi(k)1691 2450 y Fr(1)-23 b(I)1746 2472 y Fi(\033)1790 2450 y Fm(l)1788 2494 y Fn(1)1824 2472 y Fi(:::)n(\033)1932 2450 y Fm(l)1930 2495 y(i)1959 2472 y Fu(=)p Fi(\033)2059 2450 y Fm(k)2057 2494 y Fn(1)2099 2472 y Fi(:::)o(\033)2208 2450 y Fm(k)2206 2495 y(i)2273 2450 y Fr(+)20 b Fg(E)11 b Fj(\026)2474 2411 y Fh(\012)p Fu(2)2474 2477 y Fi(\014)s(;N)2614 2450 y Fr(\(1)-23 b(I)2705 2470 y Fi(\033)2749 2448 y Fn(1)2747 2491 y(1)2785 2470 y Fi(:::)n(\033)2893 2448 y Fn(1)2891 2492 y Fm(i)2929 2470 y Fu(=)p Fi(\033)3029 2448 y Fn(2)3027 2491 y(1)3065 2470 y Fi(:::)o(\033)3174 2448 y Fn(2)3172 2492 y Fm(i)3214 2450 y Fr(\))3249 2267 y Fe(1)3249 2431 y(A)3329 2267 y(1)3329 2431 y(A)3409 2291 y(\014)3409 2345 y(\014)3409 2400 y(\014)3409 2454 y(\014)3409 2509 y(\014)3464 2450 y Fr(=)25 b(0)3477 2668 y(\(1)p Fj(:)p Fr(30\))159 3031 y(It)j(is)g(quite)g(remark)-5 b(able)27 b(that)i(these)g(relations)f (su\016ce)g(to)h(determine)f(completely)g(the)g(distribution)50 3177 y(of)i(the)h(random)f(measures)1044 3154 y Fe(c)1034 3177 y Fk(W)1124 3191 y Fi(\014)1203 3177 y Fr(through)f(their)h (generalized)g(momen)m(ts.)50 3395 y Fd(Theorem)e(1.9:)47 b Fq(The)26 b(r)-5 b(andom)29 b(me)-5 b(asur)g(es)1649 3372 y Fe(c)1638 3395 y Fk(W)1728 3409 y Fi(\014)1803 3395 y Fq(ar)g(e)27 b(c)-5 b(ompletely)28 b(determine)-5 b(d)28 b(by)e(the)g(r)-5 b(elations)28 b(\(1.30\))50 3541 y(up)38 b(to)h(the)g(me)-5 b(an)40 b(value)e(of)h(the)g(two-r)-5 b(eplic)g(a)40 b(distanc)-5 b(e)40 b(distribution)g(function)e Fj(f)2935 3555 y Fi(\014)3022 3541 y Fq(given)g(in)g(Pr)-5 b(op)g(osi-)50 3686 y(tion)33 b(1.7.)50 3904 y Fa(Pro)s(of:)52 b Fr(Since)32 b(it)g(is)f(imp)s(ortan)m(t)h(to)h(understand)e(this)h (result,)g(w)m(e)h(will)d(pro)m(v)m(e)j(this)f(theorem)h(immedi-)50 4050 y(ately)-8 b(.)41 b(Note)32 b(\014rst)e(that)h(the)f(last)g(term)h (in)e(\(1.30\))k(is)c(just)122 4309 y(lim)107 4368 y Fc(N)p Fh("1)278 4309 y Fg(E)10 b Fj(\026)387 4270 y Fh(\012)p Fu(2)387 4337 y Fi(\014)t(;N)528 4309 y Fr(\(1)-23 b(I)619 4329 y Fi(\033)663 4307 y Fn(1)661 4351 y(1)699 4329 y Fi(:::)o(\033)808 4307 y Fn(1)806 4352 y Fm(i)844 4329 y Fu(=)p Fi(\033)944 4307 y Fn(2)942 4351 y(1)979 4329 y Fi(:::)o(\033)1088 4307 y Fn(2)1086 4352 y Fm(i)1129 4309 y Fr(\))25 b(=)g Fg(E)11 b Fj(\026)1395 4270 y Fh(\012)p Fu(2)1395 4337 y Fi(\014)s(;N)1536 4309 y Fr(\(1)-23 b(I)1627 4347 y Fi(d)1665 4356 y Fm(N)1723 4347 y Fu(\()p Fi(\033)n(;\033)1857 4329 y Ff(0)1882 4347 y Fu(\))p Fh(\025)1967 4286 y Fe(P)2063 4308 y Fm(i)2063 4381 y(j)s Fn(=1)2188 4347 y Fu(ln)13 b Fi(\013)2308 4356 y Fm(j)2342 4347 y Fi(=)g Fu(ln)g(2)2505 4309 y Fr(\))26 b(=)f(1)c Fk(\000)f Fg(E)10 b Fj(f)2918 4323 y Fi(\014)2973 4208 y Fe(\020)3095 4195 y Fi(i)3042 4223 y Fe(X)3045 4416 y Fi(j)t Fu(=1)3189 4309 y Fr(ln)k Fj(\013)3338 4323 y Fi(j)3376 4309 y Fj(=)h Fr(ln)g(2)3573 4208 y Fe(\021)3627 4309 y Fj(:)3477 4515 y Fr(\(1)p Fj(:)p Fr(31\))120 4661 y(Let)36 b(no)m(w)g Fj(J)p 483 4676 60 4 v 483 4693 V 42 w Fk(\021)d Fr(\()p Fj(J)p 714 4676 V 774 4683 a Fu(0)815 4661 y Fj(;)15 b(:)g(:)g(:)h(;)f(J)p 1016 4676 V 1076 4683 a Fi(n)1125 4661 y Fr(\))36 b(denote)g(a)g(nested)f(set)h(of)f (\014nite)g(subsets)f(of)i Fg(N)48 b Fr(suc)m(h)35 b(that)h(eac)m(h)g Fj(J)p 3506 4676 V 3565 4683 a Fi(r)3639 4661 y Fr(=)50 4806 y(\()p Fj(J)135 4820 y Fi(r)n(;)p Fu(1)230 4806 y Fj(;)15 b(:)g(:)g(:)h(;)f(J)481 4820 y Fi(r)n(;j)565 4829 y Fm(r)607 4806 y Fr(\))39 b(is)g(a)g(collection)g(of)h(disjoin)m (t)e(subsets)g(of)i Fg(N)6 b Fr(,)48 b(suc)m(h)39 b(that)h(for)f(eac)m (h)h Fj(J)3124 4820 y Fi(r)n(;i)3248 4806 y Fr(there)f(exists)50 4951 y Fj(J)100 4965 y Fi(r)r Fh(\000)p Fu(1)p Fi(;k)331 4951 y Fr(suc)m(h)c(that)h Fj(J)793 4965 y Fi(r)n(;i)911 4951 y Fk(\032)e Fj(J)1066 4965 y Fi(r)r Fh(\000)p Fu(1)p Fi(;k)1261 4951 y Fr(.)57 b(Let)36 b(us)f(\014x)g(the)h(notation)f (that)i Fj(J)2550 4965 y Fi(r)n(;i)2668 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5431 y Ff(j)p Fm(J)2602 5446 y(r)n(;i)2681 5431 y Ff(j)2517 5488 y Fm(r)2711 5476 y Fh(g)3477 5374 y Fr(\(1)p Fj(:)p Fr(32\))p eop %%Page: 12 12 12 11 bop 50 295 a Fo(12)3270 b Fl(Se)l(ction)27 b Fo(1)50 513 y Fr(Then)i(de\014ne,)h(with)f Fj(J)35 b Fk(\021)25 b(j)p Fj(J)p 988 528 60 4 v 1047 535 a Fu(0)1088 513 y Fk(j)p Fr(,)1478 658 y Fj(A)1546 672 y Fi(J)p 1546 687 47 4 v 1546 705 51 4 v 1622 658 a Fk(\021)49 b Fr(lim)1718 717 y Fi(N)7 b Fh("1)1908 658 y Fg(E)j Fj(\026)2017 619 y Fh(\012)p Fi(J)2017 686 y(\014)t(;N)2158 658 y Fk(A)2231 672 y Fi(J)p 2231 687 47 4 v 2231 705 51 4 v 3477 658 a Fr(\(1)p Fj(:)p Fr(33\))50 851 y(No)m(w)31 b(all)f(w)m(e)i(ha)m(v)m (e)g(to)g(realise)e(is)g(that)h(for)g(an)m(y)h(suc)m(h)e(ev)m(en)m(t)j Fk(A)2241 865 y Fi(J)p 2241 880 47 4 v 2241 897 51 4 v 2291 851 a Fr(,)e(as)h(long)e(as)h Fj(J)p 2769 866 60 4 v 2829 874 a Fu(0)2900 851 y Fr(is)f(of)h(size)g(larger)g(than)50 996 y(t)m(w)m(o,)h(there)e(are)h Fj(l)r(;)15 b(m;)g(r)s Fr(,)31 b(1)25 b Fk(\024)g Fj(r)j Fk(\024)d Fj(n)p Fr(,)30 b(suc)m(h)g(that)h(for)f(some)h Fj(J)p 2130 1011 V 2130 1029 V 2189 957 a Fh(0)2245 996 y Fr(with)e Fj(J)2511 963 y Fh(0)2562 996 y Fr(=)c Fj(J)30 b Fk(\000)19 b Fr(1,)31 b Fj(l)d Fk(2)c Fj(J)p 3069 1011 V 3069 1029 V 3129 957 a Fh(0)3184 996 y Fr(and)30 b Fj(m)25 b Fk(62)g Fj(J)p 3552 1011 V 3552 1029 V 3611 957 a Fh(0)3637 996 y Fr(:)1365 1197 y Fk(A)1438 1211 y Fi(J)p 1438 1226 47 4 v 1438 1243 51 4 v 1514 1197 a Fr(=)g Fk(A)1683 1212 y Fi(J)p 1683 1227 47 4 v 1683 1245 V 1729 1191 a Ff(0)1758 1197 y Fr(1)-23 b(I)1813 1219 y Fi(\033)1857 1197 y Fm(l)1855 1241 y Fn(1)1891 1219 y Fi(:::)o(\033)2000 1201 y Fm(l)1998 1237 y(r)2034 1219 y Fu(=)p Fi(\033)2134 1197 y Fm(m)2132 1241 y Fn(1)2195 1219 y Fi(:::)o(\033)2304 1201 y Fm(m)2302 1237 y(r)2369 1197 y Fj(:)1083 b Fr(\(1)p Fj(:)p Fr(34\))50 1398 y(Th)m(us)29 b(using)h(\(1.30\))j(with)c Fj(h)d Fr(=)f Fk(A)1236 1413 y Fi(J)p 1236 1428 V 1236 1445 V 1282 1392 a Ff(0)1342 1398 y Fr(allo)m(ws)30 b(to)h(express)g Fj(A)2110 1412 y Fi(J)p 2110 1427 V 2110 1444 51 4 v 2191 1398 a Fr(in)e(term)i(of)g Fj(A)2687 1413 y Fi(J)p 2687 1428 47 4 v 2687 1445 V 2733 1392 a Ff(0)2793 1398 y Fr(with)f Fj(J)p 3001 1413 60 4 v 3001 1430 V 3060 1358 a Fh(0)3116 1398 y Fr(that)h(ha)m(v)m(e)h(total)50 1543 y(length)j(strictly)g(less)h(than)g Fj(J)p 1042 1558 V 1042 1576 V 9 w Fr(,)h(and)f(those)h(of)f(length)f(t)m(w)m(o.)60 b(Th)m(us)35 b(it)g(is)g(ob)m(vious)h(that)h(the)f(reduction)50 1688 y(pro)s(cedure)30 b(will)f(yield)i(expressions)f(for)h(an)m(y)h Fj(A)1724 1702 y Fi(J)p 1724 1717 47 4 v 1724 1735 51 4 v 1807 1688 a Fr(as)g(function)e(of)i(those)h(in)m(v)m(olving)d(only) h(t)m(w)m(o)i(replicas.)50 1834 y(Of)38 b(course)h(there)g(are)g(only)f Fj(n)h Fr(suc)m(h)f(sets,)k(dep)s(ending)36 b(only)i(on)h(the)g(n)m(um) m(b)s(er)f(of)h(non-empt)m(y)g(lev)m(els.)50 1979 y(These)i(of)h (course)g(ha)m(v)m(e)i(an)d(immediate)g(meaning)g(in)g(terms)h(of)g (the)g(t)m(w)m(o-o)m(v)m(erlap)i(distribution)38 b(via)50 2125 y(\(1.31\).)159 2343 y(This)30 b(result)h(generalizes)g(the)i (corresp)s(onding)c(\014nding)h(in)g(the)i(REM,)h(where)e(of)h(course)g (the)g(m)m(ulti-)50 2488 y(index)d Fj(J)p 295 2503 60 4 v 295 2521 V 39 w Fr(is)h(trivial.)159 2707 y(W)-8 b(e)46 b(will)c(no)m(w)i(sho)m(w)h(that)g(the)g(n)m(um)m(b)s(ers)e Fj(A)1790 2721 y Fi(J)p 1790 2736 47 4 v 1790 2753 51 4 v 1885 2707 a Fr(fully)f(determine)i(the)h(pro)s(cesses)3134 2684 y Fe(c)3123 2707 y Fk(W)3213 2721 y Fi(\014)3307 2707 y Fr(de\014ned)e(in)50 2852 y(Theorem)35 b(1.5.)57 b(T)-8 b(o)36 b(see)g(this,)g(note)g(that)g(a)g(random)f(measure)g Fk(M)2433 2866 y Fi(N)2538 2852 y Fr(on)g Fg(R)2729 2819 y Fi(d)2813 2852 y Fr(con)m(v)m(erges)i(to)f Fk(M)p Fr(,)i(if)c(for)50 2998 y(an)m(y)28 b Fj(k)h Fk(2)24 b Fg(N)7 b Fr(,)35 b(an)m(y)28 b(con)m(tin)m(uous)g(function)e Fj(F)39 b Fr(:)25 b Fg(R)1682 2965 y Fi(k)1758 2998 y Fk(!)g Fg(R)s Fr(,)34 b(and)28 b(an)m(y)g(collection)g(of)g(con)m(tin)m(uous,)g(in)m (tegrable)50 3143 y(functions)h Fj(f)487 3157 y Fu(1)527 3143 y Fj(;)15 b(:)g(:)g(:)i(;)e(f)774 3157 y Fi(k)849 3143 y Fr(on)30 b Fg(R)1035 3110 y Fi(d)1083 3143 y Fr(,)128 3357 y Fj(F)214 3229 y Fe(\022)281 3233 y(Z)387 3357 y Fk(M)496 3371 y Fi(N)565 3357 y Fr(\()p Fj(dx)p Fr(\))p Fj(f)779 3371 y Fu(1)820 3357 y Fr(\()p Fj(x)p Fr(\))p Fj(;)15 b(:)g(:)g(:)j(;)1145 3233 y Fe(Z)1251 3357 y Fk(M)1360 3371 y Fi(N)1429 3357 y Fr(\()p Fj(dx)p Fr(\))p Fj(f)1643 3371 y Fi(k)1688 3357 y Fr(\()p Fj(x)p Fr(\))1810 3229 y Fe(\023)1903 3357 y Fk(!)25 b Fj(F)2106 3229 y Fe(\022)2173 3233 y(Z)2279 3357 y Fk(M)p Fr(\()p Fj(dx)p Fr(\))p Fj(f)2602 3371 y Fu(1)2643 3357 y Fr(\()p Fj(x)p Fr(\))p Fj(;)15 b(:)g(:)g(:)i(;)2967 3233 y Fe(Z)3074 3357 y Fk(M)p Fr(\()p Fj(dx)p Fr(\))p Fj(f)3397 3371 y Fi(k)3442 3357 y Fr(\()p Fj(x)p Fr(\))3564 3229 y Fe(\023)3477 3520 y Fr(\(1)p Fj(:)p Fr(35\))50 3666 y(\(see)29 b(e.g.)42 b([Ka,Re]\))30 b(By)f(standard)f(appro)m(ximation)f(argumen)m(ts,)j(if) e Fk(M)g Fr(has)h(momen)m(ts)g(of)g(all)f(orders,)g(it)50 3811 y(is)k(enough)i(to)g(consider)e(for)i Fj(f)1120 3825 y Fi(i)1183 3811 y Fr(all)e(monomial)h(functions)f Fj(f)10 b Fr(\()p Fj(x)p Fr(\))30 b(=)g Fj(x)2490 3771 y Fi(i)2516 3780 y Fn(1)2490 3835 y Fu(1)2571 3811 y Fj(:)15 b(:)g(:)h(x)2744 3770 y Fi(i)2770 3780 y Fm(d)2744 3839 y Fi(d)2813 3811 y Fr(,)34 b(and)f Fj(F)13 b Fr(\()p Fj(y)s Fr(\))31 b(=)g Fj(y)3419 3825 y Fu(1)3474 3811 y Fj(:)15 b(:)g(:)h(y)3640 3825 y Fi(k)3685 3811 y Fr(.)50 3957 y(No)m(w)31 b(let)f Fk(W)485 3909 y Fu(\()p Fi(m)p Fu(\))478 3984 y Fi(\014)s(;N)643 3957 y Fr(b)s(e)g(an)m(y)h(pro)s (cess)f(of)g(the)h(form)f(\(1.24\).)43 b(Then)226 4061 y Fe(Z)332 4184 y Fk(W)429 4137 y Fu(\()p Fi(m)p Fu(\))422 4212 y Fi(\014)s(;N)557 4184 y Fr(\()p Fj(dw)r Fr(\))p Fj(w)808 4144 y Fi(i)834 4153 y Fn(1)806 4209 y Fu(1)891 4184 y Fj(:)15 b(:)g(:)h(w)1079 4144 y Fi(i)1105 4154 y Fm(l)1077 4212 y Fi(l)1151 4184 y Fj(:)f(:)g(:)i(w)1340 4147 y Fi(i)1366 4156 y Fm(m)1338 4207 y Fi(m)236 4423 y Fr(=)332 4337 y Fe(X)376 4528 y Fi(\033)479 4322 y Fe(\020)626 4337 y(X)548 4536 y Fi(\033)592 4514 y Ff(0)590 4558 y Fn(2)626 4536 y Fi(;:::)n(;\033)778 4517 y Ff(0)776 4554 y Fm(m)851 4423 y Fj(\026)906 4437 y Fi(\014)s(;N)1041 4423 y Fr(\()p Fj(\033)1128 4437 y Fu(1)1169 4423 y Fj(\033)1224 4386 y Fh(0)1221 4446 y Fu(2)1262 4423 y Fj(;)e(:)g(:)g(:)h(;)f(\033) 1518 4386 y Fh(0)1515 4446 y Fi(m)1585 4423 y Fr(\))1620 4322 y Fe(\021)1675 4341 y Fi(i)1701 4350 y Fn(1)1755 4423 y Fj(:)g(:)g(:)1877 4322 y Fe(\020)2059 4337 y(X)1946 4536 y Fi(\033)1990 4514 y Ff(0)1988 4560 y Fm(l)p Fn(+1)2092 4536 y Fi(;:::)o(;\033)2245 4517 y Ff(0)2243 4554 y Fm(m)2318 4423 y Fj(\026)2373 4437 y Fi(\014)s(;N)2508 4423 y Fr(\()p Fj(\033)2595 4437 y Fu(1)2651 4423 y Fj(:)g(:)g(:)h(\033)2824 4437 y Fi(l)2852 4423 y Fj(\033)2907 4386 y Fh(0)2904 4446 y Fi(l)p Fu(+1)3039 4423 y Fj(:)f(:)g(:)h(\033)3215 4386 y Fh(0)3212 4446 y Fi(m)3281 4423 y Fr(\))3316 4322 y Fe(\021)3371 4341 y Fi(i)3397 4351 y Fm(l)3443 4423 y Fk(\001)f(\001)g(\001)559 4737 y(\001)g(\001)g(\001)681 4636 y Fe(\020)735 4737 y Fj(\026)790 4751 y Fi(\014)s(;N)925 4737 y Fr(\()p Fj(\033)1012 4751 y Fu(1)1068 4737 y Fj(:)g(:)g(:)h (\033)1241 4751 y Fi(m)1310 4737 y Fr(\))1345 4636 y Fe(\021)1400 4655 y Fi(i)1426 4664 y Fm(m)236 4937 y Fr(=)25 b Fj(\026)387 4890 y Fh(\012)p Fu(\()p Fi(i)498 4899 y Fn(1)533 4890 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(i)736 4899 y Fm(m)796 4890 y Fu(\))387 4965 y Fi(\014)s(;N)814 4836 y Fe(\020)868 4937 y Fr(1)-23 b(I)923 4972 y Fh(f)p Fi(\033)1004 4950 y Fn(1)1002 4994 y(1)1041 4972 y Fu(=)p Fh(\001\001\001)n Fu(=)p Fi(\033)1261 4936 y Fm(i)1285 4950 y Fn(1)1321 4936 y(+)p Ff(\001\001\001)o Fn(+)p Fm(i)1501 4945 y(m)1259 4994 y Fn(1)1566 4972 y Fh(g)1623 4937 y Fj(:)15 b(:)g(:)822 5110 y Fk(\001)g(\001)g(\001)i Fr(1)-23 b(I)999 5154 y Fh(f)p Fi(\033)1080 5109 y Fm(i)1104 5123 y Fn(1)1141 5109 y(+)p Ff(\001\001\001)o Fn(+)p Fm(i)1321 5125 y(l)p Ff(\000)p Fn(1)1427 5109 y(+1)1078 5176 y(1)1510 5154 y Fi(:::)n(\033)1618 5109 y Fm(i)1642 5123 y Fn(1)1678 5109 y(+)p Ff(\001\001\001)p Fn(+)p Fm(i)1859 5125 y(l)p Ff(\000)p Fn(1)1965 5109 y(+1)1616 5179 y Fm(l)2047 5154 y Fu(=)p Fh(\001\001\001)o Fu(=)p Fi(\033)2268 5119 y Fm(i)2292 5133 y Fn(1)2328 5119 y(+)p Ff(\001\001\001)o Fn(+)p Fm(i)2508 5128 y(m)2266 5176 y Fn(1)2573 5154 y Fi(:::)o(\033)2682 5119 y Fm(i)2706 5133 y Fn(1)2742 5119 y(+)p Ff(\001\001\001)o Fn(+)p Fm(i)2922 5128 y(m)2680 5179 y(l)2987 5154 y Fh(g)838 5319 y Fk(\001)15 b(\001)g(\001)h Fr(1)-23 b(I)1014 5363 y Fh(f)p Fi(\033)1095 5318 y Fm(i)1119 5332 y Fn(1)1156 5318 y(+)p Ff(\001\001\001)o Fn(+)p Fm(i)1336 5333 y(m)p Ff(\000)p Fn(1)1476 5318 y(+1)1093 5385 y(1)1559 5363 y Fi(:::)n(\033)1667 5318 y Fm(i)1691 5332 y Fn(1)1728 5318 y(+)p Ff(\001\001\001)o Fn(+)p Fm(i)1908 5333 y(m)p Ff(\000)p Fn(1)2048 5318 y(+1)1665 5375 y Fm(m)2131 5363 y Fu(=)p Fh(\001\001\001)n Fu(=)p Fi(\033)2351 5327 y Fm(i)2375 5341 y Fn(1)2411 5327 y(+)p Ff(\001\001\001)o Fn(+)p Fm(i)2591 5336 y(m)2349 5385 y Fn(1)2656 5363 y Fi(:::)o(\033)2765 5327 y Fm(i)2789 5341 y Fn(1)2825 5327 y(+)p Ff(\001\001\001)o Fn(+)p Fm(i)3005 5336 y(m)2763 5375 y(m)3070 5363 y Fh(g)3112 5219 y Fe(\021)3477 5462 y Fr(\(1)p Fj(:)p Fr(36\))p eop %%Page: 13 13 13 12 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(13)50 513 y Fr(where)30 b Fj(i)344 527 y Fi(m)439 513 y Fk(\025)25 b Fr(1)31 b(\(otherwise,)f(the)h(expression)e(w)m(ould)h(b)s(e)g (in\014nite\).)39 b(Clearly)-8 b(,)30 b(taking)g(pro)s(ducts)f(of)i (suc)m(h)50 658 y(terms,)f(the)f(resulting)f(ob)5 b(jects)30 b(are)g(precisely)e(exp)s(ectations)i(of)g(ev)m(en)m(ts)g(of)g(the)g (form)f Fk(A)3137 672 y Fi(J)p 3137 687 47 4 v 3137 705 51 4 v 3187 658 a Fr(.)40 b(This)28 b(pro)m(v)m(es)50 804 y(Theorem)i(1.9.)42 b Fk(})159 1022 y Fr(The)31 b(recursions)g(on)g (the)i Fj(A)1129 1036 y Fi(J)p 1129 1051 47 4 v 1129 1068 51 4 v 1211 1022 a Fr(ma)m(y)g(lo)s(ok)f(forbidding,)d(but)i(if)g (only)g(limited)f(information)g(is)h(sough)m(t,)50 1167 y(they)25 b(can)g(pro)m(vide)g(quic)m(kly)e(a)j(great)g(amoun)m(t)g(of) f(information.)37 b(As)25 b(a)h(\014rst)e(trivial)f(corollary)-8 b(,)26 b(they)f(yield)50 1313 y(a)31 b(v)m(ery)f(simple)f(pro)s(of)g (of)i(a)g(result)e(of)h(Ruelle)f([Ru]:)50 1531 y Fd(Theorem)40 b(1.10:)103 b Fq(A)n(ny)34 b(of)h(the)g(one-dimensional)i(pr)-5 b(o)g(c)g(esses)37 b Fk(R)2481 1498 y Fi(m)2481 1557 y(\014)2585 1531 y Fq(has)e(the)h(same)f(distribution)h(as)50 1676 y(R)n(uel)5 b(le's)32 b(pr)-5 b(o)g(c)g(ess)35 b(with)e(p)-5 b(ar)g(ameter)36 b Fg(E)10 b Fj(f)1427 1690 y Fi(\014)1482 1676 y Fr(\()p Fj(q)1558 1690 y Fi(m)1627 1676 y Fr(\))p Fq(,)33 b Fj(m)25 b Fr(=)g(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\))p Fq(.)50 1895 y Fa(Pro)s(of:)62 b Fr(Note)38 b(that)g(if)d(w)m(e)j(are)f(in)m (terested)g(only)f(in)f(the)i(one-dimensional)e(pro)s(cesses)i Fk(R)3268 1862 y Fi(m)3268 1920 y(\014)3337 1895 y Fr(,)h(w)m(e)g(only) 50 2040 y(need)30 b(to)h(con)m(trol)g(momen)m(ts)g(of)f(the)h(form)115 2137 y Fe(Z)222 2261 y Fk(R)299 2214 y Fu(\()p Fi(m)p Fu(\))299 2288 y Fi(\014)s(;N)434 2261 y Fr(\()p Fj(dw)r Fr(\))p Fj(w)685 2223 y Fi(r)752 2261 y Fr(=)848 2174 y Fe(X)892 2365 y Fi(\033)995 2160 y Fe(\020)1140 2174 y(X)1019 2373 y Fi(\033)1063 2351 y Ff(0)1061 2396 y Fm(m)p Fn(+1)1199 2373 y Fi(;:::)n(;\033)1351 2355 y Ff(0)1349 2391 y Fm(n)1303 2261 y Fj(\026)1358 2275 y Fi(\014)s(;N)1492 2261 y Fr(\()p Fj(\033)1579 2275 y Fu(1)1636 2261 y Fj(:)15 b(:)g(:)h(\033)1809 2275 y Fi(m)1878 2261 y Fj(\033)1933 2223 y Fh(0)1930 2283 y Fi(m)p Fu(+1)2106 2261 y Fj(:)f(:)g(:)h(\033)2282 2223 y Fh(0)2279 2283 y Fi(n)2329 2261 y Fr(\))2364 2160 y Fe(\021)2418 2179 y Fi(r)2484 2261 y Fr(=)25 b Fj(\026)2635 2222 y Fh(\012)p Fi(r)2635 2288 y(\014)s(;N)2770 2187 y Fe(\000)2811 2261 y Fr(1)-23 b(I)2867 2281 y Fh(f)p Fi(\033)2948 2259 y Fn(1)2946 2303 y(1)2984 2281 y Fi(:::)o(\033)3093 2262 y Fn(1)3091 2299 y Fm(m)3151 2281 y Fu(=)p Fh(\001\001\001)o Fu(=)p Fi(\033)3372 2259 y Fm(r)3370 2303 y Fn(1)3409 2281 y Fi(:::)n(\033)3517 2262 y Fm(r)3515 2299 y(m)3576 2281 y Fh(g)3618 2187 y Fe(\001)3477 2490 y Fr(\(1)p Fj(:)p Fr(37\))50 2635 y(Th)m(us)33 b(w)m(e)j(ha)m(v)m(e)f(the)g(same)g (t)m(yp)s(e)g(of)g(ob)5 b(jects)36 b(as)f(in)e(the)i(REM,)g(and)f (since)g(the)h(relations)f(\(1.30\))j(ha)m(v)m(e)50 2781 y(the)30 b(same)h(form)f(for)g(all)f Fj(m)p Fr(,)i(the)f(claimed)g (result)f(follo)m(ws)h(from)f(the)i(preceding)e(theorem.)41 b Fk(})159 2999 y Fr(T)-8 b(o)32 b(see)g(ho)m(w)g(simple)d(quan)m (tities)i(of)h(in)m(terest)g(can)g(b)s(e)f(computed)g(v)m(ery)h (explicitly)-8 b(,)30 b(consider)h(e.g.)45 b(in)50 3144 y(the)30 b(REM)1235 3290 y Fj(A)1303 3304 y Fi(n)1378 3290 y Fk(\021)48 b Fr(lim)1474 3349 y Fi(N)7 b Fh("1)1663 3290 y Fg(E)j Fj(\026)1772 3251 y Fh(\012)p Fi(n)1772 3317 y(\014)t(;N)1913 3290 y Fr(\(1)-23 b(I)2004 3306 y Fi(\033)2048 3288 y Fn(1)2084 3306 y Fu(=)p Fi(\033)2184 3288 y Fn(2)2220 3306 y Fu(=)p Fh(\001\001\001)n Fu(=)p Fi(\033)2440 3288 y Fm(n)2490 3290 y Fr(\))952 b(\(1)p Fj(:)p Fr(38\))50 3502 y(then)30 b(\(1.30\))j(with)c Fj(h)c Fr(=)g(1)-23 b(I)956 3518 y Fi(\033)1000 3500 y Fn(1)1036 3518 y Fu(=)p Fi(\033)1136 3500 y Fn(2)1171 3518 y Fu(=)p Fh(\001\001\001)o Fu(=)p Fi(\033)1392 3500 y Fm(n)1472 3502 y Fr(pro)m(vides)29 b(the)h(recursion)916 3754 y Fj(A)984 3768 y Fi(n)p Fu(+1)1151 3754 y Fr(=)1258 3692 y Fj(n)20 b Fk(\000)g Fr(1)p 1258 3733 212 4 v 1336 3816 a Fj(n)1480 3754 y(A)1548 3768 y Fi(n)1618 3754 y Fr(+)1724 3692 y(1)p 1720 3733 55 4 v 1720 3816 a Fj(n)1785 3754 y(A)1853 3768 y Fi(n)1903 3754 y Fj(A)1971 3768 y Fu(2)2037 3754 y Fr(=)25 b Fj(A)2201 3768 y Fi(n)2266 3626 y Fe(\022)2333 3754 y Fr(1)20 b Fk(\000)2500 3692 y Fr(1)h Fk(\000)f Fj(A)2725 3706 y Fu(2)p 2500 3733 266 4 v 2606 3816 a Fj(n)2777 3626 y Fe(\023)1151 4027 y Fr(=)1291 3913 y Fi(n)1255 3940 y Fe(Y)1247 4136 y Fi(k)r Fu(=2)1394 3899 y Fe(\022)1461 4027 y Fr(1)h Fk(\000)1629 3965 y Fr(1)f Fk(\000)g Fj(A)1853 3979 y Fu(2)p 1629 4006 V 1736 4089 a Fj(k)1905 3899 y Fe(\023)1987 4027 y Fj(A)2055 4041 y Fu(2)2122 4027 y Fr(=)2315 3965 y(\000\()p Fj(n)g Fr(+)g Fj(A)2641 3979 y Fu(2)2682 3965 y Fr(\))p 2228 4006 576 4 v 2228 4089 a(\000\()p Fj(n)g Fr(+)g(1\)\000\()p Fj(A)2726 4103 y Fu(2)2768 4089 y Fr(\))3477 3906 y(\(1)p Fj(:)p Fr(39\))50 4304 y(But)1080 4450 y(lim)1056 4509 y Fi(N)7 b Fh("1)1246 4450 y Fj(\026)1301 4411 y Fh(\012)p Fu(2)1301 4477 y Fi(\014)s(;N)1436 4450 y Fr(\(1)-23 b(I)1526 4466 y Fi(\033)1570 4448 y Fn(1)1606 4466 y Fu(=)p Fi(\033)1706 4448 y Fn(2)1742 4466 y Fu(=)p Fh(\001\001\001)o Fu(=)p Fi(\033)1963 4448 y Fm(n)2012 4450 y Fr(\))26 b(=)2168 4326 y Fe(Z)2275 4450 y Fk(R)2352 4464 y Fi(\014)2401 4450 y Fr(\()p Fj(dw)r Fr(\))p Fj(w)2652 4412 y Fi(n)3477 4450 y Fr(\(1)p Fj(:)p Fr(40\))50 4662 y(so)k(that)h(\(1.36\))i (implies)28 b(a)i(form)m(ula)g(for)g(the)h(mean)f(of)h(the)f Fj(n)p Fr(-th)g(momen)m(ts)h(of)g Fk(R)2906 4676 y Fi(\014)2955 4662 y 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b(arious)28 b(results)g(claimed)g(ab)s(o)m(v) m(e:)41 b(In)28 b(Section)50 658 y(3)35 b(w)m(e)h(pro)m(v)m(e)g (Theorem)f(1.3)i(and)d(1.5.)57 b(Section)35 b(4)h(con)m(tains)f(the)g (pro)s(of)g(of)g(Theorem)g(1.6.)57 b(Finally)-8 b(,)35 b(in)50 804 y(Section)23 b(5,)j(w)m(e)f(pro)m(v)m(e)f(Prop)s(osition)e (3.1)j(and)e(the)h(Ghirlanda-Guerra)e(iden)m(tities)g(stated)j(in)e (Prop)s(osition)50 949 y(1.8)50 1167 y Fs(Ac)m(kno)m(wledgemen)m(ts:)77 b Fr(A.B.)45 b(thanks)e(F)-8 b(rancis)43 b(Comets)h(and)f(the)h(Univ)m (ersit)m(\023)-43 b(e)44 b(P)m(aris)f(7)h(for)g(their)50 1313 y(hospitalit)m(y)29 b(during)f(the)j(\014nal)e(p)s(erio)s(d)f(of)i (the)h(writing)d(of)j(this)e(pap)s(er.)50 1956 y Fv(2.)55 b(Pro)s(of)41 b(Theorems)h(1.3)g(and)f(1.5.)159 2174 y Fr(In)32 b(this)f(section)i(w)m(e)g(pro)m(v)m(e)g(the)f(con)m(v)m (ergence)j(results)d(for)g(the)g(partition)g(function)f(\(Theorem)h (1.3\))50 2319 y(and)e(that)h(of)f(the)h(Gibbs)d(measures)j(to)g (Ruelle's)e(pro)s(cesses)h Fk(W)2279 2286 y Fi(m)2272 2345 y(\014)2379 2319 y Fr(\(Theorem)g(1.5\).)50 2537 y Fq(Pr)-5 b(o)g(of)42 b(of)f(The)-5 b(or)g(em)42 b(1.3.)67 b Fr(It)40 b(is)e(based)h(on)f(Lemmas)i(2.1)g(and)e(2.2)i(b)s(elo)m(w.) 66 b(Lemma)40 b(2.1)g(pro)m(v)m(es)g(the)50 2683 y(theorem)33 b(in)e(the)i(case)h Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))30 b(=)e(0.)48 b(The)32 b(pro)s(of)g(in)f(general)i(case)h(pro)s(ceeds)e (b)m(y)g(induction)f(on)h Fj(l)r Fr(\()p Fj(\014)5 b Fr(\).)49 b(W)-8 b(e)50 2828 y(represen)m(t)30 b(the)h(normalised)d (partition)h(function)g(as)i(the)f(sum)g(of)g(t)m(w)m(o)i(terms)65 3407 y Fj(e)107 3271 y Fe(P)203 3294 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))203 3367 y Fm(j)s Fn(=1)330 3259 y Fe(\000)371 3333 y Fh(\000)p Fi(\014)s(N)537 3274 y Fk(p)p 613 3274 277 4 v 59 x Fu(2)r(\026)-38 b Fi(a)688 3342 y Fm(j)735 3333 y Fu(ln)20 b(\026)-43 b Fi(\013)855 3342 y Fm(j)889 3333 y Fu(+)p Fi(\014)s(\015)1027 3342 y Fm(j)1062 3333 y Fu([ln\()p Fi(N)19 b Fu(ln)i(\026)-44 b Fi(\013)1368 3342 y Fm(j)1402 3333 y 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4216 y Fh(p)p 3016 4216 100 4 v 3016 4258 a Fi(a)3055 4267 y Fm(m)3115 4258 y Fi(X)3175 4267 y Fm(\033)3212 4281 y Fn(1)3247 4267 y Fm(:::\033)3347 4276 y(m)3437 4295 y Fk(\000)20 b Fj(C)7 b Fr(\()p Fj(\014)e Fr(\))3726 4195 y Fe(\021)90 4565 y Fr(=)25 b Fj(I)233 4528 y Fi(N)226 4588 y Fu(1)322 4565 y Fr(+)20 b Fj(I)460 4528 y Fi(N)453 4588 y Fu(2)50 5098 y Fr(with)29 b(the)h(same)h(notation)1030 5075 y(\026)1004 5098 y Fj(X)1086 5034 y Fi(\033)1128 5043 y Fn(1)1164 5034 y Fi(:::)o(\033)1271 5043 y Fm(J)1306 5059 y(l)p Ff(\000)p Fn(1)1079 5109 y Fi(\033)1121 5118 y Fm(J)1156 5134 y(l)p Ff(\000)p Fn(1)1263 5118 y(+1)1345 5109 y Fi(:::)o(\033)1452 5118 y Fm(J)1487 5134 y(l)1553 5098 y Fr(as)f(in)f(\(1.8\))j(and)e(sho)m(w)g(that)g Fj(I)2632 5065 y Fi(N)2625 5121 y Fu(1)2731 5098 y Fr(con)m(v)m(erges)j(in)28 b(distribution)50 5244 y(to)j(the)f(in)m(tegral)h(\(1.15\))h(and)e (that)h Fj(I)1332 5211 y Fi(N)1325 5266 y Fu(2)1431 5244 y Fr(con)m(v)m(erges)i(to)e(zero.)159 5462 y(F)-8 b(or)36 b(an)m(y)g Fj(x)e Fk(2)f Fg(R)44 b Fr(the)36 b(di\013erence)e(b)s(et)m (w)m(een)j Fj(I)1762 5429 y Fi(N)1755 5484 y Fu(1)1866 5462 y Fr(and)e(the)g(in)m(tegral)h Fj(I)42 b Fr(of)36 b(\(1.15\))h(can)f(b)s(e)f(written)f(as)p eop %%Page: 15 15 15 14 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(15)50 513 y Fr(follo)m(ws:)165 692 y Fj(I)212 654 y Fi(N)205 714 y Fu(1)301 692 y Fk(\000)20 b Fj(I)95 938 y Fr(=)184 817 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))181 852 y Fe(X)180 1048 y Fi(k)r Fu(=1)328 837 y Fe(h)461 852 y(X)386 1043 y Fi(\033)428 1052 y Fn(1)463 1043 y Fi(;:::)o(\033)592 1052 y Fm(J)627 1066 y Fn(1)682 938 y Fj(e)724 890 y Fi(\014)s(\015)806 899 y Fn(1)843 890 y Fi(u)886 861 y Ff(\000)p Fn(1)886 914 y(ln)d(\026)-38 b Fm(\013)991 928 y Fn(1)1026 914 y Fm(;N)1105 890 y Fu(\()1154 874 y(\026)1133 890 y Fi(X)1193 899 y Fm(\033)1230 913 y Fn(1)1266 899 y Ff(\001\001\001)p Fm(\033)1366 915 y(J)1401 929 y Fn(1)1445 890 y Fu(\))1493 938 y Fk(\001)15 b(\001)g(\001)1782 852 y Fe(X)1615 1043 y Fi(\033)1657 1052 y Fm(J)1692 1068 y(k)q Ff(\000)p Fn(1)1812 1052 y(+1)1894 1043 y Fi(:::)o(\033)2001 1052 y Fm(J)2036 1068 y(k)2095 938 y Fj(e)2137 877 y Fi(\014)s(\015)2219 887 y Fm(k)2260 877 y Fi(u)2303 848 y Ff(\000)p Fn(1)2303 901 y(ln)j(\026)-38 b Fm(\013)2409 917 y(k)2448 901 y(;N)2527 877 y Fu(\()2576 861 y(\026)2555 877 y Fi(X)2620 816 y Fm(\033)2657 830 y Fn(1)2693 816 y Fm(:::)o(\033)2792 832 y(J)2827 848 y(k)q Ff(\000)p Fn(1)2615 889 y Fm(\033)2652 905 y(J)2687 921 y(k)q Ff(\000)p Fn(1)2807 905 y(+1)2890 889 y Fm(:::)o(\033)2989 905 y(J)3024 921 y(k)3073 877 y Fu(\))367 1198 y Fk(\002)20 b Fr(1)-23 b(I)514 1246 y Fh(f8)p Fi(i)p Fu(=1)p Fi(;:::)11 b(;k)r Fh(\000)p Fu(1:)p Fi(\015)1024 1255 y Fn(1)1058 1246 y Fi(u)1101 1216 y Ff(\000)p Fn(1)1101 1270 y(ln)17 b(\026)-38 b Fm(\013)1206 1284 y Fn(1)1241 1270 y Fm(;N)1320 1246 y Fu(\()1369 1230 y(\026)1348 1246 y Fi(X)1408 1255 y Fm(\033)1445 1269 y Fn(1)1481 1255 y Ff(\001\001\001)p Fm(\033)1581 1270 y(J)1616 1284 y Fn(1)1660 1246 y Fu(\)+)p Fh(\001\001\001)o Fu(+)p Fi(\015)1903 1255 y Fm(i)1932 1246 y Fi(u)1975 1216 y Ff(\000)p Fn(1)1975 1270 y(ln)17 b(\026)-37 b Fm(\013)2081 1285 y(i)2109 1270 y(;N)2188 1246 y Fu(\()2237 1230 y(\026)2216 1246 y Fi(X)2281 1186 y Fm(\033)2318 1200 y Fn(1)2354 1186 y Fm(:::)o(\033)2453 1202 y(J)2488 1217 y(i)p Ff(\000)p Fn(1)2276 1261 y Fm(\033)2313 1276 y(J)2348 1291 y(i)p Ff(\000)p Fn(1)2457 1276 y(+1)2540 1261 y Ff(\001\001\001)o Fm(\033)2639 1276 y(J)2674 1291 y(i)2712 1246 y Fu(\))p Fh(\024)p Fu(\()p Fi(\015)2863 1255 y Fn(1)2899 1246 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(\015)3114 1255 y Fm(i)3143 1246 y Fu(\))p Fi(x)p Fh(g)367 1412 y Fk(\002)20 b Fr(1)-23 b(I)514 1461 y Fh(f)p Fi(\015)589 1470 y Fn(1)625 1461 y Fi(u)668 1432 y Ff(\000)p Fn(1)668 1486 y(ln)17 b(\026)-38 b Fm(\013)773 1500 y Fn(1)808 1486 y Fm(;N)887 1461 y Fu(\()936 1445 y(\026)915 1461 y Fi(X)975 1470 y Fm(\033)1012 1484 y Fn(1)1048 1470 y Ff(\001\001\001)o Fm(\033)1147 1486 y(J)1182 1500 y Fn(1)1227 1461 y Fu(\)+)p Fh(\001\001\001)o Fu(+)p Fi(\015)1470 1471 y Fm(k)1510 1461 y Fi(u)1553 1432 y Ff(\000)p Fn(1)1553 1486 y(ln)17 b(\026)-38 b Fm(\013)1658 1501 y(k)1698 1486 y(;N)1777 1461 y Fu(\()1825 1445 y(\026)1805 1461 y Fi(X)1870 1401 y Fm(\033)1907 1415 y Fn(1)1942 1401 y Fm(:::\033)2042 1416 y(J)2077 1432 y(k)q Ff(\000)p Fn(1)1865 1476 y Fm(\033)1902 1492 y(J)1937 1508 y(k)q Ff(\000)p Fn(1)2057 1492 y(+1)2139 1476 y Ff(\001\001\001)p Fm(\033)2239 1492 y(J)2274 1508 y(k)2323 1461 y Fu(\))p Fi(>)p Fu(\()p Fi(\015)2474 1470 y Fn(1)2510 1461 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(\015)2725 1471 y Fm(k)2764 1461 y Fu(\))p Fi(x)p Fh(g)185 1706 y Fk(\002)20 b Fj(C)7 b Fr(\()p Fj(\014)e Fr(\))545 1619 y Fe(X)369 1810 y Fi(\033)411 1819 y Fm(J)446 1835 y(k)486 1819 y Fn(+1)568 1810 y Fi(;:::)n(\033)696 1819 y Fm(J)731 1835 y(k)q Fn(+1)808 1706 y Fj(e)850 1642 y Fi(\014)s(\015)932 1652 y Fm(k)q Fn(+1)1050 1642 y Fi(u)1093 1612 y Ff(\000)p Fn(1)1093 1666 y(ln)18 b(\026)-38 b Fm(\013)1199 1680 y Fn(1)1234 1666 y Fm(;N)1313 1642 y Fu(\()1362 1626 y(\026)1341 1642 y Fi(X)1406 1589 y Fm(\033)1443 1603 y Fn(1)1479 1589 y Fm(:::)o(\033)1578 1604 y(J)1613 1620 y(k)1401 1657 y(\033)1438 1672 y(J)1473 1688 y(k)1513 1672 y Fn(+1)1595 1657 y Ff(\001\001\001)p Fm(\033)1695 1672 y(J)1730 1688 y(k)q Fn(+1)1856 1642 y Fu(\))1905 1706 y Fk(\001)15 b(\001)g(\001)2085 1619 y Fe(X)1844 1810 y Fi(\033)1886 1819 y Fm(J)1921 1838 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)2115 1819 y(+1)2197 1810 y Fi(:::)o(\033)2304 1819 y Fm(J)2339 1838 y(l)p Fn(\()p Fm(\014)r Fn(\))2337 1706 y Fj(e)2379 1638 y Fi(\014)s(\015)2461 1650 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2576 1638 y Fi(u)2619 1609 y Ff(\000)p Fn(1)2619 1663 y(ln)i(\026)-38 b Fm(\013)2724 1681 y(l)p Fn(\()p Fm(\014)r Fn(\))2838 1663 y Fm(;N)2917 1638 y Fu(\()2965 1622 y(\026)2945 1638 y Fi(X)3010 1571 y Fm(\033)3047 1585 y Fn(1)3082 1571 y Fm(:::\033)3182 1587 y(J)3217 1605 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)3005 1651 y Fm(\033)3042 1666 y(J)3077 1684 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)3271 1666 y(+1)3353 1651 y Fm(:::\033)3453 1666 y(J)3488 1684 y(l)p Fn(\()p Fm(\014)r Fn(\))3611 1638 y Fu(\))367 2018 y Fk(\000)20 b Fj(C)7 b Fr(\()p Fj(\014)e Fr(\))671 1894 y Fe(Z)1063 2037 y Fn(\()p Fm(x)1124 2051 y Fn(1)1159 2037 y Fm(;:::)11 b(;x)1311 2056 y(l)p Fn(\()p Fm(\014)r Fn(\))1424 2037 y(\))p Ff(2)p Fb(R)1521 2021 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))1639 2037 y Fm(;)744 2112 y Ff(8)p Fm(i)p Fn(=1)p Fm(;:::)f(;k)q Ff(\000)p Fn(1:)p Fm(\015)1165 2126 y Fn(1)1201 2112 y Fm(x)1237 2126 y Fn(1)1272 2112 y(+)p Ff(\001\001\001)p Fn(+)p Fm(\015)1463 2127 y(i)1491 2112 y(x)1527 2127 y(i)1556 2112 y Ff(\024)p Fn(\()p Fm(\015)1664 2126 y Fn(1)1699 2112 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)1889 2127 y(i)1918 2112 y Fn(\))p Fm(x)921 2172 y(\015)955 2186 y Fn(1)991 2172 y Fm(x)1027 2186 y Fn(1)1062 2172 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)1252 2188 y(k)1292 2172 y(x)1328 2188 y(k)1368 2172 y(>)p Fn(\()p Fm(\015)1476 2186 y Fn(1)1511 2172 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)1701 2188 y(k)1741 2172 y Fn(\))p Fm(x)2020 2018 y Fj(e)2062 1981 y Fi(\014)s(\015)2144 1990 y Fn(1)2180 1981 y Fi(x)2221 1990 y Fn(1)2257 1981 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(\014)s(\015)2516 1993 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2630 1981 y Fi(x)2671 1993 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2790 2018 y Fk(P)2860 1981 y Fu(\()p Fi(l)p Fu(\()p Fi(\014)s Fu(\)\))3047 2018 y Fr(\()p Fj(dx)3181 2032 y Fu(1)3237 2018 y Fj(:)15 b(:)g(:)h(dx)3457 2035 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))3587 2018 y Fr(\))3622 1917 y Fe(i)185 2378 y Fr(+)k Fj(C)7 b Fr(\()p Fj(\014)e Fr(\))535 2292 y Fe(X)459 2482 y Fi(\033)501 2491 y Fn(1)537 2482 y Fi(;:::)o(\033)666 2491 y Fm(J)701 2505 y Fn(1)756 2378 y Fj(e)798 2330 y Fi(\014)s(\015)880 2339 y Fn(1)916 2330 y Fi(u)959 2300 y Ff(\000)p Fn(1)959 2354 y(ln)18 b(\026)-38 b Fm(\013)1065 2368 y Fn(1)1100 2354 y Fm(;N)1179 2330 y Fu(\()1228 2314 y(\026)1207 2330 y Fi(X)1267 2339 y Fm(\033)1304 2353 y Fn(1)1340 2339 y Ff(\001\001\001)o Fm(\033)1439 2354 y(J)1474 2368 y Fn(1)1519 2330 y Fu(\))1567 2378 y Fk(\001)15 b(\001)g(\001)1839 2292 y Fe(X)1597 2482 y Fi(\033)1639 2491 y Fm(J)1674 2510 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)1869 2491 y(+1)1951 2482 y Fi(:::)o(\033)2058 2491 y Fm(J)2093 2510 y(l)p Fn(\()p Fm(\014)r Fn(\))2227 2378 y Fj(e)2269 2311 y Fi(\014)s(\015)2351 2323 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2466 2311 y Fi(u)2509 2281 y Ff(\000)p Fn(1)2509 2335 y(ln)i(\026)-38 b Fm(\013)2614 2353 y(l)p Fn(\()p Fm(\014)r Fn(\))2727 2335 y Fm(;N)2807 2311 y Fu(\()2855 2295 y(\026)2835 2311 y Fi(X)2900 2244 y Fm(\033)2937 2258 y Fn(1)2972 2244 y Fm(:::\033)3072 2259 y(J)3107 2277 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)2895 2323 y Fm(\033)2932 2338 y(J)2967 2357 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)3161 2338 y(+1)3243 2323 y Fm(:::\033)3343 2338 y(J)3378 2357 y(l)p Fn(\()p Fm(\014)r Fn(\))3501 2311 y Fu(\))367 2644 y Fk(\002)20 b Fr(1)-23 b(I)514 2692 y Fh(f8)p Fi(i)p Fu(=1)p Fi(;:::)11 b(;k)r Fu(:)p Fi(\015)931 2701 y Fn(1)965 2692 y Fi(u)1008 2662 y Ff(\000)p Fn(1)1008 2716 y(ln)17 b(\026)-38 b Fm(\013)1113 2730 y Fn(1)1148 2716 y Fm(;N)1227 2692 y Fu(\()1276 2676 y(\026)1255 2692 y Fi(X)1315 2701 y Fm(\033)1352 2715 y Fn(1)1388 2701 y Ff(\001\001\001)o Fm(\033)1487 2716 y(J)1522 2730 y Fn(1)1567 2692 y Fu(\)+)p Fh(\001\001\001)o Fu(+)p Fi(\015)1810 2701 y Fm(i)1839 2692 y Fi(u)1882 2662 y Ff(\000)p Fn(1)1882 2716 y(ln)17 b(\026)-37 b Fm(\013)1988 2731 y(i)2016 2716 y(;N)2095 2692 y Fu(\()2144 2676 y(\026)2123 2692 y Fi(X)2188 2632 y Fm(\033)2225 2646 y Fn(1)2261 2632 y Fm(:::)o(\033)2360 2648 y(J)2395 2663 y(i)p Ff(\000)p Fn(1)2183 2707 y Fm(\033)2220 2722 y(J)2255 2737 y(i)p Ff(\000)p Fn(1)2364 2722 y(+1)2447 2707 y Ff(\001\001\001)o Fm(\033)2546 2722 y(J)2581 2737 y(i)2619 2692 y Fu(\))p Fh(\024)p Fu(\()p Fi(\015)2770 2701 y Fn(1)2806 2692 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(\015)3021 2701 y Fm(i)3050 2692 y Fu(\))p Fi(x)p Fh(g)185 2905 y Fk(\000)20 b Fj(C)7 b Fr(\()p Fj(\014)e Fr(\))489 2781 y Fe(Z)830 2946 y Fn(\()p Fm(x)891 2960 y Fn(1)926 2946 y Fm(;:::)11 b(;x)1078 2965 y(l)p Fn(\()p Fm(\014)r Fn(\))1192 2946 y(\))p Ff(2)p Fb(R)1289 2930 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))1406 2946 y Fm(;)551 3021 y Ff(8)p Fm(i)p Fn(=1)p Fm(;:::)f(;k)q Fn(:)p Fm(\015)892 3035 y Fn(1)928 3021 y Fm(x)964 3035 y Fn(1)1000 3021 y(+)p Ff(\001\001\001)o Fn(+)p Fm(\015)1190 3036 y(i)1218 3021 y(x)1254 3036 y(i)1283 3021 y Ff(\024)p Fn(\()p Fm(\015)1391 3035 y Fn(1)1426 3021 y(+)p Ff(\001\001\001)p Fn(+)p Fm(\015)1617 3036 y(i)1645 3021 y Fn(\))p Fm(x)1736 2905 y Fj(e)1778 2867 y Fi(\014)s(\015)1860 2876 y Fn(1)1897 2867 y Fi(x)1938 2876 y Fn(1)1973 2867 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(\014)s(\015)2232 2879 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2346 2867 y Fi(x)2387 2879 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2506 2905 y Fk(P)2576 2867 y Fu(\()p Fi(l)p Fu(\()p Fi(\014)s Fu(\)\))2763 2905 y Fr(\()p Fj(dx)2897 2919 y Fu(1)2953 2905 y Fj(:)15 b(:)g(:)i(dx)3174 2921 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))3303 2905 y Fr(\))p Fj(:)3523 3113 y Fr(\(2)p Fj(:)p Fr(1\))50 3258 y(Giv)m(en)35 b Fj(\017)g(>)f Fr(0,)j(one)g(can)f(c)m(ho)s(ose)g Fj(x)f(<)f Fr(0)i(su\016cien)m(tly)e (large)i(b)m(y)g(absolute)f(v)-5 b(alue)36 b(suc)m(h)f(that)i(for)e (all)g Fj(N)50 3404 y Fr(starting)g(from)g(some)h Fj(N)10 b Fr(\()p Fj(x)p Fr(\),)37 b(the)f(probabilit)m(y)d(that)j(the)f(last)g (t)m(w)m(o)i(terms)f(of)f(\(2.1\))i(are)f(smaller)e(than)50 3549 y Fj(\017)p Fr(,)40 b(is)d(arbitrarily)f(close)i(to)h(1.)65 b(Namely)-8 b(,)40 b(for)e(the)h(last)f(term)g(this)f(is)g(ob)m(vious)h (from)f(Prop)s(osition)g(1.4)50 3695 y(\(ii\).)42 b(F)-8 b(or)31 b(the)h(term)f(b)s(efore)f(the)i(last)e(it)h(su\016ces)g(to)g (apply)f(Cheb)m(yshev's)g(inequalit)m(y)g(and)g(to)i(estimate)50 3840 y(its)40 b(exp)s(ectation)g(after.)72 b(Since)39 b Fj(\015)1261 3854 y Fu(1)1344 3840 y Fj(>)i Fk(\001)15 b(\001)g(\001)43 b Fj(>)f(\015)1764 3857 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1933 3840 y Fr(and)e Fj(\014)5 b(\015)2223 3857 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))2394 3840 y Fj(>)42 b Fr(1,)i(then)c(b)m(y)g(Lemma)g(2.2)i(for)e(all)50 4007 y Fj(N)35 b Fk(\025)25 b Fj(N)10 b Fr(\()p Fj(x)p Fr(\))30 b(the)g(exp)s(ectation)g(of)g(this)f(term)h(is)e(smaller)h(than)g Fj(C)7 b Fr(\()p Fj(\014)e Fr(\))p Fj(C)i(e)2558 3955 y Fi(x)2612 3893 y Fe(P)2708 3916 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2708 3989 y Fm(j)s Fn(=1)2822 3955 y Fu(\()p Fi(\014)s(\015)2932 3964 y Fm(j)2967 3955 y Fh(\000)p Fu(1\))3118 4007 y Fk(#)26 b Fr(0)k(as)g Fj(x)25 b Fk(#)h(\0001)p Fr(.)50 4153 y(Let)g(us)g(\014x)f Fj(x)h(<)e Fr(0)j(large)f(enough)g(b)m(y)g (absolute)g(v)-5 b(alue)25 b(and)h(consider)f(the)h Fj(k)s Fr(th)g(term)g(in)f(square)h(brac)m(k)m(ets,)50 4298 y Fj(k)32 b Fr(=)d(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\).)49 b(The)32 b(inequalities)e (\(1.9\))35 b(b)s(eing)c(v)m(eri\014ed)h(for)g(the)h(partition)f (\(1.12\),)j(w)m(e)f(ha)m(v)m(e)g(the)50 4444 y(p)s(oin)m(t)c(pro)s (cesses)h(con)m(v)m(ergence)j(stated)e(in)e(Theorem)h(1.1.)45 b(If)31 b Fj(k)f Fr(=)c Fj(l)r Fr(\()p Fj(\014)5 b Fr(\),)33 b(then)e(b)m(y)g(Prop)s(osition)e(1.4)k(\(i\),)50 4589 y(the)e(in)m(tegral)f(in)g(the)h Fj(l)r Fr(\()p Fj(\014)5 b Fr(\)th)32 b(term)f(is)f(o)m(v)m(er)i(a)f(\014nite)f(n)m(um)m(b)s(er) f(of)j(p)s(oin)m(ts)d(of)i Fk(P)2772 4556 y Fu(\()p Fi(l)p Fu(\()p Fi(\014)s Fu(\)\))2990 4589 y Fr(a.s.)43 b(Then,)30 b(indeed,)50 4735 y(this)37 b(term)i(con)m(v)m(erges)h(to)f(zero)h(in)d (distribution.)61 b(If)38 b(1)i Fk(\024)e Fj(k)k(<)d(l)r Fr(\()p Fj(\014)5 b Fr(\),)41 b(then)d(the)h(un-truncated)f(sums)50 4880 y(o)m(v)m(er)32 b Fj(\033)297 4894 y Fi(J)337 4904 y Fm(k)377 4894 y Fu(+1)474 4880 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(\033)743 4894 y Fi(n)824 4880 y Fr(are)32 b(in)m(v)m(olv)m(ed) f(in)f(the)h Fj(k)s Fr(th)g(term.)44 b(They)31 b(are)h(distributed)c (as)k Fj(I)3036 4847 y Fu(1)3029 4905 y Fi(N)3129 4880 y Fr(for)f(the)g(GREM)50 5025 y(of)h Fj(n)21 b Fk(\000)h Fj(J)374 5039 y Fi(k)451 5025 y Fr(lev)m(els)32 b(where)g(the)h (parameter)g Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))33 b(equals)e Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))23 b Fk(\000)e Fj(k)s Fr(.)47 b(By)33 b(induction)d(eac)m(h)j(of)g(these)g(sums)50 5171 y(con)m(v)m(erges)c(in)c(distribution)f(to)j(the)h(in)m(tegral)e (of)h Fj(e)1779 5138 y Fi(\014)s(\015)1861 5148 y Fm(k)q Fn(+1)1980 5138 y Fi(x)2021 5148 y Fm(k)q Fn(+1)2138 5138 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(\014)s(\015)2397 5150 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2511 5138 y Fi(x)2552 5150 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2698 5171 y Fr(o)m(v)m(er)h(the)g (pro)s(cess)e Fk(P)3426 5138 y Fu(\()p Fi(l)p Fu(\()p Fi(\014)s Fu(\))p Fh(\000)p Fi(k)r Fu(\))50 5316 y Fr(on)40 b Fj(R)256 5283 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))p Fh(\000)p Fi(k)482 5316 y Fr(.)72 b(Moreo)m(v)m(er,)45 b(b)m(y)c(Prop)s(osition)d (1.4)k(\(i\))e(the)h(in)m(tegral)f(in)g(the)h Fj(k)s Fr(th)f(term)h(of)g(\(2.1\))h(is)e(the)50 5462 y(linear)27 b(com)m(bination)i(of)g(a.s.)41 b(\014nite)28 b(n)m(um)m(b)s(er)g(of)h (indep)s(enden)m(t)e(in)m(tegrals)i(distributed)d(as)j(these.)41 b(These)p eop %%Page: 16 16 16 15 bop 50 295 a Fo(16)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fr(argumen)m(ts)j(together)h(imply)c(the)j(con)m(v)m(ergence)i (to)f(zero)f(of)g(the)g Fj(k)s Fr(th)f(term)h(of)g(\(2.1\))h(and)e (complete)i(the)50 658 y(analysis)e(of)h Fj(I)542 625 y Fu(1)535 683 y Fi(N)604 658 y Fr(.)159 876 y(Let)h(us)e(write)h(a)h (similar)d(represen)m(tation)i(for)g Fj(I)1826 843 y Fu(2)1819 901 y Fi(N)1888 876 y Fr(:)65 1113 y Fj(I)112 1075 y Fi(N)105 1135 y Fu(2)110 1359 y Fr(=)200 1238 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))197 1273 y Fe(X)196 1469 y Fi(k)r Fu(=1)419 1273 y Fe(X)344 1464 y Fi(\033)386 1473 y Fn(1)421 1464 y Fi(;:::)o(\033)550 1473 y Fm(J)585 1487 y Fn(1)640 1359 y Fj(e)682 1311 y Fi(\014)s(\015)764 1320 y Fn(1)800 1311 y Fi(u)843 1282 y Ff(\000)p Fn(1)843 1336 y(ln)18 b(\026)-38 b Fm(\013)949 1350 y Fn(1)984 1336 y Fm(;N)1063 1311 y Fu(\()1112 1295 y(\026)1091 1311 y Fi(X)1151 1320 y Fm(\033)1188 1334 y Fn(1)1224 1320 y Ff(\001\001\001)o Fm(\033)1323 1336 y(J)1358 1350 y Fn(1)1403 1311 y Fu(\))1451 1359 y Fk(\001)15 b(\001)g(\001)1740 1273 y Fe(X)1572 1464 y Fi(\033)1614 1473 y Fm(J)1649 1489 y(k)q Ff(\000)p Fn(1)1770 1473 y(+1)1852 1464 y Fi(:::)n(\033)1958 1473 y Fm(J)1993 1489 y(k)2053 1359 y Fj(e)2095 1299 y Fi(\014)s(\015)2177 1309 y Fm(k)2218 1299 y Fi(u)2261 1269 y Ff(\000)p Fn(1)2261 1323 y(ln)i(\026)-38 b Fm(\013)2366 1339 y(k)2406 1323 y(;N)2485 1299 y Fu(\()2533 1282 y(\026)2513 1299 y Fi(X)2578 1238 y Fm(\033)2615 1253 y(J)2650 1267 y Fn(1)2690 1238 y Fm(:::\033)2790 1253 y(J)2825 1269 y(k)q Ff(\000)p Fn(1)2573 1311 y Fm(\033)2610 1326 y(J)2645 1342 y(k)q Ff(\000)p Fn(1)2765 1326 y(+1)2847 1311 y Fm(:::\033)2947 1326 y(J)2982 1342 y(k)3031 1299 y Fu(\))383 1619 y Fk(\002)20 b Fr(1)-23 b(I)529 1667 y Fh(f8)p Fi(i)p Fu(=1)p Fi(;:::)11 b(;k)r Fh(\000)p Fu(1:)p Fi(\015)1039 1676 y Fn(1)1074 1667 y Fi(u)1117 1638 y Ff(\000)p Fn(1)1117 1691 y(ln)17 b(\026)-38 b Fm(\013)1222 1705 y Fn(1)1257 1691 y Fm(;N)1336 1667 y Fu(\()1385 1651 y(\026)1364 1667 y Fi(X)1424 1676 y Fm(\033)1461 1690 y Fn(1)1497 1676 y Ff(\001\001\001)o Fm(\033)1596 1692 y(J)1631 1706 y Fn(1)1676 1667 y Fu(\)+)p Fh(\001\001\001)o Fu(+)p Fi(\015)1919 1676 y Fm(i)1948 1667 y Fi(u)1991 1638 y Ff(\000)p Fn(1)1991 1691 y(ln)17 b(\026)-38 b Fm(\013)2096 1706 y(i)2125 1691 y(;N)2204 1667 y Fu(\()2252 1651 y(\026)2232 1667 y Fi(X)2297 1608 y Fm(\033)2334 1623 y(J)2369 1637 y Fn(1)2409 1608 y Fm(:::\033)2509 1623 y(J)2544 1638 y(i)p Ff(\000)p Fn(1)2292 1682 y Fm(\033)2329 1698 y(J)2364 1713 y(i)p Ff(\000)p Fn(1)2473 1698 y(+1)2555 1682 y Ff(\001\001\001)p Fm(\033)2655 1698 y(J)2690 1713 y(i)2728 1667 y Fu(\))p Fh(\024)p Fu(\()p Fi(\015)2879 1676 y Fn(1)2915 1667 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(\015)3130 1676 y Fm(i)3159 1667 y Fu(\))p Fi(x)p Fh(g)383 1833 y Fk(\002)20 b Fr(1)-23 b(I)529 1883 y Fh(f)p Fi(\015)604 1892 y Fn(1)640 1883 y Fi(u)683 1853 y Ff(\000)p Fn(1)683 1907 y(ln)18 b(\026)-38 b Fm(\013)789 1921 y Fn(1)824 1907 y Fm(;N)903 1883 y Fu(\()952 1866 y(\026)931 1883 y Fi(X)991 1892 y Fm(\033)1028 1906 y Fn(1)1064 1892 y Ff(\001\001\001)o Fm(\033)1163 1907 y(J)1198 1921 y Fn(1)1243 1883 y Fu(\)+)p Fh(\001\001\001)o Fu(+)p Fi(\015)1486 1893 y Fm(k)1526 1883 y Fi(u)1569 1853 y Ff(\000)p Fn(1)1569 1907 y(ln)17 b(\026)-38 b Fm(\013)1674 1923 y(k)1714 1907 y(;N)1793 1883 y Fu(\()1841 1866 y(\026)1821 1883 y Fi(X)1886 1822 y Fm(\033)1923 1837 y(J)1958 1851 y Fn(1)1998 1822 y Fm(:::)o(\033)2097 1837 y(J)2132 1853 y(k)q Ff(\000)p Fn(1)1881 1898 y Fm(\033)1918 1913 y(J)1953 1929 y(k)q Ff(\000)p Fn(1)2073 1913 y(+1)2155 1898 y Ff(\001\001\001)p Fm(\033)2255 1913 y(J)2290 1929 y(k)2339 1883 y Fu(\))p Fi(>)p Fu(\()p Fi(\015)2490 1892 y Fn(1)2526 1883 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(\015)2740 1893 y Fm(k)2780 1883 y Fu(\))p Fi(x)p Fh(g)201 2127 y Fk(\002)469 2040 y Fe(X)292 2231 y Fi(\033)334 2240 y Fm(J)369 2256 y(k)409 2240 y Fn(+1)491 2231 y Fi(;:::)o(\033)620 2240 y Fm(J)655 2256 y(k)q Fn(+1)792 2127 y Fj(e)834 2063 y Fi(\014)s(\015)916 2073 y Fm(k)q Fn(+1)1034 2063 y Fi(u)1077 2034 y Ff(\000)p Fn(1)1077 2087 y(ln)18 b(\026)-38 b Fm(\013)1183 2101 y Fn(1)1218 2087 y Fm(;N)1297 2063 y Fu(\()1346 2047 y(\026)1325 2063 y Fi(X)1390 2010 y Fm(\033)1427 2025 y(J)1462 2039 y Fn(1)1502 2010 y Fm(:::\033)1602 2025 y(J)1637 2041 y(k)1385 2078 y(\033)1422 2093 y(J)1457 2109 y(k)1497 2093 y Fn(+1)1579 2078 y Ff(\001\001\001)p Fm(\033)1679 2093 y(J)1714 2109 y(k)q Fn(+1)1841 2063 y Fu(\))1889 2127 y Fk(\001)15 b(\001)g(\001)2085 2040 y Fe(X)1843 2231 y Fi(\033)1885 2240 y Fm(J)1920 2259 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)2115 2240 y(+1)2197 2231 y Fi(:::)n(\033)2303 2240 y Fm(J)2338 2259 y(l)p Fn(\()p Fm(\014)r Fn(\))2472 2127 y Fj(e)2514 2060 y Fi(\014)s(\015)2596 2072 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2711 2060 y Fi(u)2754 2030 y Ff(\000)p Fn(1)2754 2084 y(ln)j(\026)-38 b Fm(\013)2860 2102 y(l)p Fn(\()p Fm(\014)r Fn(\))2973 2084 y Fm(;N)3052 2060 y Fu(\()3101 2043 y(\026)3080 2060 y Fi(X)3145 1992 y Fm(\033)3182 2008 y(J)3217 2022 y Fn(1)3258 1992 y Fm(:::)o(\033)3357 2008 y(J)3392 2026 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)3140 2072 y Fm(\033)3177 2087 y(J)3212 2106 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)3407 2087 y(+1)3489 2072 y Fm(:::)o(\033)3588 2087 y(J)3623 2106 y(l)p Fn(\()p Fm(\014)r Fn(\))3747 2060 y Fu(\))201 2451 y Fk(\002)292 2350 y Fe(\020)346 2451 y Fj(e)388 2377 y Fh(\000)458 2316 y Fe(P)554 2338 y Fm(n)554 2411 y(i)p Fn(=)p Fm(J)660 2430 y(l)p Fn(\()p Fm(\014)r Fn(\))774 2411 y(+1)856 2377 y Fu(\()p Fi(N)19 b Fu(ln)14 b Fi(\013)1082 2386 y Fm(i)1111 2377 y Fu(+)p Fi(N)7 b(\014)1276 2350 y Fn(2)1311 2377 y Fi(\013)1358 2386 y Fm(i)1387 2377 y Fi(=)p Fu(2\))1504 2364 y Fe(X)1433 2555 y Fi(\033)1475 2564 y Fm(J)1510 2583 y(l)p Fn(\()p Fm(\014)r Fn(\))1624 2564 y(+1)1721 2451 y Fj(e)1763 2387 y Fi(\014)1807 2347 y Fk(p)p 1884 2347 271 4 v 1884 2387 a Fi(a)1923 2396 y Fm(J)1958 2414 y(l)p Fn(\()p Fm(\014)r Fn(\))2072 2396 y(+1)2154 2387 y Fi(X)2214 2396 y Fm(\033)2251 2410 y Fn(1)2287 2396 y Fm(:::)o(\033)2386 2411 y(J)2421 2430 y(l)p Fn(\()p Fm(\014)r Fn(\))2535 2411 y(+1)2642 2451 y Fk(\001)15 b(\001)g(\001)2763 2364 y Fe(X)2777 2555 y Fi(\033)2819 2564 y Fm(m)2909 2451 y Fj(e)2951 2413 y Fi(\014)2995 2372 y Fh(p)p 3056 2372 100 4 v 3056 2413 a Fi(a)3095 2422 y Fm(m)3156 2413 y Fi(X)3216 2422 y Fm(\033)3253 2436 y Fn(1)3288 2422 y Fm(:::)o(\033)3387 2431 y(m)3477 2451 y Fk(\000)20 b Fj(C)7 b Fr(\()p Fj(\014)e Fr(\))3766 2350 y Fe(\021)201 2794 y Fr(+)367 2707 y Fe(X)292 2898 y Fi(\033)334 2907 y Fn(1)370 2898 y Fi(;:::)n(\033)498 2907 y Fm(J)533 2921 y Fn(1)589 2794 y Fj(e)631 2746 y Fi(\014)s(\015)713 2755 y Fn(1)749 2746 y Fi(u)792 2716 y Ff(\000)p Fn(1)792 2770 y(ln)17 b(\026)-37 b Fm(\013)898 2784 y Fn(1)933 2770 y Fm(;N)1012 2746 y Fu(\()1060 2730 y(\026)1040 2746 y Fi(X)1100 2755 y Fm(\033)1137 2769 y Fn(1)1172 2755 y Ff(\001\001\001)p Fm(\033)1272 2770 y(J)1307 2784 y Fn(1)1352 2746 y Fu(\))1400 2794 y Fk(\001)15 b(\001)g(\001)1641 2707 y Fe(X)1400 2898 y Fi(\033)1442 2907 y Fm(J)1477 2926 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)1671 2907 y(+1)1753 2898 y Fi(:::)o(\033)1860 2907 y Fm(J)1895 2926 y(l)p Fn(\()p Fm(\014)r Fn(\))2029 2794 y Fj(e)2071 2726 y Fi(\014)s(\015)2153 2738 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2268 2726 y Fi(u)2311 2697 y Ff(\000)p Fn(1)2311 2751 y(ln)i(\026)-38 b Fm(\013)2416 2769 y(l)p Fn(\()p Fm(\014)r Fn(\))2530 2751 y Fm(;N)2609 2726 y Fu(\()2657 2710 y(\026)2637 2726 y Fi(X)2702 2659 y Fm(\033)2739 2675 y(J)2774 2689 y Fn(1)2814 2659 y Fm(:::\033)2914 2675 y(J)2949 2693 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)2697 2739 y Fm(\033)2734 2754 y(J)2769 2772 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)2963 2754 y(+1)3045 2739 y Fm(:::\033)3145 2754 y(J)3180 2772 y(l)p Fn(\()p Fm(\014)r Fn(\))3303 2726 y Fu(\))383 3060 y Fk(\002)20 b Fr(1)-23 b(I)529 3108 y Fh(f8)p Fi(i)p Fu(=1)p Fi(;:::)11 b(;k)r Fu(:)p Fi(\015)946 3117 y Fn(1)980 3108 y Fi(u)1023 3078 y Ff(\000)p Fn(1)1023 3132 y(ln)18 b(\026)-38 b Fm(\013)1129 3146 y Fn(1)1164 3132 y Fm(;N)1243 3108 y Fu(\()1292 3092 y(\026)1271 3108 y Fi(X)1331 3117 y Fm(\033)1368 3131 y Fn(1)1404 3117 y Ff(\001\001\001)o Fm(\033)1503 3132 y(J)1538 3146 y Fn(1)1583 3108 y Fu(\)+)p Fh(\001\001\001)o Fu(+)p Fi(\015)1826 3117 y Fm(i)1855 3108 y Fi(u)1898 3078 y Ff(\000)p Fn(1)1898 3132 y(ln)17 b(\026)-38 b Fm(\013)2003 3147 y(i)2032 3132 y(;N)2111 3108 y Fu(\()2159 3092 y(\026)2139 3108 y Fi(X)2204 3048 y Fm(\033)2241 3064 y(J)2276 3078 y Fn(1)2316 3048 y Fm(:::\033)2416 3064 y(J)2451 3079 y(i)p Ff(\000)p Fn(1)2199 3120 y Fm(\033)2236 3135 y(J)2271 3150 y(i)p Ff(\000)p Fn(1)2380 3135 y(+1)2462 3120 y Fm(:::\033)2562 3135 y(J)2597 3150 y(i)2635 3108 y Fu(\))p Fh(\024)p Fu(\()p Fi(\015)2786 3117 y Fn(1)2822 3108 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(\015)3037 3117 y Fm(i)3066 3108 y Fu(\))p Fi(x)p Fh(g)201 3329 y Fk(\002)292 3228 y Fe(\020)346 3329 y Fj(e)388 3255 y Fh(\000)458 3194 y Fe(P)554 3216 y Fm(n)554 3289 y(i)p Fn(=)p Fm(J)660 3308 y(l)p Fn(\()p Fm(\014)r Fn(\))774 3289 y(+1)856 3255 y Fu(\()p Fi(N)19 b Fu(ln)14 b Fi(\013)1082 3264 y Fm(i)1111 3255 y Fu(+)p Fi(N)7 b(\014)1276 3228 y Fn(2)1311 3255 y Fi(\013)1358 3264 y Fm(i)1387 3255 y Fi(=)p Fu(2\))1489 3243 y Fe(X)1418 3434 y Fi(\033)1460 3443 y Fm(J)1495 3461 y(l)p Fn(\()p Fm(\014)r Fn(\))1609 3443 y(+1)1706 3329 y Fj(e)1748 3265 y Fi(\014)1792 3225 y Fk(p)p 1869 3225 271 4 v 1869 3265 a Fi(a)1908 3274 y Fm(J)1943 3292 y(l)p Fn(\()p Fm(\014)r Fn(\))2057 3274 y(+1)2139 3265 y Fi(X)2199 3274 y Fm(\033)2236 3288 y Fn(1)2271 3274 y Fm(:::\033)2371 3289 y(J)2406 3308 y(l)p Fn(\()p Fm(\014)r Fn(\))2520 3289 y(+1)2626 3329 y Fk(\001)15 b(\001)g(\001)2748 3243 y Fe(X)2762 3434 y Fi(\033)2804 3443 y Fm(m)2894 3329 y Fj(e)2936 3291 y Fi(\014)2980 3250 y Fh(p)p 3041 3250 100 4 v 3041 3291 a Fi(a)3080 3300 y Fm(m)3140 3291 y Fi(X)3200 3300 y Fm(\033)3237 3314 y Fn(1)3273 3300 y Fm(:::)o(\033)3372 3309 y(m)3462 3329 y Fk(\000)20 b Fj(C)7 b Fr(\()p Fj(\014)e Fr(\))3751 3228 y Fe(\021)3805 3329 y Fj(:)3523 3559 y Fr(\(2)p Fj(:)p Fr(2\))50 3704 y(Again,)39 b(b)m(y)f(Lemma)g(2.2,)j(for)c(all)g Fj(N)48 b Fk(\025)37 b Fj(N)10 b Fr(\()p Fj(x)p Fr(\))38 b(the)g(exp)s (ectation)g(of)g(the)g(absolute)g(v)-5 b(alue)37 b(of)h(the)g(last)50 3862 y(term)g(in)f(\(2.2\))j(do)s(es)e(not)g(exceed)i(2)p Fj(C)7 b(C)g Fr(\()p Fj(\014)e Fr(\))p Fj(e)1652 3809 y Fi(x)1706 3748 y Fe(P)1802 3770 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))1802 3843 y Fm(j)s Fn(=1)1916 3809 y Fu(\()p Fi(\014)s(\015)2026 3818 y Fm(j)2061 3809 y Fh(\000)p Fu(1\))2225 3862 y Fk(#)39 b Fr(0)g(as)f Fj(x)g Fk(#)h(\0001)p Fr(.)64 b(Then,)40 b(giv)m(en)e Fj(\017)g(>)g Fr(0,)50 4007 y(using)e(Cheb)m(yshev's)h (inequalit)m(y)e(one)j(can)g(\014x)f Fj(x)g(<)g Fr(0)h(suc)m(h)f(that)h (for)f(all)g Fj(N)47 b Fk(\025)37 b Fj(N)10 b Fr(\()p Fj(x)p Fr(\))38 b(the)f(last)h(term)50 4153 y(of)g(\(2.2\))h(is)e (smaller)f(than)i Fj(\017)f Fr(with)g(probabilit)m(y)e(arbitrarily)g (close)j(to)g(1.)63 b(Ha)m(ving)38 b(\014xed)f Fj(x)h(<)f Fr(0)h(with)50 4298 y(su\016cien)m(tly)d(large)i(absolute)g(v)-5 b(alue,)38 b(let)f(us)f(turn)g(to)h(the)g(sum)f(of)h(the)g(\014rst)f Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))38 b(terms.)60 b(In)36 b(the)h Fj(k)s Fr(th)50 4444 y(term)26 b(of)h(\(2.2\))i(the)d (un-truncated)g(sums)g(m)m(ultiplied)d(b)m(y)j(the)h(term)f(in)g(round) f(brac)m(k)m(ets)j(are)f(distributed)50 4589 y(as)g Fj(I)205 4556 y Fi(N)198 4612 y Fu(2)300 4589 y Fr(in)e(the)i(GREM)g(of)f Fj(n)12 b Fk(\000)g Fj(J)1164 4603 y Fi(k)1236 4589 y Fr(lev)m(els)26 b(where)g(the)h(parameter)g Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))27 b(equals)f Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))12 b Fk(\000)g Fj(k)s Fr(.)41 b(They)26 b(con)m(v)m(erge)50 4735 y(to)36 b(zero)g(in)e(probabilit)m(y)f(b)m(y)i(induction.)53 b(F)-8 b(urthermore,)36 b(for)f(an)m(y)h Fj(\016)h(>)c Fr(0)j(one)f(can)h(\014nd)e(large)h(enough)50 4880 y(n)m(um)m(b)s(ers) 29 b Fj(N)10 b Fr(\()p Fj(\016)n(;)15 b(x)p Fr(\),)33 b Fj(R)q Fr(\()p Fj(\016)n(;)15 b(x)p Fr(\))32 b(and)f Fj(X)7 b Fr(\()p Fj(\016)n(;)15 b(x)p Fr(\))33 b(suc)m(h)d(that)i(for)e (all)g Fj(N)36 b Fk(\025)26 b Fj(N)10 b Fr(\()p Fj(\016)n(;)15 b(x)p Fr(\))32 b(with)e(probabilit)m(y)e(at)j(least)50 5025 y(1)25 b Fk(\000)f Fj(\016)41 b Fr(the)d Fj(k)s Fr(th)f(term)g(of)g(\(2.2\))i(is)e(a)g(linear)f(com)m(bination)g(of)i (at)g(most)f Fj(R)q Fr(\()p Fj(\016)n(;)15 b(x)p Fr(\))39 b(suc)m(h)e(sums)f(with)g(the)50 5171 y(w)m(eigh)m(ts)26 b(at)g(most)g Fj(e)735 5138 y Fi(\014)s(X)5 b Fu(\()p Fi(\016)n(;x)p Fu(\))1000 5171 y Fr(.)39 b(The)26 b(n)m(um)m(b)s(er)e Fj(R)q Fr(\()p Fj(\016)n(;)15 b(x)p Fr(\))27 b(exists)f(b)m(y)f(Prop)s (osition)f(1.4)j(\(i\))e(and)g(of)h(the)g(n)m(um)m(b)s(er)50 5316 y Fj(X)7 b Fr(\()p Fj(\016)n(;)15 b(x)p Fr(\))27 b(do)s(es)e(b)m(y)h(Theorem)f(1.5)h(of)33 b([BK1].)40 b(This)23 b(theorem)j(asserts)g(the)g(p)s(oin)m(t)e(pro)s(cess)h(con)m (v)m(ergence)j(of)50 5462 y(extremes)23 b Fk(E)472 5476 y Fi(N)566 5462 y Fr(=)662 5389 y Fe(R)705 5494 y Fc(R)755 5476 y Fm(k)811 5462 y Fj(\016)851 5476 y Fi(\016)884 5485 y Fn(1)920 5476 y Fi(x)961 5485 y Fn(1)996 5476 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(\016)1206 5486 y Fm(k)1245 5476 y Fi(x)1286 5486 y Fm(k)1331 5462 y Fk(P)1401 5429 y Fi(k)1394 5487 y(N)1463 5462 y Fr(\()p Fj(dx)1597 5476 y Fu(1)1639 5462 y Fj(;)15 b(:)g(:)g(:)31 b(;)15 b(dx)1954 5476 y Fi(k)2000 5462 y Fr(\))25 b Fk(!)g(E)34 b Fr(=)2354 5389 y Fe(R)2397 5494 y Fc(R)2447 5476 y Fm(k)2502 5462 y Fj(\016)2542 5476 y Fi(\016)2575 5485 y Fn(1)2611 5476 y Fi(x)2652 5485 y Fn(1)2688 5476 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(\016)2897 5486 y Fm(k)2937 5476 y Fi(x)2978 5486 y Fm(k)3022 5462 y Fk(P)3092 5429 y Fi(k)3138 5462 y Fr(\()p Fj(dx)3272 5476 y Fu(1)3313 5462 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(dx)3629 5476 y Fi(k)3674 5462 y Fr(\))p eop %%Page: 17 17 17 16 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(17)50 513 y Fr(where)31 b(the)h(pro)s(cess)f Fk(E)40 b Fr(has)31 b(at)i(least)e(one)h(p)s(oin)m(t)f(in)f([)p Fj(a;)15 b Fk(1)p Fr(\))33 b(with)d(exp)s(onen)m(tially)g(small)g(probabilit)m (y)f(as)50 658 y Fj(a)c Fk(!)g(1)p Fr(.)39 b(Then)24 b(the)i Fj(k)s Fr(th)f(term)g(in)f Fj(I)1299 625 y Fi(N)1292 681 y Fu(2)1393 658 y Fr(con)m(v)m(erges)j(to)f(zero)g(in)e(probabilit) m(y)-8 b(,)24 b(as)h Fj(N)36 b Fk(")25 b(1)p Fr(,)i Fj(k)h Fr(=)d(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\).)50 804 y(Hence,)31 b Fj(I)392 771 y Fi(N)385 826 y Fu(2)492 804 y Fr(con)m(v)m(erges)h(to)f(zero)g(in)e (probabilit)m(y)-8 b(.)39 b(This)29 b(concludes)g(the)i(pro)s(of)e(of)i (the)f(theorem.)42 b Fk(})50 1022 y Fd(Lemma)c(2.1:)51 b Fq(L)-5 b(et)33 b(in)f(the)h(GREM)f Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))27 b(=)e(0)p Fq(.)42 b(Then)1290 1254 y Fj(e)1332 1210 y Fh(\000)p Fi(N)1466 1148 y Fe(P)1562 1171 y Fm(n)1562 1244 y(i)p Fn(=1)1681 1210 y Fi(\014)1725 1182 y Fn(2)1761 1210 y Fi(a)1800 1219 y Fm(i)1829 1210 y Fi(=)p Fu(2)1907 1254 y Fj(Z)1969 1268 y Fi(\014)s(;N)2146 1202 y Fh(D)2130 1254 y Fk(!)25 b Fj(C)7 b Fr(\()p Fj(\014)e Fr(\))p Fj(:)50 1485 y Fq(The)33 b(c)-5 b(onstant)34 b Fj(C)7 b Fr(\()p Fj(\014)e Fr(\))26 b(=)f(1)33 b Fq(if)f Fj(\014)5 b(\015)1187 1499 y Fu(1)1253 1485 y Fj(<)25 b Fr(1)33 b Fq(and)539 1717 y Fj(C)7 b Fr(\()p Fj(\014)e Fr(\))25 b(=)g Fj(P)929 1616 y Fe(\020)1492 1631 y(\\)1410 1815 y Fm(i)p Fn(:1)p Ff(\024)p Fm(i)p Ff(\024)p Fm(J)1640 1829 y Fn(1)1010 1873 y(\()p Fm(a)1070 1887 y Fn(1)1105 1873 y(+)p Ff(\001\001\001)o Fn(+)p Fm(a)1296 1888 y(i)1325 1873 y Fn(\))p Fm(=)r Fn(\026)-33 b Fm(a)1418 1889 y(J)1453 1903 y Fn(1)1493 1873 y(=ln)o(\()p Fm(\013)1659 1887 y Fn(1)1694 1873 y Ff(\001\001\001)p Fm(\013)1799 1888 y(i)1827 1873 y Fn(\))p Fm(=)11 b Fn(ln)17 b(\026)-37 b Fm(\013)2002 1889 y(J)2037 1903 y Fn(1)2087 1717 y Fr(\()2122 1654 y Fk(p)p 2198 1654 89 4 v 63 x Fj(a)2246 1731 y Fu(1)2287 1717 y Fj(Z)2349 1731 y Fu(1)2410 1717 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)2718 1654 y Fk(p)p 2794 1654 79 4 v 63 x Fj(a)2842 1731 y Fi(i)2872 1717 y Fj(Z)2934 1731 y Fi(i)2990 1717 y Fj(<)k Fr(0\))3166 1616 y Fe(\021)3523 1717 y Fr(\(2)p Fj(:)p Fr(3\))50 2070 y Fq(if)32 b Fj(\014)5 b(\015)241 2084 y Fu(1)307 2070 y Fr(=)25 b(1)33 b Fq(wher)-5 b(e)34 b Fj(Z)800 2084 y Fu(1)841 2070 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(Z)1120 2084 y Fi(J)1160 2093 y Fn(1)1233 2070 y Fq(ar)-5 b(e)33 b(indep)-5 b(endent)35 b(standar)-5 b(d)35 b(Gaussian)f(r.v.)50 2288 y(Pr)-5 b(o)g(of.)73 b Fr(By)41 b(Prop)s(osition)e(1.4)j(of)48 b([BK1])42 b Fj(A)1632 2302 y Fu(1)p Fi(;j)1771 2288 y Fk(\024)g Fj(A)1952 2302 y Fu(1)p Fi(;J)2050 2311 y Fn(1)2133 2288 y Fr(=)g Fj(\015)2298 2255 y Fu(2)2293 2311 y(1)2380 2288 y Fr(for)f(all)e Fj(j)49 b Fr(=)42 b(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(J)3177 2302 y Fu(1)3260 2288 y Fr(and)40 b(b)m(y)h(the)50 2434 y(de\014nition)34 b(of)i Fj(J)619 2448 y Fu(1)696 2434 y Fj(A)764 2448 y Fu(1)p Fi(;j)895 2434 y Fj(<)f(A)1069 2448 y Fu(1)p Fi(;J)1167 2457 y Fn(1)1242 2434 y Fr(=)g Fj(\015)1400 2401 y Fu(2)1395 2456 y(1)1476 2434 y Fr(for)h Fj(j)41 b(>)34 b(J)1854 2448 y Fu(1)1896 2434 y Fr(.)58 b(Therefore,)37 b(if)e Fj(\014)5 b(\015)2615 2448 y Fu(1)2691 2434 y Fj(<)35 b Fr(1,)j(then)e Fj(\014)3174 2401 y Fu(2)3215 2434 y Fj(A)3283 2448 y Fu(1)p Fi(;j)3414 2434 y Fj(<)e Fr(1)j(for)50 2579 y(all)31 b Fj(j)j Fr(=)29 b(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(n)p Fr(;)33 b(if)e Fj(\014)5 b(\015)998 2593 y Fu(1)1068 2579 y Fr(=)28 b(1,)34 b(then)e Fj(\014)1536 2546 y Fu(2)1577 2579 y Fj(A)1645 2593 y Fu(1)p Fi(;j)1770 2579 y Fk(\024)c Fr(1)33 b(for)f(all)f Fj(j)j Fr(=)29 b(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(n)32 b Fr(and)g(the)g(equalit)m(y)g(tak)m(es)50 2725 y(place)e(only)g(for)g(those)h Fj(j)g Fk(\024)25 b Fj(J)1073 2739 y Fu(1)1144 2725 y Fr(for)30 b(whic)m(h)f(\()p Fj(a)1626 2739 y Fu(1)1688 2725 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)2044 2739 y Fi(j)2082 2725 y Fr(\))p Fj(=)q Fr(\026)-46 b Fj(a)2210 2739 y Fi(J)2250 2748 y Fn(1)2316 2725 y Fr(=)25 b(ln)o(\()p Fj(\013)2581 2739 y Fu(1)2638 2725 y Fk(\001)15 b(\001)g(\001)h Fj(\013)2817 2739 y Fi(j)2855 2725 y Fr(\))p Fj(=)f Fr(ln)24 b(\026)-54 b Fj(\013)3100 2739 y Fi(J)3140 2748 y Fn(1)3181 2725 y Fr(.)159 2943 y(First,)33 b(w)m(e)g(study)e(the)i(case)h(of)f Fj(\014)5 b(\015)1360 2957 y Fu(1)1430 2943 y Fj(<)28 b Fr(1,)34 b(in)d(whic)m(h)h(w)m(e)h(sho)m(w)f(that)h Fj(Z)2628 2957 y Fi(\014)s(;N)2764 2943 y Fj(=)p Fg(E)11 b Fj(Z)2926 2957 y Fi(\014)s(;N)3112 2891 y Fh(D)3096 2943 y Fk(!)29 b Fr(1)j(as)h Fj(N)39 b Fk(")30 b(1)p Fr(.)50 3088 y(Let)h(us)e(\014x)h Fj(\017)25 b(>)g Fr(0)31 b(suc)m(h)f(that)872 3320 y(ln)o(\()p Fj(\013)1041 3334 y Fu(1)1097 3320 y Fk(\001)15 b(\001)g(\001)h Fj(\013)1276 3334 y Fi(j)1315 3320 y Fr(\))k Fk(\000)g Fr(\()p Fj(\014)1552 3282 y Fu(2)1614 3320 y Fk(\000)g Fr(\()p Fj(\014)26 b Fk(\000)19 b Fj(\017)p Fr(\))1979 3282 y Fu(2)2021 3320 y Fj(=)p Fr(2\)\()p Fj(a)2229 3334 y Fu(1)2291 3320 y Fr(+)h Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)2647 3334 y Fi(j)2686 3320 y Fr(\))25 b Fj(>)g Fr(0)636 b(\(2)p Fj(:)p Fr(4\))50 3552 y(for)26 b(all)g Fj(j)31 b Fr(=)25 b(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(n)27 b Fr(b)s(ecause)g(of)g(the)g (strict)g(inequalit)m(y)e Fj(\014)2111 3519 y Fu(2)2152 3552 y Fj(A)2220 3566 y Fu(1)p Fi(;j)2341 3552 y Fj(<)g Fr(1.)40 b(W)-8 b(e)28 b(use)f(truncation)f(tec)m(hniques)50 3697 y(originally)i(in)m(tro)s(duced)g(b)m(y)j(M.)g(T)-8 b(alagrand.)40 b(Let)31 b(us)f(de\014ne)f(the)i(truncated)f(partition)f (function)623 3906 y Fj(Z)692 3868 y Fi(T)685 3928 y(\014)s(;N)846 3906 y Fr(=)c(2)987 3868 y Fh(\000)p Fi(N)1128 3819 y Fe(X)1155 4010 y Fi(\033)1197 4019 y Fn(1)1274 3906 y Fj(e)1316 3868 y Fi(\014)1360 3826 y Fh(p)p 1421 3826 75 4 v 1421 3868 a Fi(a)1460 3877 y Fn(1)1496 3868 y Fi(X)1556 3877 y Fm(\033)1593 3891 y Fn(1)1652 3906 y Fk(\001)15 b(\001)g(\001)1774 3819 y Fe(X)1796 4010 y Fi(\033)1838 4019 y Fm(n)1920 3906 y Fj(e)1962 3868 y Fi(\014)2006 3826 y Fh(p)p 2067 3826 V 2067 3868 a Fi(a)2106 3877 y Fn(1)2141 3868 y Fi(X)2201 3877 y Fm(\033)2238 3891 y Fn(1)2274 3877 y Fm(\033)2311 3891 y Fn(2)2346 3877 y Fm(:::\033)2446 3886 y(n)1022 4142 y Fk(\002)20 b Fr(1)-23 b(I)1169 4171 y Fh(f8)1247 4180 y Fm(i)p Fn(=1)p Fm(;:::)11 b(;n)1509 4130 y Fh(p)p 1569 4130 V 41 x Fi(a)1608 4180 y Fn(1)1644 4171 y Fi(X)1704 4180 y Fm(\033)1741 4194 y Fn(1)1780 4171 y Fu(+)p Fh(\001\001\001)o Fu(+)1957 4130 y Fh(p)p 2017 4130 69 4 v 41 x Fi(a)2056 4180 y Fm(i)2085 4171 y Fi(X)2145 4180 y Fm(\033)2182 4194 y Fn(1)2217 4180 y Fm(:::)o(\033)2316 4195 y(i)2350 4171 y Fi(<)p Fu(\()p Fi(\014)s Fu(+)p Fi(\017)p Fu(\)\()p Fi(a)2660 4180 y Fn(1)2697 4171 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(a)2913 4180 y Fm(i)2942 4171 y Fu(\))2970 4117 y Fh(p)p 3030 4117 65 4 v 54 x Fi(N)c Fh(g)3523 4025 y Fr(\(2)p Fj(:)p Fr(5\))50 4353 y(and)30 b(set)1487 4476 y(\026)1468 4499 y Fj(Z)1537 4461 y Fi(T)1530 4521 y(\014)s(;N)1690 4499 y Fk(\021)25 b Fj(Z)1848 4513 y Fi(\014)s(;N)2004 4499 y Fk(\000)20 b Fj(Z)2164 4461 y Fi(T)2157 4521 y(\014)s(;N)3523 4499 y Fr(\(2)p Fj(:)p Fr(6\))50 4698 y(W)-8 b(e)31 b(can)g(then)f(write)1178 4795 y Fj(Z)1240 4809 y Fi(\014)s(;N)p 1147 4836 258 4 v 1147 4919 a Fg(E)11 b Fj(Z)1264 4933 y Fi(\014)s(;N)1441 4857 y Fr(=)1578 4783 y Fj(Z)1647 4750 y Fi(T)1640 4809 y(\014)s(;N)p 1548 4836 V 1548 4922 a Fg(E)g Fj(Z)1672 4890 y Fi(T)1665 4949 y(\014)s(;N)1837 4857 y Fk(\002)1939 4783 y Fg(E)f Fj(Z)2062 4750 y Fi(T)2055 4809 y(\014)t(;N)p 1939 4836 V 1939 4919 a Fg(E)g Fj(Z)2055 4933 y Fi(\014)t(;N)2228 4857 y Fr(+)2379 4760 y(\026)2360 4783 y Fj(Z)2429 4750 y Fi(T)2422 4809 y(\014)s(;N)p 2329 4836 V 2329 4919 a Fg(E)h Fj(Z)2446 4933 y Fi(\014)s(;N)2598 4857 y Fj(:)900 b Fr(\(2)p Fj(:)p Fr(7\))50 5086 y(It)30 b(is)g(easy)h(to)g(compute)244 5294 y Fg(E)10 b Fj(Z)367 5256 y Fi(T)360 5316 y(\014)t(;N)527 5294 y Fr(=)25 b Fj(e)665 5256 y Fi(\014)709 5229 y Fn(2)745 5256 y Fi(N)q(=)p Fu(2)882 5294 y Fg(P)p Fr(\()p Fk(8)p Fj(i)h Fr(=)f(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(n)25 b Fr(:)1568 5230 y Fk(p)p 1644 5230 89 4 v 64 x Fj(a)1692 5308 y Fu(1)1733 5294 y Fj(Z)1795 5308 y Fu(1)1856 5294 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)2164 5230 y Fk(p)p 2240 5230 79 4 v 64 x Fj(a)2288 5308 y Fi(i)2318 5294 y Fj(Z)2380 5308 y Fi(i)2436 5294 y Fj(<)k(\017)p Fr(\()p Fj(a)2652 5308 y Fu(1)2713 5294 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)3069 5308 y Fi(i)3100 5294 y Fr(\))3135 5212 y Fk(p)p 3211 5212 83 4 v 82 x Fj(N)10 b Fr(\))527 5466 y Fk(\030)25 b Fj(e)665 5429 y Fi(\014)709 5401 y Fn(2)745 5429 y Fi(N)q(=)p Fu(2)907 5466 y Fr(=)g Fg(E)10 b Fj(Z)1119 5480 y Fi(\014)t(;N)3523 5371 y Fr(\(2)p Fj(:)p Fr(8\))p eop %%Page: 18 18 18 17 bop 50 295 a Fo(18)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fr(where)39 b Fj(Z)384 527 y Fu(1)425 513 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(Z)704 527 y Fi(n)794 513 y Fr(are)40 b(indep)s(enden)m(t)e(standard)h(Gaussian)g(r.v.)69 b(Th)m(us)38 b(the)i(second)g(term)g(in)f(\(2.7\))50 658 y(con)m(v)m(erges)28 b(to)e(zero)h(in)e Fj(L)911 672 y Fu(1)977 658 y Fr(and)g(in)g(the)h (\014rst)f(term)h(the)g(factor)h Fg(E)11 b Fj(Z)2331 625 y Fi(T)2324 684 y(\014)s(;N)2465 658 y Fj(=)p Fg(E)g Fj(Z)2627 672 y Fi(\014)t(;N)2794 658 y Fr(con)m(v)m(erges)28 b(to)e(1.)40 b(Finally)-8 b(,)50 804 y(for)30 b(an)m(y)h Fj(\016)e(>)c Fr(0)171 1103 y Fg(P)p Fr(\()p Fk(j)p Fj(Z)355 1066 y Fi(T)348 1126 y(\014)s(;N)485 1103 y Fj(=)p Fg(E)11 b Fj(Z)654 1066 y Fi(T)647 1126 y(\014)s(;N)808 1103 y Fk(\000)20 b Fr(1)p Fk(j)26 b Fj(>)f(\016)s Fr(\))h Fk(\024)1302 1030 y Fg(E)11 b Fr(\()p Fj(Z)1461 997 y Fi(T)1454 1055 y(\014)t(;N)1615 1030 y Fk(\000)20 b Fg(E)11 b Fj(Z)1830 997 y Fi(T)1823 1055 y(\014)s(;N)1964 1030 y Fr(\))1999 997 y Fu(2)p 1302 1082 738 4 v 1487 1168 a Fr(\()p Fg(E)g Fj(Z)1646 1137 y Fi(T)1639 1195 y(\014)s(;N)1780 1168 y Fr(\))1815 1142 y Fu(2)197 1426 y Fk(\024)304 1270 y Fe(P)399 1292 y Fi(N)399 1365 y(j)t Fu(=1)530 1338 y Fr(\()p Fj(\013)623 1352 y Fu(1)679 1338 y Fk(\001)k(\001)g (\001)h Fj(\013)858 1352 y Fi(j)897 1338 y Fr(\))932 1305 y Fh(\000)p Fi(N)1058 1338 y Fg(E)11 b Fj(e)1155 1305 y Fu(2)p Fi(\014)1236 1250 y Fh(p)p 1301 1250 65 4 v 1301 1305 a Fi(N)c Fu(\()1394 1263 y Fh(p)p 1454 1263 75 4 v 42 x Fi(a)1493 1314 y Fn(1)1529 1305 y Fi(X)1589 1314 y Fn(1)1624 1305 y Fu(+)p Fh(\001\001\001)o Fu(+)1801 1267 y Fh(p)p 1860 1267 74 4 v 1860 1305 a Fi(a)1899 1314 y Fm(j)1933 1305 y Fi(X)1993 1314 y Fm(j)2027 1305 y Fu(\))2060 1338 y Fr(1)-23 b(I)2116 1367 y Fh(f)2153 1325 y(p)p 2213 1325 75 4 v 42 x Fi(a)2252 1376 y Fn(1)2287 1367 y Fi(X)2347 1376 y Fn(1)2383 1367 y Fu(+)p Fh(\001\001\001)n Fu(+)2559 1330 y Fh(p)p 2619 1330 74 4 v 37 x Fi(a)2658 1376 y Fm(j)2692 1367 y Fi(X)2752 1376 y Fm(j)2786 1367 y Fi(<)p Fu(\()p Fi(\014)s Fu(+)p Fi(\017)p Fu(\)\()p Fi(a)3096 1376 y Fn(1)3133 1367 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(a)3349 1376 y Fm(j)3383 1367 y Fu(\))3411 1313 y Fh(p)p 3471 1313 65 4 v 54 x Fi(N)7 b Fh(g)p 304 1405 3274 4 v 1418 1491 a Fj(\016)1461 1465 y Fu(2)1503 1491 y Fj(e)1545 1463 y Fi(\014)1589 1445 y Fn(2)1625 1463 y Fi(N)g Fu(\()p Fi(a)1757 1472 y Fn(1)1793 1463 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(a)2008 1472 y Fm(j)2042 1463 y Fu(\))2075 1491 y Fr(\(1)21 b(+)f Fj(o)p Fr(\(1\)\))197 1716 y Fk(\024)25 b Fr(\(1)c(+)f Fj(o)p Fr(\(1\)\))p Fj(\016)722 1679 y Fh(\000)p Fu(2)870 1603 y Fi(N)837 1630 y Fe(X)839 1824 y Fi(j)t Fu(=1)968 1716 y Fr(\()p Fj(\013)1061 1730 y Fu(1)1117 1716 y Fk(\001)15 b(\001)g(\001)i Fj(\013)1297 1730 y Fi(j)1335 1716 y Fr(\))1370 1679 y Fh(\000)p Fi(N)1496 1716 y Fj(e)1538 1679 y Fi(\014)1582 1651 y Fn(2)1619 1679 y Fi(N)7 b Fu(\()p Fi(a)1751 1688 y Fn(1)1786 1679 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(a)2002 1688 y Fm(j)2036 1679 y Fu(\))2069 1716 y Fg(P)2124 1643 y Fe(\000)2166 1716 y Fj(Z)32 b(<)25 b Fr(\()p Fk(\000)p Fj(\014)g Fr(+)20 b Fj(\017)p Fr(\))2701 1609 y Fe(q)p 2793 1609 666 4 v 2793 1716 a Fj(N)10 b Fr(\()p Fj(a)2959 1730 y Fi(j)3017 1716 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)22 b Fr(+)e Fj(a)3374 1730 y Fi(n)3423 1716 y Fr(\))3458 1643 y Fe(\001)197 2048 y Fk(\024)25 b Fr(\(1)c(+)f Fj(o)p Fr(\(1\)\))p Fj(\016)722 2010 y Fh(\000)p Fu(2)870 1934 y Fi(N)837 1961 y Fe(X)839 2155 y Fi(j)t Fu(=1)968 2048 y Fr(\()p Fj(\013)1061 2062 y Fu(1)1117 2048 y Fk(\001)15 b(\001)g(\001)i Fj(\013)1297 2062 y Fi(j)1335 2048 y Fr(\))1370 2010 y Fh(\000)p Fi(N)1496 2048 y Fj(e)1538 2010 y Fi(\014)1582 1983 y Fn(2)1619 2010 y Fi(N)7 b Fu(\()p Fi(a)1751 2019 y Fn(1)1786 2010 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(a)2002 2019 y Fm(j)2036 2010 y Fu(\))2069 2048 y Fj(e)2111 2010 y Fh(\000)p Fi(N)g Fu(\()p Fi(\014)s Fh(\000)p Fi(\017)p Fu(\))2420 1983 y Fn(2)2456 2010 y Fu(\()p Fi(a)2523 2019 y Fm(j)2558 2010 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(a)2774 2019 y Fm(n)2819 2010 y Fu(\))p Fi(=)p Fu(2)3523 2254 y Fr(\(2)p Fj(:)p Fr(9\))50 2399 y(where)36 b Fj(X)394 2413 y Fu(1)435 2399 y Fj(;)15 b(:)g(:)g(:)31 b(;)15 b(X)726 2413 y Fi(n)813 2399 y Fr(and)36 b Fj(Z)42 b Fr(are)37 b(indep)s(enden)m(t)d(standard)i(Gaussian)f(r.v.)58 b(In)36 b(view)f(of)i(the)f(c)m(hoice)i(of)50 2545 y Fj(\017)i(>)g Fr(0,)j(\(2.9\))e(con)m(v)m(erges)g(to)f(zero)h(as)e Fj(N)51 b Fk(")40 b(1)p Fr(.)68 b(Then)39 b(the)g(\014rst)g(factor)h (in)f(the)g(\014rst)g(term)g(of)h(\(2.7\))50 2690 y Fj(Z)119 2657 y Fi(T)112 2716 y(\014)s(;N)247 2690 y Fj(=)p Fg(E)11 b Fj(Z)416 2657 y Fi(T)409 2716 y(\014)t(;N)575 2690 y Fk(!)25 b Fr(1)31 b(in)e(probabilit)m(y)f(\014nishing)f(the)k(pro)s (of)f(of)g(the)h(lemma)f(in)f(the)h(case)i Fj(\014)5 b(\015)3208 2704 y Fu(1)3274 2690 y Fj(<)25 b Fr(1.)159 2908 y(Assume)30 b(no)m(w)h(that)g Fj(\014)5 b(\015)986 2922 y Fu(1)1053 2908 y Fr(=)25 b(1.)42 b(W)-8 b(e)32 b(\014x)e Fj(\017)h Fr(according)g(to)g(\(2.4\))i(for)d(all)g Fj(j)36 b Fr(suc)m(h)30 b(that)h Fj(\014)3151 2875 y Fu(2)3192 2908 y Fj(A)3260 2922 y Fu(1)p Fi(;j)3383 2908 y Fj(<)25 b Fr(1.)42 b(\(In)50 3054 y(particular,)33 b(the)h(last)g(inequalit)m(y)f(holds)f(for)i(all)f Fj(j)k(>)31 b(J)2001 3068 y Fu(1)2042 3054 y Fr(.\))52 b(F)-8 b(or)35 b(those)f Fj(j)j Fk(\024)31 b Fj(J)2788 3068 y Fu(1)2863 3054 y Fr(for)j(whic)m(h)f Fj(\014)3326 3021 y Fu(2)3367 3054 y Fj(A)3435 3068 y Fu(1)p Fi(;j)3562 3054 y Fr(=)e(1)50 3199 y(w)m(e)g(truncate)g(the)f(partition)f(function)g(b)m(y)h Fj(u)1617 3216 y Fu(ln)q(\()p Fi(\013)1753 3225 y Fn(1)1789 3216 y Fh(\001\001\001)o Fi(\013)1901 3225 y Fm(j)1935 3216 y Fu(\))p Fi(;N)2054 3199 y Fr(\(0\))2169 3140 y Fk(p)p 2246 3140 504 4 v 2246 3199 a Fj(a)2294 3213 y Fu(1)2355 3199 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)2711 3213 y Fi(j)2750 3199 y Fr(.)40 b(Then)255 3461 y Fj(Z)324 3423 y Fi(T)317 3483 y(\014)s(;N)478 3461 y Fr(=)o(2)593 3423 y Fh(\000)p Fi(N)735 3375 y Fe(X)762 3566 y Fi(\033)804 3575 y Fn(1)881 3461 y Fj(e)923 3423 y Fi(\014)967 3382 y Fh(p)p 1028 3382 75 4 v 1028 3423 a Fi(a)1067 3432 y Fn(1)1103 3423 y Fi(X)1163 3432 y Fm(\033)1200 3446 y Fn(1)1259 3461 y Fk(\001)15 b(\001)g(\001)1380 3375 y Fe(X)1403 3566 y Fi(\033)1445 3575 y Fm(n)1527 3461 y Fj(e)1569 3423 y Fi(\014)1613 3382 y Fh(p)p 1673 3382 V 41 x Fi(a)1712 3432 y Fn(1)1748 3423 y Fi(X)1808 3432 y Fm(\033)1845 3446 y Fn(1)1880 3432 y Fm(\033)1917 3446 y Fn(2)1953 3432 y Fm(:::)o(\033)2052 3441 y(n)750 3697 y Fk(\002)20 b Fr(1)-23 b(I)897 3727 y Fh(f8)p Fi(j)t Fu(=1)p Fi(;:::)11 b(;n;)29 b(A)1373 3736 y Fn(1)p Fm(;j)1459 3727 y Fi()p Fu(0)p Fh(g)445 3056 y Fr(\()p Fj(\013)538 3070 y Fi(j)t Fu(+1)684 3056 y Fk(\001)15 b(\001)g(\001)h Fj(\013)863 3070 y Fi(n)912 3056 y Fr(\))947 3019 y Fh(\000)p Fi(N)1074 3056 y Fj(e)1116 3019 y Fh(\000)p Fi(\014)1217 2991 y Fn(2)1253 3019 y Fu(\()p Fi(a)1320 3028 y Fm(j)s Fn(+1)1432 3019 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(a)1648 3028 y Fm(n)1693 3019 y Fu(\))p Fi(N)q(=)p Fu(2)1884 2970 y Fe(X)1873 3161 y Fi(\033)1915 3170 y Fm(j)s Fn(+1)2042 3056 y Fj(e)2084 3014 y Fi(\014)2128 2977 y Fh(p)p 2188 2977 152 4 v 37 x Fi(a)2227 3023 y Fm(j)s Fn(+1)2339 3014 y Fi(X)2399 3023 y Fm(\033)2436 3037 y Fn(1)2471 3023 y Fm(:::\033)2571 3038 y(j)s Fn(+1)2707 3056 y Fk(\001)f(\001)g (\001)2828 2970 y Fe(X)2850 3161 y Fi(\033)2892 3170 y Fm(n)2974 3056 y Fj(e)3016 3019 y Fi(\014)3060 2977 y Fh(p)p 3121 2977 85 4 v 3121 3019 a Fi(a)3160 3028 y Fm(n)3205 3019 y Fi(X)3265 3028 y Fm(\033)3302 3042 y Fn(1)3337 3028 y Fm(\033)3374 3042 y Fn(2)3410 3028 y Fm(:::)o(\033)3509 3037 y(n)3563 3056 y Fj(:)3477 3265 y Fr(\(2)p Fj(:)p Fr(10\))50 3411 y(The)31 b(\014rst)f(sum)h(in)f(this) g(estimate)i(o)m(v)m(er)g Fj(j)37 b Fr(with)30 b Fj(A)1815 3425 y Fu(1)p Fi(;j)1938 3411 y Fj(<)d(A)2104 3425 y Fu(1)p Fi(;J)2202 3434 y Fn(1)2273 3411 y Fr(con)m(v)m(erges)34 b(to)e(zero)g(in)e Fj(L)3156 3425 y Fu(1)3228 3411 y Fr(as)h(the)h(mean)50 3556 y(of)37 b(its)f Fj(j)5 b Fr(th)38 b(term)f(is)f(of)h(the)g(order)f Fj(e)1340 3523 y Fh(\000)p Fi(\017)1427 3496 y Fn(2)1463 3523 y Fu(\()p Fi(a)1530 3532 y Fn(1)1566 3523 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(a)1782 3532 y Fm(j)1816 3523 y Fu(\))p Fi(N)q(=)p Fu(2)1981 3556 y Fr(.)61 b(In)36 b(the)h Fj(j)5 b Fr(th)37 b(term)g(of)g(the)g(last)g(sum)f(w)m(e)i(ha)m(v)m(e)50 3702 y(for)c(all)f Fj(i)g Fr(=)f(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(j)40 b(A)983 3716 y Fu(1)p Fi(;i)1104 3702 y Fk(\024)32 b Fj(A)1275 3716 y Fu(1)p Fi(;J)1373 3725 y Fn(1)1445 3702 y Fr(=)g Fj(A)1616 3716 y Fu(1)p Fi(;j)1746 3702 y Fr(b)m(y)j(Prop)s(osition)d(1.4)k(from)e ([BK1].)54 b(This)33 b(means)h(that)50 3847 y(inequalities)28 b(ensuring)h(the)i(con)m(v)m(ergence)i(to)e(the)g(P)m(oisson)g(p)s(oin) m(t)f(pro)s(cess)g(\(see)i(\(1.6\))g(of)f(Theorem)g(1.1)50 3993 y(in)39 b([BK1]\))i(are)g(v)m(eri\014ed)e(where)h Fj(a)1283 4007 y Fi(i)1313 3993 y Fj(=)1373 3924 y Fe(P)1470 3947 y Fi(j)1470 4020 y(i)p Fu(=1)1607 3993 y Fj(a)1655 4007 y Fi(i)1726 3993 y Fr(substitute)f Fj(a)2209 4007 y Fi(i)2239 3993 y Fr(,)52 b(\026)-54 b Fj(\013)43 b Fr(=)2520 3924 y Fe(Q)2606 3947 y Fi(j)2606 4020 y(i)p Fu(=1)2743 3993 y Fj(\013)2801 4007 y Fi(i)2832 3993 y Fr(,)g(and)c(Theorem)h(1.1)i(of)50 4138 y([BK1])35 b(applies.)50 b(Therefore,)34 b(the)h(p)s(oin)m(t)e(pro)s(cess)1828 4070 y Fe(P)1924 4165 y Fi(\033)1966 4174 y Fn(1)2002 4165 y Fi(;:::)n(\033)2130 4174 y Fm(j)2184 4138 y Fj(\016)2224 4166 y Fi(u)2267 4137 y Ff(\000)p Fn(1)2267 4193 y(ln\()p Fm(\013)2387 4207 y Fn(1)2421 4193 y Ff(\001\001\001)p Fm(\013)2526 4208 y(j)2559 4193 y Fn(\))p Fm(;N)2663 4166 y Fu(\()2712 4150 y(\026)2691 4166 y Fi(X)2751 4175 y Fm(\033)2788 4189 y Fn(1)2824 4175 y Ff(\001\001\001)o Fm(\033)2923 4190 y(j)2962 4166 y Fu(\))3029 4138 y Fr(con)m(v)m(erges) j(w)m(eakly)50 4298 y(to)i(the)g(P)m(oisson)f(p)s(oin)m(t)g(pro)s(cess) g(on)h Fg(R)46 b Fr(with)36 b(the)i(in)m(tensit)m(y)f(measure)h Fj(e)2634 4265 y Fh(\000)p Fi(x)2736 4298 y Fj(dx)g Fr(ha)m(ving)g (almost)f(surely)50 4444 y(a)h(\014nite)e(n)m(um)m(b)s(er)g(of)i(p)s (oin)m(ts)e(in)g([0)p Fj(;)15 b Fk(1)p Fr(\).)63 b(Then)36 b(for)i(an)m(y)f Fj(\017)g(>)g Fr(0)h(one)f(can)h(\014nd)e Fj(K)7 b Fr(\()p Fj(\017)p Fr(\))38 b(and)e Fj(R)q Fr(\()p Fj(\017)p Fr(\))i(suc)m(h)50 4589 y(that)j(the)f(n)m(um)m(b)s(er)f(of)h (its)g(p)s(oin)m(ts)f(in)g([0)p Fj(;)15 b Fk(1)p Fr(\))41 b(is)e(smaller)g(than)h Fj(K)7 b Fr(\()p Fj(\017)p Fr(\))41 b(and)e(eac)m(h)i(of)g(them)f(is)f(smaller)50 4735 y(than)30 b Fj(R)q Fr(\()p Fj(\017)p Fr(\))g(with)f(probabilit)m(y)f(at)j(least)f (1)20 b Fk(\000)g Fj(\017)30 b Fr(for)g(all)f Fj(N)40 b Fr(large)30 b(enough.)41 b(The)29 b(sum)g(o)m(v)m(er)j Fj(\033)3261 4749 y Fi(j)t Fu(+1)3391 4735 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(\033)3660 4749 y Fi(n)50 4880 y Fr(attac)m(hed)39 b(to)g(eac)m(h)f(p)s(oin)m(t)f(of)h(this)e(pro)s (cess)h(is)g(the)g(prop)s(erly)f(normalised)f(partition)i(function)f (of)i(the)50 5025 y(GREM)h(with)f(the)i(righ)m(t)e(partition)g (\(1.12\))k Fj(J)1692 5039 y Fu(1)1759 5025 y Fk(\000)26 b Fj(j;)15 b(J)1983 5039 y Fu(2)2025 5025 y Fj(;)g(:)g(:)g(:)32 b(;)15 b(J)2292 5039 y Fi(m)2401 5025 y Fr(and)38 b Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))41 b(=)f(0.)67 b(If)39 b Fj(J)3180 5039 y Fu(1)3247 5025 y Fk(\000)26 b Fj(j)46 b Fr(=)39 b(0,)j(it)50 5171 y(con)m(v)m(erges)36 b(to)g(a)f(constan)m(t)h(b)m(y)e (the)h(statemen)m(t)h(of)f(the)g(lemma)f(just)g(pro)m(v)m(ed)h(in)e (the)i(previous)e(case)i(as)50 5316 y Fj(\014)106 5283 y Fu(2)147 5316 y Fj(A)215 5330 y Fi(J)255 5339 y Fn(1)291 5330 y Fu(+1)p Fi(;J)445 5339 y Fn(2)511 5316 y Fj(<)25 b(\014)663 5283 y Fu(2)704 5316 y Fj(A)772 5330 y Fu(1)p Fi(;J)870 5339 y Fn(1)937 5316 y Fr(=)g(1.)43 b(Otherwise,)30 b(start)h(induction)e(pro)s(cedure)h(on)g(the)h(size)g(of)g(the)g (\014rst)f(cell.)50 5462 y(Then)24 b(this)f(sum)h(o)m(v)m(er)i Fj(\033)881 5476 y Fi(j)t Fu(+1)1011 5462 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(\033)1280 5476 y Fi(n)1355 5462 y Fr(with)23 b(the)i(\014rst)f(cell)g(smaller)f(than)i Fj(J)2610 5476 y Fu(1)2676 5462 y Fr(con)m(v)m(erges)h(to)g(a)f(constan)m(t)h(b)m(y)p eop %%Page: 20 20 20 19 bop 50 295 a Fo(20)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fr(induction)32 b(assumption.)50 b(It)35 b(follo)m(ws)e(from)h (these)h(argumen)m(ts)f(that)h(the)f(whole)g(sum)f(o)m(v)m(er)j Fj(\033)3362 527 y Fu(1)3402 513 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(\033)3671 527 y Fi(j)50 658 y Fr(is)33 b(b)s(ounded)g(in)g (probabilit)m(y:)47 b(for)34 b(an)m(y)h Fj(\017)d(>)g Fr(0)j(there)g(exits)f Fj(M)10 b Fr(\()p Fj(\017)p Fr(\))35 b(that)h(it)e(is)f(bigger)i(than)f Fj(M)10 b Fr(\()p Fj(\017)p Fr(\))35 b(with)50 804 y(probabilit)m(y)f(smaller)i(than)h Fj(\017)f Fr(for)h(all)f Fj(N)47 b Fr(su\016cien)m(tly)35 b(large.)61 b(Finally)35 b(the)i(prefactor)h Fj(e)3187 771 y Fh(\000)12 b Fu(ln)h Fi(N)q(=)p Fu(2)3503 804 y Fr(mak)m(e)50 949 y(it)32 b(con)m(v)m(erge)i(in)e(probabilit)m(y)e(to)j (zero.)48 b(This)30 b(pro)m(v)m(es)k(the)e(con)m(v)m(ergence)j(in)c (probabilit)m(y)f(to)k(zero)f(of)g(the)50 1095 y(second)d(term)h(in)e (\(2.7\))j(and)e(completes)g(the)h(pro)s(of)f(of)g(the)h(lemma.)40 b Fk(})p Fr(.)50 1313 y Fd(Lemma)i(2.2:)61 b Fq(L)-5 b(et)36 b Fr(0)c Fj(<)f(a)1086 1327 y Fi(i)1148 1313 y Fj(<)g Fr(1)p Fq(,)37 b Fj(a)1408 1327 y Fi(i)1470 1313 y Fj(>)31 b Fr(1)p Fq(,)37 b Fj(\015)1729 1327 y Fi(i)1791 1313 y Fr(=)1893 1235 y Fe(p)p 1984 1235 435 4 v 78 x Fj(a)2032 1327 y Fi(i)2062 1313 y Fj(=)p Fr(\(2)15 b(ln)g Fj(\013)2352 1327 y Fi(i)2383 1313 y Fr(\))p Fq(,)37 b Fj(i)31 b Fr(=)h(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(m)36 b Fq(b)-5 b(e)36 b(the)g(numb)-5 b(ers)50 1458 y(such)43 b(that)h Fj(\015)510 1472 y Fu(1)595 1458 y Fj(>)f Fk(\001)15 b(\001)g(\001)45 b Fj(>)f(\015)1021 1472 y Fi(m)1090 1458 y Fq(,)h Fj(\014)5 b(\015)1266 1472 y Fi(m)1379 1458 y Fj(>)44 b Fr(1)p Fq(.)73 b(L)-5 b(et)43 b Fj(X)1882 1472 y Fu(1)1923 1458 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(X)2215 1472 y Fi(m)2327 1458 y Fq(b)-5 b(e)43 b(indep)-5 b(endent)45 b(standar)-5 b(d)45 b(Gaussian)50 1604 y(r)-5 b(andom)35 b(variables.)43 b(Then)33 b(for)g(any)g Fj(x)25 b Fk(2)g Fg(R)41 b Fq(ther)-5 b(e)34 b(exists)f Fj(N)10 b Fr(\()p Fj(x)p Fr(\))33 b Fq(such)g(that)g(for)h(al)5 b(l)33 b Fj(N)i Fk(\025)25 b Fj(N)10 b Fr(\()p Fj(x)p Fr(\))144 1830 y Fg(E)17 b Fj(e)247 1783 y Fi(\014)s(\015)329 1792 y Fn(1)365 1783 y Fi(u)408 1753 y Ff(\000)p Fn(1)408 1807 y(ln)11 b Fm(\013)514 1821 y Fn(1)549 1807 y Fm(;N)628 1783 y Fu(\()p Fi(X)716 1792 y Fn(1)751 1783 y Fu(\)+)p Fh(\001\001\001)o Fu(+)p Fi(\014)s(\015)1038 1792 y Fm(m)1099 1783 y Fi(u)1142 1753 y Ff(\000)p Fn(1)1142 1807 y(ln)g Fm(\013)1248 1816 y(m)1308 1807 y(;N)1387 1783 y Fu(\()p Fi(X)1475 1792 y Fm(m)1535 1783 y Fu(\))1568 1830 y Fr(1)-23 b(I)1623 1859 y Fh(f8)p Fi(i)p Fu(=1)p Fi(;:::)11 b(;m)p Fu(:)p Fi(\015)2064 1868 y Fn(1)2098 1859 y Fi(u)2141 1829 y Ff(\000)p Fn(1)2141 1883 y(ln)g Fm(\013)2247 1897 y Fn(1)2282 1883 y Fm(;N)2361 1859 y Fu(\()p Fi(X)2449 1868 y Fn(1)2485 1859 y Fu(\)+)p Fh(\001\001\001)o Fu(+)p Fi(\015)2728 1868 y Fm(i)2757 1859 y Fi(u)2800 1829 y Ff(\000)p Fn(1)2800 1883 y(ln)f Fm(\013)2905 1898 y(i)2933 1883 y(;N)3013 1859 y Fu(\()p Fi(X)3101 1868 y Fm(i)3129 1859 y Fu(\))p Fi(<)p Fu(\()p Fi(\015)3280 1868 y Fn(1)3317 1859 y Fu(+)p Fh(\001\001\001)n Fi(\015)3475 1868 y Fm(i)3504 1859 y Fu(\))p Fi(x)p Fh(g)230 2048 y Fk(\024)25 b Fj(C)7 b Fr(\()p Fj(\013)491 2062 y Fu(1)547 2048 y Fk(\001)15 b(\001)g(\001)h Fj(\013)726 2062 y Fi(m)795 2048 y Fr(\))830 2011 y Fh(\000)p Fi(N)956 2048 y Fj(e)998 1995 y Fi(x)1052 1934 y Fe(P)1148 1956 y Fm(m)1148 2029 y(j)s Fn(=1)1260 1995 y Fu(\()p Fi(\014)s(\015)1370 2004 y Fm(j)1405 1995 y Fh(\000)p Fu(1\))3477 2157 y Fr(\(2)p Fj(:)p Fr(11\))50 2303 y Fq(wher)-5 b(e)34 b(the)f(c)-5 b(onstant)34 b Fj(C)39 b Fq(do)-5 b(es)34 b(not)f(dep)-5 b(end)34 b(on)f Fj(x)p Fq(.)50 2521 y(Pr)-5 b(o)g(of.)42 b Fr(The)29 b(r.h.s.)41 b(of)30 b(\(2.11\))j(equals)168 2770 y Fj(e)210 2656 y Fe(P)307 2678 y Fm(m)307 2751 y(j)s Fn(=1)431 2644 y Fe(\000)472 2717 y Fh(\000)p Fi(\014)s(N)638 2658 y Fk(p)p 714 2658 277 4 v 59 x Fu(2)p Fi(a)789 2726 y Fm(j)836 2717 y Fu(ln)13 b Fi(\013)956 2726 y Fm(j)990 2717 y Fu(+)p Fi(\014)s(\015)1128 2726 y Fm(j)1163 2717 y Fu([ln\()p Fi(N)19 b Fu(ln)14 b Fi(\013)1470 2726 y Fm(j)1504 2717 y Fu(\)+ln)f(4)p Fi(\031)r Fu(])p Fi(=)p Fu(2)1836 2644 y Fe(\001)371 2943 y Fk(\002)19 b Fg(E)11 b Fj(e)558 2905 y Fi(\014)603 2864 y Fh(p)p 669 2864 75 4 v 669 2905 a Fi(a)708 2914 y Fn(1)743 2905 y Fi(X)803 2914 y Fn(1)839 2905 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(\014)1059 2864 y Fh(p)p 1119 2864 100 4 v 41 x Fi(a)1158 2914 y Fm(m)1219 2905 y Fi(X)1279 2914 y Fm(m)1344 2943 y Fr(1)1389 2959 y Fh(8)p Fi(i)p Fu(=1)p Fi(;:::)g(;m)p Fu(:)1755 2918 y Fh(p)p 1813 2918 75 4 v 1813 2959 a Fi(a)1852 2968 y Fn(1)1888 2959 y Fi(X)1948 2968 y Fn(1)1983 2959 y Fu(+)p Fh(\001\001\001)o Fu(+)2160 2918 y Fh(p)p 2219 2918 69 4 v 2219 2959 a Fi(a)2258 2968 y Fm(i)2287 2959 y Fi(X)2347 2968 y Fm(i)2376 2959 y Fi(<)2433 2918 y Fh(p)p 2492 2918 75 4 v 2492 2959 a Fi(a)2531 2968 y Fn(1)2567 2959 y Fi(u)2610 2969 y Fn(ln)g Fm(\013)2716 2983 y Fn(1)2751 2969 y Fm(;N)2830 2959 y Fu(\()p Fi(x)p Fu(\)+)p Fh(\001\001\001)p Fu(+)3105 2918 y Fh(p)p 3164 2918 69 4 v 3164 2959 a Fi(a)3203 2968 y Fm(i)3232 2959 y Fi(u)3275 2969 y Fn(ln)g Fm(\013)3381 2984 y(i)3409 2969 y(;N)3489 2959 y Fu(\()p Fi(x)p Fu(\))194 3150 y Fr(=)25 b Fj(e)332 3036 y Fe(P)428 3058 y Fm(m)428 3131 y(j)s Fn(=1)552 3024 y Fe(\000)594 3097 y Fh(\000)p Fi(\014)s(N)760 3038 y Fk(p)p 835 3038 277 4 v 835 3097 a Fu(2)p Fi(a)910 3106 y Fm(j)957 3097 y Fu(ln)13 b Fi(\013)1077 3106 y Fm(j)1112 3097 y Fu(+)p Fi(\014)s(\015)1250 3106 y Fm(j)1284 3097 y Fu([ln\()p Fi(N)20 b Fu(ln)13 b Fi(\013)1591 3106 y Fm(j)1625 3097 y Fu(\)+ln)g(4)p Fi(\031)r Fu(])p Fi(=)p Fu(2+)p Fi(N)7 b(\014)s(a)2161 3070 y Fn(2)2161 3115 y Fm(j)2197 3097 y Fi(=)p Fu(2)2270 3024 y Fe(\001)371 3323 y Fk(\002)19 b Fg(P)516 3222 y Fe(\020)571 3323 y Fk(8)p Fj(i)25 b Fr(=)g(1)p Fj(;)15 b(:)g(:)g(:)33 b(;)15 b(m)25 b Fr(:)1193 3260 y Fk(p)p 1268 3260 89 4 v 1268 3323 a Fj(a)1316 3337 y Fu(1)1357 3323 y Fj(X)1432 3337 y Fu(1)1493 3323 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)22 b Fr(+)1801 3260 y Fk(p)p 1877 3260 79 4 v 63 x Fj(a)1925 3337 y Fi(i)1956 3323 y Fj(X)2031 3337 y Fi(i)557 3523 y Fj(<)653 3460 y Fk(p)p 729 3460 89 4 v 63 x Fj(a)777 3537 y Fu(1)818 3523 y Fr(\()p Fj(u)905 3537 y Fu(ln)13 b Fi(\013)1025 3546 y Fn(1)1061 3537 y Fi(;N)1152 3523 y Fr(\()p Fj(x)p Fr(\))21 b Fk(\000)f Fj(\014)1442 3439 y Fe(p)p 1533 3439 172 4 v 84 x Fj(a)1581 3537 y Fu(1)1622 3523 y Fj(N)9 b Fr(\))21 b(+)f Fk(\001)15 b(\001)g(\001)21 b Fr(+)2068 3460 y Fk(p)p 2144 3460 79 4 v 63 x Fj(a)2192 3537 y Fi(i)2222 3523 y Fr(\()p Fj(u)2309 3537 y Fu(ln)14 b Fi(\013)2430 3546 y Fm(i)2459 3537 y Fi(;N)2549 3523 y Fr(\()p Fj(x)p Fr(\))21 b Fk(\000)f Fj(\014)2839 3439 y Fe(p)p 2930 3439 162 4 v 84 x Fj(a)2978 3537 y Fi(i)3009 3523 y Fj(N)9 b Fr(\))3126 3422 y Fe(\021)3181 3523 y Fj(:)3477 3659 y Fr(\(2)p Fj(:)p Fr(12\))50 3805 y(No)m(w)31 b(Prop)s(osition)d(2.3)k(b)s(elo)m(w)d(will)f(b)s(e)i(applied)e(with) 297 4038 y Fj(c)336 4052 y Fi(i)367 4038 y Fr(\()p Fj(N)10 b Fr(\))26 b(=)f Fj(u)694 4052 y Fu(ln)13 b Fi(\013)814 4061 y Fm(i)843 4052 y Fi(;N)934 4038 y Fr(\()p Fj(x)p Fr(\))21 b Fk(\000)e Fj(\014)1223 3954 y Fe(p)p 1315 3954 V 1315 4038 a Fj(a)1363 4052 y Fi(i)1393 4038 y Fj(N)35 b Fr(=)25 b(\()1632 3953 y Fe(p)p 1723 3953 240 4 v 85 x Fr(2)15 b(ln)g Fj(\013)1933 4052 y Fi(i)1983 4038 y Fk(\000)20 b Fj(\014)2130 3974 y Fk(p)p 2206 3974 79 4 v 64 x Fj(a)2254 4052 y Fi(i)2285 4038 y Fr(\))2320 3956 y Fk(p)p 2396 3956 83 4 v 82 x Fj(N)30 b Fr(+)20 b Fj(o)p Fr(\(1\))p Fj(;)107 b(i)26 b Fr(=)f(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(m)50 4270 y Fr(where)30 b Fj(c)352 4284 y Fi(i)409 4270 y Fr(=)505 4200 y Fk(p)p 581 4200 240 4 v 70 x Fr(2)15 b(ln)g Fj(\013)791 4284 y Fi(i)841 4270 y Fk(\000)21 b Fj(\014)989 4212 y Fk(p)p 1065 4212 79 4 v 58 x Fj(a)1113 4284 y Fi(i)1169 4270 y Fj(<)26 b Fr(0)31 b(as)g Fj(\014)5 b(\015)1557 4284 y Fi(i)1613 4270 y Fj(>)26 b(\014)5 b(\015)1813 4284 y Fi(m)1908 4270 y Fj(>)26 b Fr(1)31 b(and)2258 4215 y Fk(p)p 2334 4215 172 4 v 55 x Fj(a)2382 4284 y Fi(i)p Fh(\000)p Fu(1)2505 4270 y Fj(c)2544 4284 y Fi(i)2595 4270 y Fk(\000)2687 4212 y(p)p 2762 4212 79 4 v 2762 4270 a Fj(a)2810 4284 y Fi(i)2841 4270 y Fj(c)2880 4284 y Fi(i)p Fh(\000)p Fu(1)3029 4270 y Fj(>)26 b Fr(0)31 b(as)g Fj(\015)3361 4284 y Fi(i)p Fh(\000)p Fu(1)3511 4270 y Fj(>)25 b(\015)3654 4284 y Fi(i)3685 4270 y Fr(.)50 4416 y(Th)m(us)k(\(2.12\))k(do)s(es)d(not)g(exceed)205 4665 y(\()240 4584 y Fk(p)p 316 4584 83 4 v 81 x Fj(N)10 b Fr(\))434 4628 y Fh(\000)p Fi(m)580 4642 y Fr(~)560 4665 y Fj(C)c(e)673 4551 y Fe(P)770 4573 y Fm(m)770 4646 y(j)s Fn(=1)894 4539 y Fe(\000)935 4612 y Fh(\000)p Fi(\014)s(N)1101 4553 y Fk(p)p 1177 4553 277 4 v 59 x Fu(2)p Fi(a)1252 4621 y Fm(j)1299 4612 y Fu(ln)13 b Fi(\013)1419 4621 y Fm(j)1453 4612 y Fu(+)p Fi(\014)s(\015)1591 4621 y Fm(j)1626 4612 y Fu([ln\()p Fi(N)19 b Fu(ln)13 b Fi(\013)1932 4621 y Fm(j)1967 4612 y Fu(\)+ln)g(4)p Fi(\031)r Fu(])p Fi(=)p Fu(2+)p Fi(N)7 b(\014)s(a)2503 4585 y Fn(2)2503 4631 y Fm(j)2539 4612 y Fi(=)p Fu(2)p Fh(\000)p Fu(\()p Fi(u)2740 4622 y Fn(ln)k Fm(\013)2846 4637 y(j)2879 4622 y(;N)2959 4612 y Fu(\()p Fi(x)p Fu(\))p Fh(\000)p Fi(\014)3157 4554 y Fk(p)p 3234 4554 138 4 v 3234 4612 a Fi(a)3273 4621 y Fm(j)3307 4612 y Fi(N)c Fu(\))3400 4585 y Fn(2)3435 4612 y Fi(=)p Fu(2)3508 4539 y Fe(\001)230 4907 y Fr(=)25 b(\()361 4825 y Fk(p)p 437 4825 83 4 v 82 x Fj(N)10 b Fr(\))555 4869 y Fh(\000)p Fi(m)702 4884 y Fr(~)681 4907 y Fj(C)d(e)795 4776 y Fe(P)891 4799 y Fm(m)891 4872 y(j)s Fn(=1)1015 4737 y Fe(\020)1069 4838 y Fh(\000)p Fi(u)1169 4810 y Fn(2)1169 4856 y(ln)j Fm(\013)1274 4871 y(j)1308 4856 y(;N)1387 4838 y Fu(\()p Fi(x)p Fu(\))p Fi(=)p Fu(2+)p Fi(\014)s(\015)1695 4847 y Fm(j)1731 4838 y Fi(x)1772 4737 y Fe(\021)1856 4907 y Fk(\024)25 b Fj(C)7 b Fr(\()p Fj(\013)2117 4921 y Fu(1)2173 4907 y Fk(\001)15 b(\001)g(\001)i Fj(\013)2353 4921 y Fi(m)2422 4907 y Fr(\))2457 4869 y Fh(\000)p Fi(N)2583 4907 y Fj(e)2625 4854 y Fi(x)2679 4793 y Fe(P)2775 4815 y Fm(m)2775 4888 y(j)s Fn(=1)2886 4854 y Fu(\()p Fi(\014)s(\015)2996 4863 y Fm(j)3031 4854 y Fh(\000)p Fu(1\))50 5098 y Fr(for)30 b(all)f Fj(N)40 b Fr(large)31 b(enough.)40 b Fk(})50 5316 y Fd(Prop)s(osition)k(2.3:)52 b Fq(Assume)37 b(that)i(the)f(numb)-5 b(ers)38 b Fr(0)d Fj(<)f(a)2195 5330 y Fi(i)2259 5316 y Fj(<)g Fr(1)p Fq(,)39 b Fj(c)2515 5330 y Fi(i)2580 5316 y Fj(<)34 b Fr(0)p Fq(,)39 b Fj(i)34 b Fr(=)g(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)p Fq(,)39 b(ar)-5 b(e)39 b(such)50 5462 y(that)241 5407 y Fk(p)p 317 5407 172 4 v 55 x Fj(a)365 5476 y Fi(i)p Fh(\000)p Fu(1)488 5462 y Fj(c)527 5476 y Fi(i)582 5462 y Fk(\000)677 5403 y(p)p 753 5403 79 4 v 59 x Fj(a)801 5476 y Fi(i)831 5462 y Fj(c)870 5476 y Fi(i)p Fh(\000)p Fu(1)1029 5462 y Fj(>)d Fr(0)i Fq(for)h(al)5 b(l)39 b Fj(i)c Fr(=)h(2)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)p Fq(.)58 b(L)-5 b(et)39 b Fj(c)2309 5476 y Fi(i)2340 5462 y Fr(\()p Fj(N)10 b Fr(\))36 b(=)f Fj(c)2674 5476 y Fi(i)2704 5385 y Fk(p)p 2780 5385 83 4 v 77 x Fj(N)f Fr(+)24 b Fj(o)p Fr(\(1\))40 b Fq(as)f Fj(N)45 b Fk(")36 b(1)i Fq(b)-5 b(e)p eop %%Page: 21 21 21 20 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(21)50 513 y Fq(the)36 b(functions)f(of)h Fj(N)45 b Fq(with)37 b(le)-5 b(ading)36 b(c)-5 b(onstants)37 b Fj(c)1780 527 y Fi(i)1811 513 y Fq(.)50 b(L)-5 b(et)36 b(also)h Fj(X)2313 527 y Fu(1)2354 513 y Fj(;)15 b(:)g(:)g(:)31 b(;)15 b(X)2645 527 y Fi(m)2751 513 y Fq(b)-5 b(e)35 b(indep)-5 b(endent)37 b(standar)-5 b(d)50 658 y(Gaussian)34 b(r)-5 b(andom)35 b(variables.)42 b(Then)33 b(for)h(al)5 b(l)33 b(su\016ciently)g(lar)-5 b(ge)33 b Fj(N)324 907 y Fg(P)379 807 y Fe(\020)434 907 y Fk(8)p Fj(i)25 b Fr(=)g(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)26 b Fr(:)1055 844 y Fk(p)p 1131 844 89 4 v 63 x Fj(a)1179 921 y Fu(1)1220 907 y Fj(X)1295 921 y Fu(1)1356 907 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)1664 844 y Fk(p)p 1740 844 79 4 v 63 x Fj(a)1788 921 y Fi(i)1818 907 y Fj(X)1893 921 y Fi(i)1949 907 y Fj(<)2045 844 y Fk(p)p 2121 844 89 4 v 63 x Fj(a)2169 921 y Fu(1)2210 907 y Fj(c)2249 921 y Fu(1)2290 907 y Fr(\()p Fj(N)10 b Fr(\))21 b(+)e Fk(\001)c(\001)g(\001)22 b Fr(+)2772 844 y Fk(p)p 2847 844 79 4 v 2847 907 a Fj(a)2895 921 y Fi(i)2926 907 y Fj(c)2965 921 y Fi(i)2995 907 y Fr(\()p Fj(N)10 b Fr(\))3148 807 y Fe(\021)349 1126 y Fk(\024)25 b Fr(\()480 1044 y Fk(p)p 556 1044 83 4 v 82 x Fj(N)10 b Fr(\))674 1088 y Fh(\000)p Fi(m)821 1103 y Fr(~)801 1126 y Fj(C)c(e)914 1073 y Fh(\000)984 1012 y Fe(P)1080 1034 y Fm(N)1080 1107 y(j)s Fn(=1)1204 1073 y Fi(c)1237 1045 y Fn(2)1237 1091 y Fm(j)1272 1073 y Fu(\()p Fi(N)h Fu(\))p Fi(=)p Fu(2)3477 1003 y Fr(\(2)p Fj(:)p Fr(13\))50 1368 y Fq(wher)-5 b(e)34 b(the)f(c)-5 b(onstant)841 1345 y Fr(~)820 1368 y Fj(C)39 b Fq(dep)-5 b(ends)34 b(only)g(on)f(the)g(c)-5 b(onstants)34 b Fj(c)2180 1382 y Fu(1)2221 1368 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(c)2477 1382 y Fi(m)2547 1368 y Fq(.)50 1586 y(Pr)-5 b(o)g(of.)65 b Fr(It)38 b(is)f(a)i(rather)f(cum)m(b)s(ersome)f (computation)h(where)g(sev)m(eral)h(induction)c(pro)s(cedures)i(are)i (in-)50 1732 y(v)m(olv)m(ed.)i(It)29 b(is)g(based)g(on)h(the)f(elemen)m (tary)h(estimate)h(for)e(the)h(densit)m(y)f(of)g(standard)g(random)g(v) -5 b(ariables:)50 1877 y(for)30 b(an)m(y)h Fj(x)25 b(>)g Fr(0)1052 1941 y Fe(\020)1121 1980 y Fr(1)p 1117 2021 52 4 v 1117 2104 a Fj(x)1200 2042 y Fk(\000)1326 1980 y Fr(1)p 1302 2021 93 4 v 1302 2104 a Fj(x)1354 2078 y Fu(3)1406 1941 y Fe(\021)1460 2042 y Fj(e)1502 2004 y Fh(\000)p Fi(x)1600 1977 y Fn(2)1636 2004 y Fi(=)p Fu(2)1739 2042 y Fk(\024)1835 1918 y Fe(Z)1926 1940 y Fh(1)1886 2124 y Fi(x)2018 2042 y Fj(e)2060 2004 y Fh(\000)p Fi(t)2144 1977 y Fn(2)2181 2004 y Fi(=)p Fu(2)2284 2042 y Fk(\024)2394 1980 y Fr(1)p 2391 2021 52 4 v 2391 2104 a Fj(x)2454 2042 y(e)2496 2004 y Fh(\000)p Fi(x)2594 1977 y Fn(2)2629 2004 y Fi(=)p Fu(2)3477 2042 y Fr(\(2)p Fj(:)p Fr(14\))159 2347 y(In)32 b(the)g(case)i Fj(m)29 b Fr(=)f(1)33 b(\(2.13\))i(follo)m(ws)d(directly)f(from)h(\(2.14\).)50 b(Assume)32 b(that)h(it)f(is)f(pro)m(v)m(ed)i(to)h Fj(m)21 b Fk(\000)g Fr(1)50 2492 y(and)30 b(let)g(us)g(mak)m(e)h(an)f (induction)e(step)j(to)g Fj(m)p Fr(.)40 b(Denote)32 b(b)m(y)50 2756 y Fj(D)125 2770 y Fi(m)219 2756 y Fr(=)25 b Fk(f)p Fr(\()p Fj(x)447 2770 y Fu(1)489 2756 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(x)758 2770 y Fi(m)827 2756 y Fr(\))26 b Fk(2)f Fg(R)1033 2718 y Fi(m)1134 2756 y Fr(:)45 b Fk(8)p Fj(i)25 b Fr(=)g(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)1770 2693 y Fk(p)p 1846 2693 89 4 v 63 x Fj(a)1894 2770 y Fu(1)1934 2756 y Fj(X)2009 2770 y Fu(1)2051 2756 y Fr(+)p Fk(\001)g(\001)g(\001)p Fr(+)2298 2693 y Fk(p)p 2374 2693 79 4 v 63 x Fj(a)2422 2770 y Fi(i)2452 2756 y Fj(X)2527 2770 y Fi(i)2583 2756 y Fj(<)2679 2693 y Fk(p)p 2755 2693 89 4 v 63 x Fj(a)2803 2770 y Fu(1)2844 2756 y Fj(c)2883 2770 y Fu(1)2924 2756 y Fr(\()p Fj(N)10 b Fr(\)+)p Fk(\001)15 b(\001)g(\001)q Fr(+)3325 2693 y Fk(p)p 3401 2693 79 4 v 63 x Fj(a)3449 2770 y Fi(i)3479 2756 y Fj(c)3518 2770 y Fi(i)3548 2756 y Fr(\()p Fj(N)10 b Fr(\))p Fk(g)p Fj(:)50 3019 y Fr(F)-8 b(or)31 b(an)m(y)g Fj(\016)e(>)c Fr(0)30 b(b)m(y)h(\(2.14\))h(w)m(e)f (ma)m(y)g(write:)65 3274 y Fg(P)120 3173 y Fe(\020)175 3274 y Fk(8)p Fj(i)25 b Fr(=)g(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)26 b Fr(:)796 3211 y Fk(p)p 872 3211 89 4 v 63 x Fj(a)920 3288 y Fu(1)961 3274 y Fj(X)1036 3288 y Fu(1)1097 3274 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)1405 3211 y Fk(p)p 1481 3211 79 4 v 63 x Fj(a)1529 3288 y Fi(i)1559 3274 y Fj(X)1634 3288 y Fi(i)1690 3274 y Fj(<)1786 3211 y Fk(p)p 1862 3211 89 4 v 63 x Fj(a)1910 3288 y Fu(1)1950 3274 y Fj(c)1989 3288 y Fu(1)2031 3274 y Fr(\()p Fj(N)10 b Fr(\))20 b(+)g Fk(\001)15 b(\001)g(\001)22 b Fr(+)2513 3211 y Fk(p)p 2588 3211 79 4 v 2588 3274 a Fj(a)2636 3288 y Fi(i)2667 3274 y Fj(c)2706 3288 y Fi(i)2736 3274 y Fr(\()p Fj(N)10 b Fr(\))2889 3173 y Fe(\021)90 3499 y Fk(\024)330 3377 y(p)p 405 3377 118 4 v 405 3435 a Fj(a)453 3449 y Fi(m)p 197 3478 459 4 v 197 3497 a Fe(p)p 288 3497 324 4 v 77 x Fr(\(2)p Fj(\031)s Fr(\))458 3548 y Fi(m)528 3574 y Fj(N)g(\016)681 3376 y Fe(Z)732 3582 y Fi(D)791 3591 y Fm(m)p Ff(\000)p Fn(1)951 3499 y Fj(e)993 3462 y Fu(\()1021 3420 y Fh(p)p 1082 3420 75 4 v 1082 3462 a Fi(a)1121 3471 y Fn(1)1157 3462 y Fi(c)1190 3471 y Fn(1)1225 3462 y Fu(\()p Fi(N)d Fu(\)+)p Fh(\001\001\001)n Fu(+)1522 3420 y Fh(p)p 1582 3420 100 4 v 42 x Fi(a)1621 3471 y Fm(m)1682 3462 y Fi(c)1715 3471 y Fm(m)1775 3462 y Fu(\()p Fi(N)g Fu(\))p Fh(\000)1953 3420 y(p)p 2012 3420 75 4 v 2012 3462 a Fi(a)2051 3471 y Fn(1)2087 3462 y Fi(x)2128 3471 y Fn(1)2164 3462 y Fh(\000\001\001\001)o(\000)2343 3424 y(p)p 2402 3424 180 4 v 2402 3462 a Fi(a)2441 3471 y Fm(m)p Ff(\000)p Fn(1)2582 3462 y Fi(x)2623 3471 y Fm(m)p Ff(\000)p Fn(1)2764 3462 y Fu(\))2792 3434 y Fn(2)2828 3462 y Fi(=)p Fu(\(2)p Fi(a)2968 3471 y Fm(m)3030 3462 y Fu(\))994 3733 y Fk(\002)20 b Fj(e)1127 3695 y Fh(\000)p Fu(\()p Fi(x)1253 3668 y Fn(2)1253 3713 y Fm(m)p Ff(\000)p Fn(1)1394 3695 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(x)1612 3668 y Fn(2)1612 3713 y(1)1648 3695 y Fu(\))p Fi(=)p Fu(2)1754 3733 y Fj(dx)1853 3747 y Fi(m)p Fh(\000)p Fu(1)2031 3733 y Fj(:)15 b(:)g(:)h(dx)2251 3747 y Fu(1)85 3905 y Fr(+)k Fg(P)231 3804 y Fe(\020)286 3905 y Fk(8)p Fj(i)25 b Fr(=)g(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)21 b Fk(\000)f Fr(1)25 b(:)1064 3842 y Fk(p)p 1139 3842 89 4 v 1139 3905 a Fj(a)1187 3919 y Fu(1)1228 3905 y Fj(X)1303 3919 y Fu(1)1365 3905 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)1673 3842 y Fk(p)p 1748 3842 79 4 v 1748 3905 a Fj(a)1796 3919 y Fi(i)1827 3905 y Fj(X)1902 3919 y Fi(i)1958 3905 y Fj(<)2054 3842 y Fk(p)p 2129 3842 89 4 v 2129 3905 a Fj(a)2177 3919 y Fu(1)2218 3905 y Fj(c)2257 3919 y Fu(1)2298 3905 y Fr(\()p Fj(N)10 b Fr(\))21 b(+)f Fk(\001)15 b(\001)g(\001)21 b Fr(+)2780 3842 y Fk(p)p 2856 3842 79 4 v 63 x Fj(a)2904 3919 y Fi(i)2934 3905 y Fj(c)2973 3919 y Fi(i)3004 3905 y Fr(\()p Fj(N)10 b Fr(\))247 4042 y Fk(p)p 323 4042 89 4 v 63 x Fj(a)371 4119 y Fu(1)411 4105 y Fj(X)486 4119 y Fu(1)548 4105 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)856 4046 y Fk(p)p 932 4046 211 4 v 59 x Fj(a)980 4119 y Fi(m)p Fh(\000)p Fu(1)1142 4105 y Fj(X)1217 4119 y Fi(m)p Fh(\000)p Fu(1)1404 4105 y Fj(<)1500 4042 y Fk(p)p 1576 4042 89 4 v 63 x Fj(a)1624 4119 y Fu(1)1665 4105 y Fj(c)1704 4119 y Fu(1)1745 4105 y Fr(\()p Fj(N)10 b Fr(\))21 b(+)f Fk(\001)15 b(\001)g(\001)21 b Fr(+)2227 4046 y Fk(p)p 2303 4046 211 4 v 59 x Fj(a)2351 4119 y Fi(m)p Fh(\000)p Fu(1)2513 4105 y Fj(c)2552 4119 y Fi(m)p Fh(\000)p Fu(1)2714 4105 y Fr(\()p Fj(N)10 b Fr(\))21 b(+)2979 4042 y Fk(p)p 3055 4042 118 4 v 63 x Fj(a)3103 4119 y Fi(m)3172 4105 y Fj(c)3211 4119 y Fi(m)3280 4105 y Fr(\()p Fj(N)10 b Fr(\))21 b(+)f Fj(\016)3588 4024 y Fk(p)p 3665 4024 83 4 v 3665 4105 a Fj(N)3748 4005 y Fe(\021)90 4279 y Fk(\021)25 b Fj(I)233 4242 y Fu(1)226 4302 y Fi(N)s(;m)397 4279 y Fr(+)20 b Fj(I)535 4242 y Fu(2)528 4302 y Fi(N)s(;m)679 4279 y Fj(:)50 4515 y Fr(Let)34 b(us)f(\014rst)g(sho)m(w)h(that)g(with) e Fj(\016)j(>)30 b Fr(0)35 b(small)d(enough)h Fj(I)2026 4482 y Fu(2)2019 4540 y Fi(N)s(;m)2204 4515 y Fr(is)g(of)h(exp)s(onen)m (tially)e(smaller)g(order)h(than)50 4661 y(the)d(b)s(ound)f(\(2.13\).) 43 b(F)-8 b(or)31 b(this)e(purp)s(ose)g(w)m(e)i(\014nd)d Fj(k)34 b Fr(suc)m(h)c(that)116 4888 y(\()p Fj(a)199 4902 y Fi(m)p Fh(\000)p Fu(1)382 4888 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)738 4902 y Fi(m)p Fh(\000)p Fi(i)p Fu(+1)982 4888 y Fr(\))p Fj(c)1056 4902 y Fi(m)p Fh(\000)p Fi(i)1233 4888 y Fk(\025)1329 4829 y(p)p 1405 4829 200 4 v 59 x Fj(a)1453 4902 y Fi(m)p Fh(\000)p Fi(i)1605 4888 y Fr(\()1640 4829 y Fk(p)p 1716 4829 292 4 v 59 x Fj(a)1764 4902 y Fi(m)p Fh(\000)p Fi(i)p Fu(+1)2008 4888 y Fj(c)2047 4902 y Fi(m)p Fh(\000)p Fi(i)p Fu(+1)2311 4888 y Fr(+)g Fk(\001)15 b(\001)g(\001)21 b Fr(+)2619 4825 y Fk(p)p 2695 4825 118 4 v 63 x Fj(a)2743 4902 y Fi(m)2812 4888 y Fj(c)2851 4902 y Fi(m)2920 4888 y Fr(\))91 b Fk(8)p Fj(i)25 b Fr(=)g(2)p Fj(;)15 b Fr(3)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(k)s Fr(;)86 5061 y(\()p Fj(a)169 5075 y Fi(m)p Fh(\000)p Fu(1)352 5061 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)708 5075 y Fi(m)p Fh(\000)p Fi(k)874 5061 y Fr(\))p Fj(c)948 5075 y Fi(m)p Fh(\000)p Fi(k)r Fh(\000)p Fu(1)1233 5061 y Fj(<)1329 5001 y Fk(p)p 1405 5001 308 4 v 60 x Fj(a)1453 5075 y Fi(m)p Fh(\000)p Fi(k)r Fh(\000)p Fu(1)1712 5061 y Fr(\()1747 5001 y Fk(p)p 1823 5001 215 4 v 60 x Fj(a)1871 5075 y Fi(m)p Fh(\000)p Fi(k)2037 5061 y Fj(c)2076 5075 y Fi(m)p Fh(\000)p Fi(k)2263 5061 y Fr(+)g Fk(\001)15 b(\001)g(\001)21 b Fr(+)2571 4998 y Fk(p)p 2647 4998 118 4 v 63 x Fj(a)2695 5075 y Fi(m)2764 5061 y Fj(c)2803 5075 y Fi(m)2872 5061 y Fr(\))p Fj(:)3477 5170 y Fr(\(2)p Fj(:)p Fr(15\))50 5315 y(\(If)40 b(\(2.15\))j(fails)c (already)h(for)g Fj(i)i Fr(=)g(2)f(w)m(e)f(put)g Fj(k)45 b Fr(=)d(1,)h(if)d(\(2.15\))i(holds)d(up)h(to)h Fj(i)h Fr(=)g Fj(m)26 b Fk(\000)h Fr(1,)44 b(w)m(e)c(put)50 5461 y Fj(k)e Fr(=)c Fj(m)24 b Fk(\000)f Fr(1.\))58 b(Then)35 b(the)i(term)f Fj(I)1277 5428 y Fu(2)1270 5486 y Fi(N)s(;m)1457 5461 y Fr(is)f(b)s(ounded)e(b)m(y)j(the)g(probabilit)m(y)e(of)i(the)g (follo)m(wing)e(ev)m(en)m(t)k(for)p eop %%Page: 22 22 22 21 bop 50 295 a Fo(22)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fr(whic)m(h)i(the)h(estimate)i(\(2.13\))g(applies)d(b)m(y)h (induction:)65 766 y Fg(P)120 665 y Fe(\020)175 766 y Fk(8)p Fj(i)25 b Fr(=)g(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)20 b Fk(\000)g Fj(k)k Fk(\000)c Fr(1)25 b(:)1114 703 y Fk(p)p 1190 703 89 4 v 63 x Fj(a)1238 780 y Fu(1)1279 766 y Fj(X)1354 780 y Fu(1)1415 766 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)1723 703 y Fk(p)p 1799 703 79 4 v 63 x Fj(a)1847 780 y Fi(i)1877 766 y Fj(X)1952 780 y Fi(i)2008 766 y Fj(<)2104 703 y Fk(p)p 2180 703 89 4 v 63 x Fj(a)2228 780 y Fu(1)2268 766 y Fr(\()p Fj(c)2342 780 y Fu(1)2384 766 y Fr(\()p Fj(N)10 b Fr(\))21 b(+)f Fj(\016)2692 684 y Fk(p)p 2768 684 83 4 v 82 x Fj(N)10 b(=)2896 703 y Fk(p)p 2972 703 89 4 v 63 x Fj(a)3020 780 y Fu(1)3061 766 y Fr(\))21 b(+)f Fk(\001)15 b(\001)g(\001)21 b Fr(+)3425 703 y Fk(p)p 3501 703 79 4 v 63 x Fj(a)3549 780 y Fi(i)3579 766 y Fj(c)3618 780 y Fi(i)3649 766 y Fr(\()p Fj(N)10 b Fr(\))247 881 y Fk(p)p 323 881 89 4 v 63 x Fj(a)371 958 y Fu(1)411 944 y Fj(X)486 958 y Fu(1)548 944 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)856 884 y Fk(p)p 932 884 308 4 v 60 x Fj(a)980 958 y Fi(m)p Fh(\000)p Fi(k)r Fh(\000)p Fu(1)1239 944 y Fj(X)1314 958 y Fi(m)p Fh(\000)p Fi(k)r Fh(\000)p Fu(1)1593 944 y Fr(+)1684 868 y Fe(p)p 1775 868 753 4 v 76 x Fj(a)1823 958 y Fi(m)p Fh(\000)p Fi(k)2009 944 y Fr(+)f Fk(\001)15 b(\001)g(\001)22 b Fr(+)e Fj(a)2366 958 y Fi(m)p Fh(\000)p Fu(1)2528 944 y Fj(X)2603 958 y Fi(m)p Fh(\000)p Fi(k)454 1117 y Fj(<)550 1053 y Fk(p)p 626 1053 89 4 v 64 x Fj(a)674 1131 y Fu(1)714 1117 y Fr(\()p Fj(c)788 1131 y Fu(1)830 1117 y Fr(\()p Fj(N)10 b Fr(\))21 b(+)f Fj(\016)1138 1035 y Fk(p)p 1214 1035 83 4 v 82 x Fj(N)10 b(=)1342 1053 y Fk(p)p 1418 1053 89 4 v 64 x Fj(a)1466 1131 y Fu(1)1507 1117 y Fr(\))21 b(+)f Fk(\001)15 b(\001)g(\001)21 b Fr(+)1871 1057 y Fk(p)p 1947 1057 211 4 v 60 x Fj(a)1995 1131 y Fi(m)p Fh(\000)p Fu(1)2157 1117 y Fj(c)2196 1131 y Fi(m)p Fh(\000)p Fu(1)2358 1117 y Fr(\()p Fj(N)10 b Fr(\))21 b(+)2623 1053 y Fk(p)p 2699 1053 118 4 v 64 x Fj(a)2747 1131 y Fi(m)2816 1117 y Fj(c)2855 1131 y Fi(m)2924 1117 y Fr(\()p Fj(N)10 b Fr(\))3077 1016 y Fe(\021)90 1357 y Fk(\024)380 1273 y Fr(~)359 1296 y Fj(C)p 197 1336 396 4 v 197 1431 a Fr(\()232 1355 y Fk(p)p 308 1355 83 4 v 76 x Fj(N)g Fr(\))426 1405 y Fi(m)p Fh(\000)p Fi(k)603 1357 y Fj(e)645 1320 y Fh(\000)p Fu([\()p Fi(c)784 1329 y Fn(1)820 1320 y Fu(\()p Fi(N)d Fu(\)+)p Fi(\016)1032 1265 y Fh(p)p 1092 1265 65 4 v 55 x Fi(N)f(=)1193 1278 y Fh(p)p 1253 1278 75 4 v 42 x Fi(a)1292 1329 y Fn(1)1328 1320 y Fu(\))1356 1292 y Fn(2)1392 1320 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(c)1601 1292 y Fn(2)1601 1338 y Fm(m)p Ff(\000)p Fm(k)q Ff(\000)p Fn(1)1826 1320 y Fu(\()p Fi(N)h Fu(\)])p Fi(=)p Fu(2)631 1585 y Fk(\002)20 b Fj(e)764 1547 y Fh(\000)p Fu([)842 1510 y Fh(p)p 901 1510 185 4 v 901 1547 a Fi(a)940 1557 y Fm(m)p Ff(\000)p Fm(k)1085 1547 y Fi(c)1118 1557 y Fm(m)p Ff(\000)p Fm(k)1263 1547 y Fu(\()p Fi(N)7 b Fu(\)+)p Fh(\001\001\001)n Fu(+)1560 1506 y Fh(p)p 1620 1506 100 4 v 41 x Fi(a)1659 1556 y Fm(m)1720 1547 y Fi(c)1753 1556 y Fm(m)1812 1547 y Fu(\()p Fi(N)g Fu(\)])1954 1520 y Fn(2)1990 1547 y Fi(=)p Fu([2\()p Fi(a)2151 1557 y Fm(m)p Ff(\000)p Fm(k)2296 1547 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(a)2512 1556 y Fm(m)p Ff(\000)p Fn(1)2653 1547 y Fu(\)])2706 1585 y Fj(:)3477 1694 y Fr(\(2)p Fj(:)p Fr(16\))50 1839 y(No)m(w)31 b(it)f(su\016ces)g(to)h(note)g(that)304 2101 y(\()339 2042 y Fk(p)p 415 2042 215 4 v 59 x Fj(a)463 2115 y Fi(m)p Fh(\000)p Fi(k)629 2101 y Fj(c)668 2115 y Fi(m)p Fh(\000)p Fi(k)855 2101 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)1163 2038 y Fk(p)p 1239 2038 118 4 v 63 x Fj(a)1287 2115 y Fi(m)1356 2101 y Fj(c)1395 2115 y Fi(m)1464 2101 y Fr(\))1499 2064 y Fu(2)1566 2101 y Fj(>)k Fr(\()p Fj(a)1745 2115 y Fi(m)p Fh(\000)p Fi(k)1931 2101 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)22 b Fr(+)e Fj(a)2288 2115 y Fi(m)p Fh(\000)p Fu(1)2450 2101 y Fr(\)\()p Fj(c)2559 2064 y Fu(2)2559 2124 y Fi(m)p Fh(\000)p Fi(k)2746 2101 y Fr(+)g Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(c)3093 2064 y Fu(2)3093 2124 y Fi(m)3162 2101 y Fr(\))p Fj(:)255 b Fr(\(2)p Fj(:)p Fr(17\))50 2363 y(This)26 b(is)h(v)m(eri\014ed)g(for) h Fj(k)h Fr(=)c(1)j(from)1255 2309 y Fk(p)p 1330 2309 211 4 v 1330 2363 a Fj(a)1378 2377 y Fi(m)p Fh(\000)p Fu(1)1541 2363 y Fj(c)1580 2377 y Fi(m)1674 2363 y Fj(>)1770 2305 y Fk(p)p 1846 2305 118 4 v 58 x Fj(a)1894 2377 y Fi(m)1963 2363 y Fj(c)2002 2377 y Fi(m)p Fh(\000)p Fu(1)2193 2363 y Fr(and)f(for)h Fj(k)g(>)d Fr(1)k(b)m(y)f(induction)d(from)j Fj(k)19 b Fk(\000)c Fr(1:)325 2600 y(\()360 2541 y Fk(p)p 436 2541 215 4 v 59 x Fj(a)484 2614 y Fi(m)p Fh(\000)p Fi(k)650 2600 y Fj(c)689 2614 y Fi(m)p Fh(\000)p Fi(k)876 2600 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)1184 2537 y Fk(p)p 1260 2537 118 4 v 63 x Fj(a)1308 2614 y Fi(m)1377 2600 y Fj(c)1416 2614 y Fi(m)1485 2600 y Fr(\))1520 2563 y Fu(2)350 2773 y Fj(>)k(a)494 2787 y Fi(m)p Fh(\000)p Fi(k)660 2773 y Fj(c)699 2735 y Fu(2)699 2795 y Fi(m)p Fh(\000)p Fi(k)886 2773 y Fr(+)20 b(\()1012 2714 y Fk(p)p 1088 2714 307 4 v 59 x Fj(a)1136 2787 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)1394 2773 y Fj(c)1433 2787 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)1712 2773 y Fr(+)g Fk(\001)15 b(\001)g(\001)21 b Fr(+)2020 2710 y Fk(p)p 2096 2710 118 4 v 63 x Fj(a)2144 2787 y Fi(m)2213 2773 y Fj(c)2252 2787 y Fi(m)2321 2773 y Fr(\))2356 2735 y Fu(2)861 2946 y Fr(+)f(2)997 2886 y Fk(p)p 1073 2886 215 4 v 60 x Fj(a)1121 2960 y Fi(m)p Fh(\000)p Fi(k)1287 2946 y Fj(c)1326 2960 y Fi(m)p Fh(\000)p Fi(k)1492 2946 y Fr(\()1527 2886 y Fk(p)p 1603 2886 307 4 v 60 x Fj(a)1651 2960 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)1910 2946 y Fj(c)1949 2960 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)2227 2946 y Fr(+)g Fk(\001)15 b(\001)g(\001)21 b Fr(+)2535 2883 y Fk(p)p 2611 2883 118 4 v 63 x Fj(a)2659 2960 y Fi(m)2728 2946 y Fj(c)2767 2960 y Fi(m)2837 2946 y Fr(\))350 3118 y Fj(>)k(a)494 3132 y Fi(m)p Fh(\000)p Fi(k)660 3118 y Fj(c)699 3081 y Fu(2)699 3141 y Fi(m)p Fh(\000)p Fi(k)886 3118 y Fr(+)20 b(\()p Fj(a)1060 3132 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)1339 3118 y Fr(+)f Fk(\001)c(\001)g(\001)22 b Fr(+)e Fj(a)1695 3132 y Fi(m)p Fh(\000)p Fu(1)1857 3118 y Fr(\)\()p Fj(c)1966 3081 y Fu(2)1966 3141 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)2245 3118 y Fr(+)g Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(c)2592 3081 y Fu(2)2592 3141 y Fi(m)2662 3118 y Fr(\))861 3291 y(+)g Fj(c)991 3254 y Fu(2)991 3314 y Fi(m)p Fh(\000)p Fi(k)1157 3291 y Fr(\()p Fj(a)1240 3305 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)1519 3291 y Fr(+)g Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)1875 3305 y Fi(m)p Fh(\000)p Fu(1)2037 3291 y Fr(\))g(+)g Fj(a)2231 3305 y Fi(m)p Fh(\000)p Fi(k)2397 3291 y Fr(\()p Fj(c)2471 3254 y Fu(2)2471 3314 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)2750 3291 y Fr(+)g Fk(\001)15 b(\001)g(\001)22 b Fr(+)e Fj(c)3098 3254 y Fu(2)3098 3314 y Fi(m)3167 3291 y Fr(\))350 3464 y(=)25 b(\()p Fj(a)529 3478 y Fi(m)p Fh(\000)p Fi(k)716 3464 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)1072 3478 y Fi(m)p Fh(\000)p Fu(1)1234 3464 y Fr(\)\()p Fj(c)1343 3426 y Fu(2)1343 3486 y Fi(m)p Fh(\000)p Fi(k)1530 3464 y Fr(+)g Fk(\001)15 b(\001)g(\001)22 b Fr(+)d Fj(c)1877 3426 y Fu(2)1877 3486 y Fi(m)1947 3464 y Fr(\))3477 3031 y(\(2)p Fj(:)p Fr(18\))50 3690 y(where)31 b(w)m(e)h(used)f(\(2.15\))j(with)c Fj(i)e Fr(=)f Fj(k)35 b Fr(and)c(the)h(assumptions)e(on)i Fj(c)2390 3704 y Fi(i)2452 3690 y Fr(and)f Fj(a)2678 3704 y Fi(i)2740 3690 y Fr(of)h(the)g(prop)s(osition.)42 b(Note)50 3835 y(also)33 b(that)g(b)m(y)g(\(2.17\))i(and)d(\(2.18\))k(the)d(c)m(hoice) g(of)g(an)g(appropriate)f Fj(\016)i(>)29 b Fr(0)k(in)f(\(2.16\))j(dep)s (ends)c(only)h(on)50 3981 y Fj(c)89 3995 y Fi(i)119 3981 y Fr(,)f Fj(a)223 3995 y Fi(i)284 3981 y Fr(for)f Fj(i)25 b Fr(=)g Fj(k)s(;)15 b(:)g(:)g(:)32 b(;)15 b(m)p Fr(.)159 4199 y(Next,)29 b(let)f(us)f(concen)m(trate)k(on)c(the)i(analysis)d(of) i Fj(I)1898 4166 y Fu(1)1891 4224 y Fi(N)s(;m)2042 4199 y Fr(.)40 b(W)-8 b(e)29 b(start)g(another)f(induction)d(pro)s(cedure)i (to)50 4344 y(pro)m(v)m(e)i(that)g(it)f(ob)s(eys)g(the)h(b)s(ound)d (\(2.13\))31 b(under)c(assumptions)f(of)j(the)f(prop)s(osition.)38 b(It)29 b(is)e(not)i(di\016cult)50 4490 y(to)f(c)m(hec)m(k)h(this)d (for)h Fj(m)e Fr(=)g(2)i(and)g(w)m(e)h(skip)d(it.)40 b(Assume)26 b(that)i(this)e(holds)g(true)h(up)f(to)i Fj(m)14 b Fk(\000)g Fr(1)27 b(and)g(consider)50 4635 y(the)j(case)i(of)e Fj(m)p Fr(.)41 b(Let)31 b(us)e(write)h Fj(I)1207 4602 y Fu(1)1200 4660 y Fi(N)s(;m)1381 4635 y Fr(as)281 4952 y Fj(I)328 4915 y Fu(1)321 4975 y Fi(N)s(;m)498 4952 y Fr(=)737 4829 y Fk(p)p 813 4829 V 59 x Fj(a)861 4902 y Fi(m)p 605 4931 459 4 v 605 4949 a Fe(p)p 696 4949 324 4 v 78 x Fr(\(2)p Fj(\031)s Fr(\))866 5001 y Fi(m)936 5027 y Fj(N)10 b(\016)1089 4828 y Fe(Z)1139 5035 y Fi(D)1198 5044 y Fm(m)p Ff(\000)p Fn(2)1359 4952 y Fj(e)1412 4846 y Fn([)1430 4819 y Ff(p)p 1480 4819 71 3 v 27 x Fm(a)1515 4860 y Fn(1)1551 4846 y Fm(c)1581 4860 y Fn(1)1616 4846 y(\()p Fm(N)5 b Fn(\)+)p Ff(\001\001\001)n Fn(+)1875 4817 y Ff(p)p 1925 4817 96 3 v 29 x Fm(a)1960 4855 y(m)2020 4846 y(c)2050 4855 y(m)2110 4846 y Fn(\()p Fm(N)g Fn(\))p Ff(\000)2263 4819 y(p)p 2313 4819 71 3 v 27 x Fm(a)2348 4860 y Fn(1)2383 4846 y Fm(x)2419 4860 y Fn(1)2455 4846 y Ff(\000\001\001\001)o(\000)2615 4815 y Fh(p)p 2675 4815 176 4 v 31 x Fm(a)2710 4861 y(m)p Ff(\000)p Fn(2)2851 4846 y Fm(x)2887 4861 y(m)p Ff(\000)p Fn(2)3028 4846 y(])3046 4827 y(2)p 1412 4882 1670 4 v 2048 4920 a(2\()p Fm(a)2139 4935 y(m)p Ff(\000)p Fn(1)2279 4920 y(+)p Fm(a)2361 4929 y(m)2421 4920 y Fn(\))1574 5186 y Fk(\002)20 b Fj(e)1707 5148 y Fh(\000)p Fu(\()p Fi(x)1833 5121 y Fn(2)1833 5166 y Fm(m)p Ff(\000)p Fn(2)1975 5148 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(x)2192 5121 y Fn(2)2192 5166 y(1)2228 5148 y Fu(\))p Fi(=)p Fu(2)2334 5186 y Fj(dx)2433 5200 y Fi(m)p Fh(\000)p Fu(2)2611 5186 y Fj(:)15 b(:)g(:)h(dx)2831 5200 y Fu(1)297 5284 y Fe(Z)387 5306 y Fi(E)440 5315 y Fm(m)p Ff(\000)p Fn(1)347 5491 y Fh(\0001)601 5408 y Fj(e)654 5301 y Fn([\()p Fm(a)732 5316 y(m)p Ff(\000)p Fn(1)873 5301 y(+)p Fm(a)955 5310 y(m)1015 5301 y Fn(\))p Fm(x)1076 5316 y(m)p Ff(\000)p Fn(1)1216 5301 y Ff(\000)1265 5270 y Fh(p)p 1325 5270 176 4 v 31 x Fm(a)1360 5316 y(m)p Ff(\000)p Fn(1)1501 5301 y(\()1526 5275 y Ff(p)p 1576 5275 71 3 v 26 x Fm(a)1611 5315 y Fn(1)1646 5301 y Fm(c)1676 5315 y Fn(1)1711 5301 y(\()p Fm(N)5 b Fn(\)+)p Ff(\001\001\001)o Fn(+)1971 5272 y Ff(p)p 2020 5272 96 3 v 2020 5301 a Fm(a)2055 5310 y(m)2116 5301 y(c)2146 5310 y(m)2206 5301 y Fn(\()p Fm(N)g Fn(\))p Ff(\000)2359 5275 y(p)p 2408 5275 71 3 v 2408 5301 a Fm(a)2443 5315 y Fn(1)2479 5301 y Fm(x)2515 5315 y Fn(1)2550 5301 y Ff(\000\001\001\001\000)2711 5270 y Fh(p)p 2771 5270 176 4 v 31 x Fm(a)2806 5316 y(m)p Ff(\000)p Fn(2)2946 5301 y Fm(x)2982 5316 y(m)p Ff(\000)p Fn(2)3123 5301 y(\)])3166 5283 y(2)p 654 5338 2548 4 v 1681 5376 a(2)p Fm(a)1747 5385 y(m)1807 5376 y Fn(\()p Fm(a)1867 5391 y(m)p Ff(\000)p Fn(1)2008 5376 y(+)p Fm(a)2090 5385 y(m)2150 5376 y Fn(\))3217 5408 y Fj(dx)3316 5422 y Fi(m)p Fh(\000)p Fu(1)p eop %%Page: 23 23 23 22 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(23)50 513 y Fr(with)29 b(the)h(notation)209 798 y Fj(E)276 812 y Fi(m)p Fh(\000)p Fu(1)463 798 y Fk(\021)25 b Fr(\()594 735 y Fk(p)p 670 735 89 4 v 63 x Fj(a)718 812 y Fu(1)759 798 y Fj(c)798 812 y Fu(1)839 798 y Fr(\()p Fj(N)10 b Fr(\))21 b(+)f Fk(\001)15 b(\001)g(\001)21 b Fr(+)1321 739 y Fk(p)p 1397 739 211 4 v 59 x Fj(a)1445 812 y Fi(m)p Fh(\000)p Fu(1)1607 798 y Fj(c)1646 812 y Fi(m)p Fh(\000)p Fu(1)1809 798 y Fr(\()p Fj(N)10 b Fr(\))20 b Fk(\000)2073 735 y(p)p 2149 735 89 4 v 63 x Fj(a)2197 812 y Fu(1)2238 798 y Fj(x)2290 812 y Fu(1)2351 798 y Fk(\000)g(\001)15 b(\001)g(\001)21 b(\000)2659 739 y(p)p 2735 739 211 4 v 59 x Fj(a)2783 812 y Fi(m)p Fh(\000)p Fu(2)2945 798 y Fj(x)2997 812 y Fi(m)p Fh(\000)p Fu(2)3159 798 y Fr(\))p Fj(=)3239 739 y Fk(p)p 3315 739 V 59 x Fj(a)3363 812 y Fi(m)p Fh(\000)p Fu(1)3526 798 y Fj(:)50 1084 y Fr(Whenev)m(er)469 1025 y Fk(p)p 545 1025 89 4 v 59 x Fj(a)593 1098 y Fu(1)633 1084 y Fj(x)685 1098 y Fu(1)730 1084 y Fr(+)t Fk(\001)15 b(\001)g(\001)5 b Fr(+)990 1029 y Fk(p)p 1066 1029 211 4 v 55 x Fj(a)1114 1098 y Fi(m)p Fh(\000)p Fu(2)1276 1084 y Fj(x)1328 1098 y Fi(m)p Fh(\000)p Fu(2)1515 1084 y Fj(>)1611 1025 y Fk(p)p 1687 1025 89 4 v 59 x Fj(a)1735 1098 y Fu(1)1776 1084 y Fj(c)1815 1098 y Fu(1)1856 1084 y Fr(\()p Fj(N)10 b Fr(\))t(+)t Fk(\001)15 b(\001)g(\001)6 b Fr(+)2274 1029 y Fk(p)p 2350 1029 211 4 v 55 x Fj(a)2398 1098 y Fi(m)p Fh(\000)p Fu(2)2560 1084 y Fj(c)2599 1098 y Fi(m)p Fh(\000)p Fu(2)2761 1084 y Fr(\()p Fj(N)k Fr(\))t(+)2993 1029 y Fk(p)p 3070 1029 V 3070 1084 a Fj(a)3118 1098 y Fi(m)p Fh(\000)p Fu(1)3280 1084 y Fj(c)3319 1098 y Fi(m)p Fh(\000)p Fu(1)3481 1084 y Fr(\()p Fj(N)g Fr(\))t Fk(\000)50 1229 y Fj(a)98 1243 y Fi(m)p Fh(\000)p Fu(1)260 1229 y Fr(\()p Fj(c)334 1243 y Fi(m)404 1229 y Fr(\()p Fj(N)g Fr(\))18 b Fk(\000)g Fj(\016)707 1196 y Fh(0)733 1152 y Fk(p)p 809 1152 83 4 v 77 x Fj(N)10 b Fr(\))p Fj(=)972 1171 y Fk(p)p 1048 1171 118 4 v 58 x Fj(a)1096 1243 y Fi(m)1194 1229 y Fr(for)29 b(an)m(y)h Fj(\016)1546 1196 y Fh(0)1597 1229 y Fj(>)25 b Fr(0,)30 b(w)m(e)g(are)g(allo)m(w)m (ed)f(to)g(apply)f(to)i(the)f(inner)f(in)m(tegral)h(the)50 1375 y(estimate)i(\(2.14\),)i(otherwise)d(let)g(us)g(just)f(estimate)j (it)d(b)m(y)2116 1300 y Fk(p)p 2191 1300 101 4 v 2191 1375 a Fr(2)p Fj(\031)t Fr(:)90 1704 y Fj(I)137 1667 y Fu(1)130 1727 y Fi(N)s(;m)306 1704 y Fk(\024)529 1582 y(p)p 605 1582 328 4 v 55 x Fj(a)653 1651 y Fi(m)722 1637 y Fj(a)770 1651 y Fi(m)p Fh(\000)p Fu(1)932 1637 y Fj(e)974 1604 y Fh(\000)p Fi(c)1064 1576 y Fn(2)1064 1622 y Fm(m)1124 1604 y Fu(\()p Fi(N)7 b Fu(\))p Fi(=)p Fu(2)p 388 1684 1078 4 v 388 1702 a Fe(p)p 479 1702 874 4 v 77 x Fr(\(2)p Fj(\031)s Fr(\))649 1753 y Fi(m)719 1779 y Fj(N)802 1753 y Fu(2)843 1779 y Fr(\()p Fj(a)926 1793 y Fi(m)p Fh(\000)p Fu(1)1109 1779 y Fr(+)20 b Fj(a)1248 1793 y Fi(m)1317 1779 y Fr(\))p Fj(\016)s(\016)1438 1753 y Fh(0)1491 1581 y Fe(Z)1542 1787 y Fi(D)1601 1796 y Fm(m)p Ff(\000)p Fn(2)1761 1704 y Fj(e)1814 1600 y Fn([)1832 1573 y Ff(p)p 1883 1573 71 3 v 1883 1600 a Fm(a)1918 1614 y Fn(1)1954 1600 y Fm(c)1984 1614 y Fn(1)2019 1600 y(\()p Fm(N)5 b Fn(\)+)p Ff(\001\001\001)n Fn(+)2278 1569 y Fh(p)p 2337 1569 176 4 v 2337 1600 a Fm(a)2372 1615 y(m)p Ff(\000)p Fn(1)2513 1600 y Fm(c)2543 1615 y(m)p Ff(\000)p Fn(1)2683 1600 y(\()p Fm(N)g Fn(\))p Ff(\000)2836 1573 y(p)p 2885 1573 71 3 v 2885 1600 a Fm(a)2920 1614 y Fn(1)2956 1600 y Fm(x)2992 1614 y Fn(1)3027 1600 y Ff(\000\001\001\001\000)3188 1569 y Fh(p)p 3248 1569 176 4 v 31 x Fm(a)3283 1615 y(m)p Ff(\000)p Fn(2)3424 1600 y Fm(x)3460 1615 y(m)p Ff(\000)p Fn(2)3600 1600 y(])3618 1582 y(2)p 1815 1636 1840 4 v 2631 1672 a(2)p Fm(a)2697 1687 y(m)p Ff(\000)p Fn(1)1670 1938 y Fk(\002)20 b Fj(e)1803 1900 y Fh(\000)p Fu(\()p Fi(x)1929 1873 y Fn(2)1929 1919 y Fm(m)p Ff(\000)p Fn(2)2070 1900 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(x)2288 1873 y Fn(2)2288 1919 y(1)2323 1900 y Fu(\))p Fi(=)p Fu(2)2430 1938 y Fj(dx)2529 1952 y Fi(m)p Fh(\000)p Fu(2)2706 1938 y Fj(:)15 b(:)g(:)h(dx)2926 1952 y Fu(1)397 2160 y Fr(+)678 2037 y Fk(p)p 754 2037 118 4 v 59 x Fj(a)802 2110 y Fi(m)p 499 2139 552 4 v 499 2157 a Fe(p)p 590 2157 417 4 v 78 x Fr(\(2)p Fj(\031)s Fr(\))760 2209 y Fi(m)p Fh(\000)p Fu(1)923 2235 y Fj(N)10 b(\016)1076 2036 y Fe(Z)1127 2243 y Fi(D)1186 2252 y Fm(m)p Ff(\000)p Fn(3)1327 2243 y Fh(\\)p Fi(G)1433 2221 y Fm(\016)1464 2208 y Ff(0)1433 2265 y Fm(m;)p Fn(2)1564 2160 y Fj(e)1617 2054 y Fn([)1635 2027 y Ff(p)p 1685 2027 71 3 v 27 x Fm(a)1720 2068 y Fn(1)1756 2054 y Fm(c)1786 2068 y Fn(1)1821 2054 y(\()p Fm(N)5 b Fn(\)+)p Ff(\001\001\001)n Fn(+)2080 2025 y Ff(p)p 2130 2025 96 3 v 29 x Fm(a)2165 2063 y(m)2226 2054 y(c)2256 2063 y(m)2316 2054 y Fn(\()p Fm(N)g Fn(\))p Ff(\000)2469 2027 y(p)p 2518 2027 71 3 v 2518 2054 a Fm(a)2553 2068 y Fn(1)2588 2054 y Fm(x)2624 2068 y Fn(1)2660 2054 y Ff(\000\001\001\001\000)2821 2022 y Fh(p)p 2881 2022 176 4 v 32 x Fm(a)2916 2069 y(m)p Ff(\000)p Fn(2)3056 2054 y Fm(x)3092 2069 y(m)p Ff(\000)p Fn(2)3233 2054 y(])3251 2035 y(2)p 1617 2090 1670 4 v 2253 2128 a(2\()p Fm(a)2344 2143 y(m)p Ff(\000)p Fn(1)2484 2128 y(+)p Fm(a)2566 2137 y(m)2626 2128 y Fn(\))1670 2404 y Fk(\002)20 b Fj(e)1803 2367 y Fh(\000)p Fu(\()p Fi(x)1929 2339 y Fn(2)1929 2385 y Fm(m)p Ff(\000)p Fn(2)2070 2367 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(x)2288 2339 y Fn(2)2288 2385 y(1)2323 2367 y Fu(\))p Fi(=)p Fu(2)2430 2404 y Fj(dx)2529 2418 y Fi(m)p Fh(\000)p Fu(2)2706 2404 y Fj(:)15 b(:)g(:)h(dx)2926 2418 y Fu(1)402 2577 y Fk(\021)25 b Fj(J)557 2540 y Fu(1)548 2600 y Fi(N)s(;m)720 2577 y Fr(+)20 b Fj(J)870 2540 y Fu(2)861 2600 y Fi(N)s(;m)3477 2689 y Fr(\(2)p Fj(:)p Fr(19\))50 2835 y(where)65 3088 y Fj(G)137 3051 y Fi(\016)172 3023 y Ff(0)137 3111 y Fi(m;)p Fu(2)289 3088 y Fr(=)385 2988 y Fe(n)445 3088 y Fr(\()p Fj(x)532 3102 y Fu(1)573 3088 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(x)842 3102 y Fi(m)911 3088 y Fr(\))26 b(:)1023 3025 y Fk(p)p 1098 3025 89 4 v 1098 3088 a Fj(a)1146 3102 y Fu(1)1187 3088 y Fj(x)1239 3102 y Fu(1)1300 3088 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)1608 3025 y Fk(p)p 1684 3025 V 63 x Fj(a)1732 3102 y Fu(2)1773 3088 y Fj(x)1825 3102 y Fi(m)p Fh(\000)p Fu(2)379 3289 y Fj(<)475 3225 y Fk(p)p 551 3225 V 64 x Fj(a)599 3303 y Fu(1)640 3289 y Fj(c)679 3303 y Fu(1)720 3289 y Fr(\()p Fj(N)10 b Fr(\))21 b(+)f Fk(\001)15 b(\001)g(\001)21 b Fr(+)1202 3229 y Fk(p)p 1278 3229 211 4 v 60 x Fj(a)1326 3303 y Fi(m)p Fh(\000)p Fu(2)1488 3289 y Fj(c)1527 3303 y Fi(m)p Fh(\000)p Fu(2)1689 3289 y Fr(\()p Fj(N)10 b Fr(\))21 b(+)1954 3229 y Fk(p)p 2030 3229 V 60 x Fj(a)2078 3303 y Fi(m)p Fh(\000)p Fu(1)2240 3289 y Fj(c)2279 3303 y Fi(m)p Fh(\000)p Fu(1)2442 3289 y Fr(\()p Fj(N)10 b Fr(\))20 b Fk(\000)g Fj(a)2754 3303 y Fi(m)p Fh(\000)p Fu(1)2917 3289 y Fj(c)2956 3303 y Fi(m)3025 3289 y Fr(\()p Fj(N)10 b Fr(\))p Fj(=)3223 3225 y Fk(p)p 3300 3225 118 4 v 3300 3289 a Fj(a)3348 3303 y Fi(m)3437 3289 y Fr(+)20 b Fj(\016)3571 3251 y Fh(0)3597 3207 y Fk(p)p 3673 3207 83 4 v 82 x Fj(N)3756 3188 y Fe(o)3817 3289 y Fj(:)50 3571 y Fr(The)25 b(term)i Fj(J)505 3538 y Fu(1)496 3596 y Fi(N)s(;m)673 3571 y Fr(has)f(the)h(form)e(of)i Fj(I)1341 3538 y Fu(1)1334 3596 y Fi(N)s(;m)p Fh(\000)p Fu(1)1604 3571 y Fr(with)e(parameters)h Fj(a)2321 3585 y Fi(i)2352 3571 y Fr(,)h Fj(c)2443 3585 y Fi(i)2473 3571 y Fr(,)p Fj(i)f Fr(=)f(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)d Fk(\000)g Fr(1.)40 b(By)26 b(induction)50 3716 y(on)g Fj(I)219 3683 y Fu(1)212 3741 y Fi(N)s(;m)389 3716 y Fr(it)g(ob)s(eys)g(the)g(b)s(ound)e(claimed)h(in)g(\(2.13\).)42 b(In)25 b(fact,)j(this)d(term)h(brings)f(the)h(main)f(con)m(tribution) 50 3862 y(to)30 b Fj(I)207 3829 y Fu(1)200 3887 y Fi(N)s(;m)380 3862 y Fr(and)e(to)i(the)f(probabilit)m(y)e(of)i(\(2.13\).)42 b(W)-8 b(e)31 b(will)26 b(sho)m(w)j(under)e(assumptions)h(of)h(the)g (prop)s(osition)50 4007 y(that)38 b(with)e(an)h(appropriate)g(c)m (hoice)h(of)g Fj(\016)1530 3974 y Fh(0)1593 4007 y Fj(>)f Fr(0)g(dep)s(ending)e(only)i(on)g Fj(a)2615 4021 y Fi(i)2683 4007 y Fr(and)g Fj(c)2906 4021 y Fi(i)2936 4007 y Fr(,)j Fj(i)d Fr(=)g(2)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)p Fr(,)40 b(the)50 4153 y(con)m(tribution)d(of)j Fj(J)748 4120 y Fu(2)739 4178 y Fi(N)s(;m)929 4153 y Fr(is)e(of)h(exp)s(onen)m (tially)f(smaller)f(order)i(than)g(the)g(b)s(ound)e(of)i(\(2.13\).)69 b(Observ)m(e)50 4298 y(that)37 b(with)f(notations)g Fj(a)920 4265 y Fh(0)920 4322 y Fi(i)987 4298 y Fr(=)f Fj(a)1141 4312 y Fi(i)1171 4298 y Fr(,)k Fj(c)1274 4265 y Fh(0)1274 4322 y Fi(i)1340 4298 y Fr(=)d Fj(c)1486 4312 y Fi(i)1553 4298 y Fr(for)h Fj(i)f Fr(=)f(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)25 b Fk(\000)f Fr(3,)39 b Fj(a)2491 4265 y Fh(0)2491 4321 y Fi(m)p Fh(\000)p Fu(2)2689 4298 y Fr(=)d Fj(a)2844 4312 y Fi(m)p Fh(\000)p Fu(2)3006 4298 y Fr(,)i Fj(c)3108 4265 y Fh(0)3108 4321 y Fi(m)p Fh(\000)p Fu(2)3307 4298 y Fr(=)d Fj(c)3452 4312 y Fi(m)p Fh(\000)p Fu(2)3639 4298 y Fr(+)50 4444 y(\()85 4389 y Fk(p)p 161 4389 211 4 v 55 x Fj(a)209 4458 y Fi(m)p Fh(\000)p Fu(1)371 4444 y Fj(c)410 4458 y Fi(m)p Fh(\000)p Fu(1)593 4444 y Fk(\000)21 b Fj(a)733 4458 y Fi(m)p Fh(\000)p Fu(1)895 4444 y Fj(c)934 4458 y Fi(m)1004 4444 y Fj(=)1049 4385 y Fk(p)p 1125 4385 118 4 v 59 x Fj(a)1173 4458 y Fi(m)1242 4444 y Fr(\))p Fj(=)1322 4389 y Fk(p)p 1399 4389 211 4 v 1399 4444 a Fj(a)1447 4458 y Fi(m)p Fh(\000)p Fu(2)1609 4444 y Fr(,)32 b Fj(a)1714 4411 y Fh(0)1714 4466 y Fi(m)p Fh(\000)p Fu(1)1903 4444 y Fr(=)27 b Fj(a)2049 4458 y Fi(m)p Fh(\000)p Fu(1)2232 4444 y Fr(+)20 b Fj(a)2371 4458 y Fi(m)2440 4444 y Fr(,)32 b Fj(c)2536 4411 y Fh(0)2536 4466 y Fi(m)p Fh(\000)p Fu(1)2726 4444 y Fr(=)2824 4382 y Fk(p)p 2899 4382 439 4 v 2899 4444 a Fj(a)2947 4458 y Fi(m)p Fh(\000)p Fu(1)3130 4444 y Fr(+)20 b Fj(a)3269 4458 y Fi(m)3338 4444 y Fj(c)3377 4458 y Fi(m)3446 4444 y Fj(=)3491 4385 y Fk(p)p 3567 4385 118 4 v 59 x Fj(a)3615 4458 y Fi(m)3685 4444 y Fr(,)50 4589 y Fj(J)109 4556 y Fu(2)100 4614 y Fi(N)s(;m)293 4589 y Fr(tak)m(es)44 b(the)e(form)g(of)g Fj(I)1095 4556 y Fu(1)1088 4614 y Fi(N)s(;m)p Fh(\000)p Fu(1)1374 4589 y Fr(as)h(w)m(ell.)75 b(Had)42 b(the)g(assumptions)2656 4513 y Fe(p)p 2747 4513 79 4 v 76 x Fj(a)2795 4558 y Fh(0)2795 4614 y Fi(i)2825 4589 y Fj(c)2864 4556 y Fh(0)2864 4613 y Fi(i)p Fh(\000)p Fu(1)3016 4589 y Fk(\000)3114 4513 y Fe(p)p 3205 4513 V 76 x Fj(a)3253 4558 y Fh(0)3253 4614 y Fi(i)3284 4589 y Fj(c)3323 4556 y Fh(0)3323 4613 y Fi(i)3398 4589 y Fj(<)i Fr(0)f(for)50 4735 y Fj(i)25 b Fr(=)g(2)p Fj(;)15 b(:)g(:)g(:)33 b(;)15 b(m)k Fk(\000)f Fr(1)30 b(b)s(een)f(v)m(eri\014ed,)g(then)h(b)m(y)f(induction)f(on)h Fj(I)2193 4702 y Fu(1)2186 4760 y Fi(N)s(;m)2367 4735 y Fr(this)f(term)i(could)f(b)s(e)g(estimated)h(b)m(y)f(a)50 4880 y(go)s(o)s(d)24 b(b)s(ound)d(upp)s(er)h Fj(e)835 4844 y Fh(\000)p Fu(\()p Fi(c)953 4817 y Ff(0)978 4814 y Fn(2)978 4863 y(1)1013 4844 y Fu(\()p Fi(N)7 b Fu(\)+)p Fh(\001\001\001)o Fu(+)p Fi(c)1344 4817 y Ff(0)1368 4814 y Fn(2)1368 4863 y Fm(m)p Ff(\000)p Fn(1)1509 4844 y Fu(\()p Fi(N)g Fu(\)\))p Fi(=)p Fu(2)1760 4880 y Fr(as)24 b Fj(c)1904 4847 y Fh(0)1930 4834 y Fu(2)1930 4902 y(1)1978 4880 y Fr(+)7 b Fk(\001)15 b(\001)g(\001)9 b Fr(+)e Fj(c)2287 4847 y Fh(0)2313 4834 y Fu(2)2313 4902 y Fi(m)p Fh(\000)p Fu(1)2500 4880 y Fj(>)25 b(c)2635 4847 y Fu(2)2635 4903 y(1)2683 4880 y Fr(+)7 b Fk(\001)15 b(\001)g(\001)9 b Fr(+)e Fj(c)2992 4847 y Fu(2)2992 4902 y Fi(m)3085 4880 y Fr(and)23 b(the)h(pro)s(of)g(of)50 5025 y(the)h(prop)s(osition)e(w)m (ould)i(ha)m(v)m(e)h(b)s(een)f(\014nished.)36 b(Unfortunately)25 b(the)g(conditions)2881 4950 y Fe(p)p 2972 4950 V 75 x Fj(a)3020 4994 y Fh(0)3020 5051 y Fi(i)3051 5025 y Fj(c)3090 4992 y Fh(0)3090 5049 y Fi(i)p Fh(\000)p Fu(1)3223 5025 y Fk(\000)3304 4950 y Fe(p)p 3395 4950 V 75 x Fj(a)3443 4994 y Fh(0)3443 5051 y Fi(i)3474 5025 y Fj(c)3513 4992 y Fh(0)3513 5049 y Fi(i)3568 5025 y Fj(<)g Fr(0)50 5171 y(are)39 b(v)m(eri\014ed)g(for)g(all)f Fj(i)i Fr(=)g(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)27 b Fk(\000)e Fr(1)40 b(but)f(not)g(necessarily)f(for)h Fj(i)i Fr(=)e Fj(m)26 b Fk(\000)g Fr(2.)68 b(If)39 b(for)g Fj(i)h Fr(=)g Fj(m)26 b Fk(\000)f Fr(2)50 5316 y(this)31 b(inequalit)m(y)f(fails,)h(then)h(w) m(e)h(are)f(obliged)f(to)i(con)m(tin)m(ue)f(the)g(pro)s(of.)45 b(This)30 b(leads)i(us)f(to)i(sho)m(wing)e(b)m(y)50 5462 y(induction)h(subsequen)m(tly)g(for)i Fj(k)h Fr(=)c Fj(m)23 b Fk(\000)f Fr(1)p Fj(;)15 b(:)g(:)g(:)33 b(;)15 b Fr(2)35 b(that)f(under)f(assumptions)2849 5403 y Fk(p)p 2924 5403 V 2924 5462 a Fj(a)2972 5476 y Fi(i)3003 5462 y Fj(c)3042 5476 y Fi(i)p Fh(\000)p Fu(1)3188 5462 y Fk(\000)3281 5403 y(p)p 3357 5403 V 59 x Fj(a)3405 5476 y Fi(i)3436 5462 y Fj(c)3475 5476 y Fi(i)3537 5462 y Fj(<)e Fr(0,)p eop %%Page: 24 24 24 23 bop 50 295 a Fo(24)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fj(i)e Fr(=)g(2)p Fj(;)15 b(:)g(:)g(:)33 b(;)15 b(m)p Fr(,)30 b(for)g(all)g Fj(N)40 b Fr(large)30 b(enough)533 821 y Fj(L)595 783 y Fi(k)595 843 y(N)s(;m)771 821 y Fr(=)857 697 y Fe(Z)907 903 y Fi(D)966 913 y Fm(m)p Ff(\000)p Fm(k)q Ff(\000)p Fn(1)1192 903 y Fh(\\)p Fi(G)1298 913 y Fm(m;k)1433 821 y Fj(e)1486 712 y Fn([)1504 686 y Ff(p)p 1555 686 71 3 v 1555 712 a Fm(a)1590 726 y Fn(1)1625 712 y Fm(c)1655 726 y Fn(1)1690 712 y(\()p Fm(N)5 b Fn(\)+)p Ff(\001\001\001)o Fn(+)1950 683 y Ff(p)p 1999 683 96 3 v 1999 712 a Fm(a)2034 721 y(m)2095 712 y(c)2125 721 y(m)2185 712 y Fn(\()p Fm(N)g Fn(\))p Ff(\000)2338 686 y(p)p 2387 686 71 3 v 2387 712 a Fm(a)2422 726 y Fn(1)2458 712 y Fm(x)2494 726 y Fn(1)2529 712 y Ff(\000\001\001\001\000)2690 682 y Fh(p)p 2750 682 181 4 v 30 x Fm(a)2785 728 y(m)p Ff(\000)p Fm(k)2930 712 y(x)2966 728 y(m)p Ff(\000)p Fm(k)3111 712 y Fn(])3129 694 y(2)p 1487 750 1678 4 v 1991 787 a(2\()p Fm(a)2082 803 y(m)p Ff(\000)p Fm(k)q Fn(+1)2304 787 y(+)p Ff(\001\001\001)o Fn(+)p Fm(a)2495 802 y(m)p Ff(\000)p Fn(1)2636 787 y(\))1387 1053 y Fj(e)1429 1015 y Fh(\000)p Fu(\()p Fi(x)1555 988 y Fn(2)1555 1034 y Fm(m)p Ff(\000)p Fm(k)1701 1015 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(x)1918 988 y Fn(2)1918 1033 y(1)1954 1015 y Fu(\))p Fi(=)p Fu(2)2060 1053 y Fj(dx)2159 1067 y Fi(m)p Fh(\000)p Fi(k)2341 1053 y Fj(:)15 b(:)g(:)h(dx)2561 1067 y Fu(1)2627 1053 y Fk(\024)25 b Fj(e)2765 1015 y Fh(\000)p Fi(B)s(N)7 b Fu(\(1+)p Fi(o)p Fu(\(1\)\))3477 897 y Fr(\(2)p Fj(:)p Fr(20\))50 1287 y(where)20 b Fj(G)375 1301 y Fi(m;k)530 1287 y Fr(=)25 b Fk(f)p Fr(\()p Fj(x)758 1301 y Fu(1)800 1287 y Fj(;)15 b(:)g(:)g(:)31 b(;)15 b(x)1068 1301 y Fi(m)1138 1287 y Fr(\))25 b(:)1249 1228 y Fk(p)p 1325 1228 89 4 v 59 x Fj(a)1373 1301 y Fu(1)1414 1287 y Fj(x)1466 1301 y Fu(1)1506 1287 y Fr(+)p Fk(\001)15 b(\001)g(\001)q Fr(+)1754 1232 y Fk(p)p 1830 1232 215 4 v 55 x Fj(a)1878 1301 y Fi(m)p Fh(\000)p Fi(k)2044 1287 y Fj(x)2096 1301 y Fi(m)p Fh(\000)p Fi(k)2287 1287 y Fj(<)2383 1228 y Fk(p)p 2459 1228 89 4 v 59 x Fj(a)2507 1301 y Fu(1)2547 1287 y Fj(c)2586 1301 y Fu(1)2628 1287 y Fr(\()p Fj(N)10 b Fr(\)+)p Fk(\001)15 b(\001)g(\001)q Fr(+)3029 1232 y Fk(p)p 3104 1232 215 4 v 3104 1287 a Fj(a)3152 1301 y Fi(m)p Fh(\000)p Fi(k)3319 1287 y Fj(c)3358 1301 y Fi(m)p Fh(\000)p Fi(k)3524 1287 y Fr(\()p Fj(N)10 b Fr(\)+)50 1432 y Fk(\001)15 b(\001)g(\001)25 b Fr(+)274 1377 y Fk(p)p 350 1377 211 4 v 55 x Fj(a)398 1446 y Fi(m)p Fh(\000)p Fu(1)560 1432 y Fj(c)599 1446 y Fi(m)p Fh(\000)p Fu(1)762 1432 y Fr(\()p Fj(N)10 b Fr(\))24 b Fk(\000)g Fr(\()p Fj(a)1117 1446 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)1399 1432 y Fr(+)g Fk(\001)15 b(\001)g(\001)25 b Fr(+)f Fj(a)1767 1446 y Fi(m)p Fh(\000)p Fu(1)1929 1432 y Fr(\))p Fj(c)2003 1446 y Fi(m)2073 1432 y Fr(\()p Fj(N)10 b Fr(\))p Fj(=)2271 1374 y Fk(p)p 2347 1374 118 4 v 58 x Fj(a)2395 1446 y Fi(m)2465 1432 y Fk(g)36 b Fr(and)f(where)h(the)g(constan)m(t)h Fj(B)i(>)50 1578 y(c)89 1545 y Fu(2)89 1600 y(1)153 1578 y Fr(+)22 b Fk(\001)15 b(\001)g(\001)24 b Fr(+)f Fj(c)508 1545 y Fu(2)508 1600 y Fi(m)577 1578 y Fr(.)53 b(Then)33 b(the)i(estimate)g(\(2.20\))h(in)d(the)i(case)g Fj(k)g Fr(=)d(2)j(pro)m(v)m(es)g(that)f(with)f(an)i(appropriate)50 1723 y(c)m(hoice)h(of)f Fj(\016)476 1690 y Fh(0)536 1723 y Fj(>)d Fr(0)k(the)f(con)m(tribution)f(of)h Fj(J)1571 1690 y Fu(2)1562 1748 y Fi(N)s(;m)1749 1723 y Fr(to)h(\(2.13\))h(is)d (negligible)f(and)i(concludes)f(the)h(pro)s(of)g(of)50 1869 y(the)30 b(prop)s(osition.)159 2087 y(Finally)35 b(w)m(e)i(are)g(left)g(to)g(sho)m(w)g(\(2.20\).)62 b(Let)38 b(us)e(in)m(tro)s(duce)g(the)h(notations:)53 b Fj(a)2967 2054 y Fh(0)2967 2111 y Fi(i)3034 2087 y Fk(\021)35 b Fj(a)3188 2101 y Fi(i)3219 2087 y Fr(,)j Fj(c)3321 2054 y Fh(0)3321 2111 y Fi(i)3388 2087 y Fk(\021)e Fj(c)3534 2101 y Fi(i)3601 2087 y Fr(for)50 2232 y Fj(i)25 b Fr(=)g(1)p Fj(;)15 b(:)g(:)g(:)33 b(;)15 b(m)20 b Fk(\000)g Fj(k)j Fk(\000)d Fr(1,)31 b Fj(a)966 2199 y Fh(0)966 2258 y Fi(m)p Fh(\000)p Fi(k)1158 2232 y Fk(\021)25 b Fj(a)1302 2246 y Fi(m)p Fh(\000)p Fi(k)1468 2232 y Fr(,)50 2503 y Fj(c)89 2466 y Fh(0)89 2526 y Fi(m)p Fh(\000)p Fi(k)280 2503 y Fk(\021)g Fj(c)415 2517 y Fi(m)p Fh(\000)p Fi(k)585 2503 y Fr(+)s(\()694 2444 y Fk(p)p 770 2444 307 4 v 59 x Fj(a)818 2517 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)1076 2503 y Fj(c)1115 2517 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)1376 2503 y Fr(+)s Fk(\001)15 b(\001)g(\001)t Fr(+)s Fj(a)1681 2517 y Fi(m)p Fh(\000)p Fu(1)1843 2503 y Fj(c)1882 2517 y Fi(m)p Fh(\000)p Fu(1)2047 2503 y Fk(\000)s Fr(\()p Fj(a)2204 2517 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)2465 2503 y Fr(+)s Fk(\001)g(\001)g(\001)t Fr(+)s Fj(a)2770 2517 y Fi(m)p Fh(\000)p Fu(1)2932 2503 y Fr(\))p Fj(c)3006 2517 y Fi(m)3076 2503 y Fj(=)3121 2440 y Fk(p)p 3197 2440 118 4 v 63 x Fj(a)3245 2517 y Fi(m)3314 2503 y Fr(\))p Fj(=)3394 2444 y Fk(p)p 3470 2444 215 4 v 59 x Fj(a)3518 2517 y Fi(k)r Fh(\000)p Fi(m)3685 2503 y Fj(;)1278 2790 y(a)1326 2752 y Fh(0)1326 2812 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)1609 2790 y Fk(\021)25 b Fj(a)1753 2804 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)2031 2790 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)22 b Fr(+)e Fj(a)2388 2804 y Fi(m)2457 2790 y Fj(;)1063 3031 y(c)1102 2994 y Fh(0)1102 3054 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)1386 3031 y Fk(\021)1482 2956 y Fe(p)p 1573 2956 752 4 v 75 x Fj(a)1621 3045 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)1899 3031 y Fr(+)g Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)2255 3045 y Fi(m)2324 3031 y Fj(c)2363 3045 y Fi(m)2433 3031 y Fj(=)2478 2968 y Fk(p)p 2554 2968 118 4 v 63 x Fj(a)2602 3045 y Fi(m)2671 3031 y Fj(:)50 3257 y Fr(One)k(c)m(hec)m(ks)i(that)702 3181 y Fe(p)p 793 3181 79 4 v 76 x Fj(a)841 3225 y Fh(0)841 3282 y Fi(i)871 3257 y Fj(c)910 3224 y Fh(0)910 3280 y Fi(i)p Fh(\000)p Fu(1)1042 3257 y Fk(\000)1121 3181 y Fe(p)p 1212 3181 V 76 x Fj(a)1260 3225 y Fh(0)1260 3282 y Fi(i)1290 3257 y Fj(c)1329 3224 y Fh(0)1329 3280 y Fi(i)1385 3257 y Fj(<)f Fr(0)g(for)f(all)f Fj(i)j Fr(=)f(2)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)8 b Fk(\000)g Fj(k)j Fr(+)d(1)25 b(except)h(ma)m(y)f(b)s(e)f(for)g Fj(i)h Fr(=)g Fj(m)8 b Fk(\000)g Fj(k)s Fr(.)50 3402 y(If)29 b(this)f(is)g(also)h(true)g(for)h Fj(i)25 b Fr(=)g Fj(m)18 b Fk(\000)g Fj(k)32 b Fr(or)d(if)g Fj(k)f Fr(=)d Fj(m)18 b Fk(\000)g Fr(1)29 b(\(as)h(in)e(this)g(case)j Fj(a)2627 3369 y Fh(0)2627 3428 y Fi(m)p Fh(\000)p Fi(k)r Fh(\000)p Fu(1)2886 3402 y Fr(,)f Fj(c)2980 3369 y Fh(0)2980 3428 y Fi(m)p Fh(\000)p Fi(k)r Fh(\000)p Fu(1)3268 3402 y Fr(are)g(absen)m(t\))50 3547 y(then)40 b Fj(L)329 3514 y Fi(k)329 3573 y(N)s(;m)520 3547 y Fr(tak)m(es)i(the)f(form)f(of)h Fj(I)1316 3514 y Fu(1)1309 3573 y Fi(N)s(;m)p Fh(\000)p Fi(k)r Fu(+1)1690 3547 y Fr(and)e(b)m(y)i(induction)d(on)i Fj(I)2610 3514 y Fu(1)2603 3573 y Fi(N)s(;m)2795 3547 y Fr(it)g(do)s(es)g(not)h(exceed)h(the)50 3693 y(exp)s(onen)m(tial)29 b(b)s(ound)f(\(2.20\))33 b(with)c Fj(B)h Fr(=)25 b Fj(c)1521 3660 y Fh(0)1547 3647 y Fu(2)1547 3715 y(1)1608 3693 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(c)1955 3660 y Fh(0)1981 3647 y Fu(2)1981 3715 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)2239 3693 y Fr(.)40 b(Then)30 b(the)g(observ)-5 b(ation)148 3949 y Fj(c)187 3911 y Fh(0)213 3898 y Fu(2)213 3971 y(1)274 3949 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(c)621 3911 y Fh(0)647 3898 y Fu(2)647 3971 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)925 3949 y Fk(\000)g Fr(\()p Fj(c)1090 3911 y Fu(2)1090 3971 y(1)1152 3949 y Fr(+)g Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(c)1499 3911 y Fu(2)1499 3971 y Fi(m)1568 3949 y Fr(\))173 4184 y Fk(\025)25 b Fr(2)p Fj(c)353 4198 y Fi(m)p Fh(\000)p Fi(k)535 4070 y(k)r Fh(\000)p Fu(1)536 4098 y Fe(X)543 4291 y Fi(i)p Fu(=1)669 4184 y Fr(\()704 4125 y Fk(p)p 780 4125 200 4 v 59 x Fj(a)828 4198 y Fi(m)p Fh(\000)p Fi(i)979 4184 y Fj(c)1018 4198 y Fi(m)p Fh(\000)p Fi(i)1191 4184 y Fk(\000)19 b Fj(a)1329 4198 y Fi(m)p Fh(\000)p Fi(i)1481 4184 y Fj(c)1520 4198 y Fi(m)1590 4184 y Fj(=)1635 4121 y Fk(p)p 1711 4121 118 4 v 63 x Fj(a)1759 4198 y Fi(m)1828 4184 y Fr(\))p Fj(=)1908 4125 y Fk(p)p 1984 4125 215 4 v 59 x Fj(a)2032 4198 y Fi(m)p Fh(\000)p Fi(k)2219 4184 y Fr(+)2310 4070 y Fi(k)r Fh(\000)p Fu(1)2311 4098 y Fe(X)2317 4291 y Fi(i)p Fu(=1)2458 4184 y Fj(a)2506 4198 y Fi(m)p Fh(\000)p Fi(i)2658 4184 y Fj(c)2697 4146 y Fu(2)2697 4206 y Fi(m)2766 4184 y Fj(=a)2859 4198 y Fi(m)2949 4184 y Fk(\000)3040 4070 y Fi(k)r Fh(\000)p Fu(1)3041 4098 y Fe(X)3047 4291 y Fi(i)p Fu(=1)3188 4184 y Fj(c)3227 4146 y Fu(2)3227 4206 y Fi(m)p Fh(\000)p Fi(i)173 4504 y Fj(>)25 b(c)308 4518 y Fi(m)p Fh(\000)p Fi(k)490 4390 y(k)r Fh(\000)p Fu(1)491 4417 y Fe(X)498 4611 y Fi(i)p Fu(=1)623 4504 y Fr(\()658 4444 y Fk(p)p 734 4444 200 4 v 60 x Fj(a)782 4518 y Fi(m)p Fh(\000)p Fi(i)934 4504 y Fj(c)973 4518 y Fi(m)p Fh(\000)p Fi(i)1145 4504 y Fk(\000)20 b Fj(a)1284 4518 y Fi(m)p Fh(\000)p Fi(i)1436 4504 y Fj(c)1475 4518 y Fi(m)1544 4504 y Fj(=)1589 4440 y Fk(p)p 1665 4440 118 4 v 64 x Fj(a)1713 4518 y Fi(m)1782 4504 y Fr(\))p Fj(=)1862 4444 y Fk(p)p 1939 4444 215 4 v 1939 4504 a Fj(a)1987 4518 y Fi(m)p Fh(\000)p Fi(k)2173 4504 y Fr(+)2264 4390 y Fi(k)r Fh(\000)p Fu(1)2265 4417 y Fe(X)2272 4611 y Fi(i)p Fu(=1)2413 4504 y Fj(a)2461 4518 y Fi(m)p Fh(\000)p Fi(i)2612 4504 y Fj(c)2651 4466 y Fu(2)2651 4526 y Fi(m)2721 4504 y Fj(=a)2814 4518 y Fi(m)2148 4823 y Fk(\000)2239 4710 y Fi(k)r Fh(\000)p Fu(1)2240 4737 y Fe(X)2247 4930 y Fi(i)p Fu(=1)2387 4764 y Fk(p)p 2463 4764 200 4 v 59 x Fj(a)2511 4837 y Fi(m)p Fh(\000)p Fi(i)2663 4823 y Fj(c)2702 4837 y Fi(m)2771 4823 y Fj(c)2810 4837 y Fi(m)p Fh(\000)p Fi(i)2962 4823 y Fj(=)3007 4760 y Fk(p)p 3083 4760 118 4 v 63 x Fj(a)3131 4837 y Fi(m)173 5143 y Fr(=)269 5029 y Fi(k)r Fh(\000)p Fu(1)270 5057 y Fe(X)277 5250 y Fi(i)p Fu(=1)403 5143 y Fr(\()438 5084 y Fk(p)p 514 5084 200 4 v 59 x Fj(a)562 5157 y Fi(m)p Fh(\000)p Fi(i)713 5143 y Fj(c)752 5157 y Fi(m)p Fh(\000)p Fi(i)925 5143 y Fk(\000)f Fj(a)1063 5157 y Fi(m)p Fh(\000)p Fi(i)1215 5143 y Fj(c)1254 5157 y Fi(m)1324 5143 y Fj(=)1369 5080 y Fk(p)p 1445 5080 118 4 v 63 x Fj(a)1493 5157 y Fi(m)1562 5143 y Fr(\)\()p Fj(c)1671 5157 y Fi(m)p Fh(\000)p Fi(k)1838 5143 y Fj(=)1883 5084 y Fk(p)p 1959 5084 215 4 v 59 x Fj(a)2007 5157 y Fi(m)p Fh(\000)p Fi(k)2193 5143 y Fk(\000)h Fj(c)2323 5157 y Fi(m)2393 5143 y Fj(=)2438 5080 y Fk(p)p 2514 5080 118 4 v 63 x Fj(a)2562 5157 y Fi(m)2631 5143 y Fr(\))26 b Fj(>)f Fr(0)3477 4581 y(\(2)p Fj(:)p Fr(21\))50 5462 y(\014nishes)j(the)j(pro)s(of)e(of)i(\(2.20\))h(in)d(this)h(case.)41 b(In)30 b(particular)f(this)g(pro)m(v)m(es)i(\(2.20\))i(for)d Fj(k)e Fr(=)d Fj(m)20 b Fk(\000)g Fr(1.)p eop %%Page: 25 25 25 24 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(25)159 527 y Fr(Supp)s(ose)18 b(no)m(w)j(that)g Fj(k)28 b(>)d Fr(1,)e(\(2.20\))f(holds)d(up)g(to)i Fj(k)s Fk(\000)p Fr(1)g(and)f(that)2378 429 y Fe(q)p 2469 429 308 4 v 98 x Fj(a)2517 496 y Fh(0)2517 555 y Fi(m)p Fh(\000)p Fi(k)r Fh(\000)p Fu(1)2777 527 y Fj(c)2816 494 y Fh(0)2816 553 y Fi(m)p Fh(\000)p Fi(k)3007 527 y Fk(\024)3103 429 y Fe(q)p 3194 429 215 4 v 98 x Fj(a)3242 496 y Fh(0)3242 555 y Fi(m)p Fh(\000)p Fi(k)3408 527 y Fj(c)3447 494 y Fh(0)3447 553 y Fi(m)p Fh(\000)p Fi(k)r Fh(\000)p Fu(1)3707 527 y Fr(.)50 673 y(Let)31 b(us)e(\014nd)g Fj(l)e Fk(\025)e Fj(k)34 b Fr(suc)m(h)c(that)50 1039 y(\()p Fj(a)133 1001 y Fh(0)133 1061 y Fi(m)p Fh(\000)p Fi(k)316 1039 y Fr(+)16 b Fk(\001)f(\001)g(\001)j Fr(+)e Fj(a)661 1001 y Fh(0)661 1061 y Fi(m)p Fh(\000)p Fi(i)813 1039 y Fr(\))p Fj(c)887 1001 y Fh(0)887 1061 y Fi(m)p Fh(\000)p Fi(i)p Fh(\000)p Fu(1)1157 1039 y Fk(\025)1253 935 y Fe(q)p 1344 935 293 4 v 104 x Fj(a)1392 1007 y Fh(0)1392 1064 y Fi(m)p Fh(\000)p Fi(i)p Fh(\000)p Fu(1)1637 1039 y Fr(\()1672 935 y Fe(q)p 1763 935 200 4 v 104 x Fj(a)1811 1007 y Fh(0)1811 1064 y Fi(m)p Fh(\000)p Fi(i)1963 1039 y Fj(c)2002 1001 y Fh(0)2002 1061 y Fi(m)p Fh(\000)p Fi(i)2170 1039 y Fr(+)h Fk(\001)e(\001)g(\001)i Fr(+)2467 936 y Fe(q)p 2558 936 215 4 v 103 x Fj(a)2606 1007 y Fh(0)2606 1066 y Fi(m)p Fh(\000)p Fi(k)2772 1039 y Fj(c)2811 1001 y Fh(0)2811 1061 y Fi(m)p Fh(\000)p Fi(k)2978 1039 y Fr(\))91 b Fk(8)p Fj(i)25 b Fr(=)g(2)p Fj(;)15 b Fr(3)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(l)r Fr(;)3477 1184 y(\(2)p Fj(:)p Fr(22\))121 1441 y(\()p Fj(a)204 1404 y Fh(0)204 1464 y Fi(m)p Fh(\000)p Fi(k)391 1441 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)747 1404 y Fh(0)747 1464 y Fi(m)p Fh(\000)p Fi(l)p Fh(\000)p Fu(1)989 1441 y Fr(\))p Fj(c)1063 1404 y Fh(0)1063 1464 y Fi(m)p Fh(\000)p Fi(l)p Fh(\000)p Fu(2)1331 1441 y Fj(<)1427 1339 y Fe(q)p 1518 1339 291 4 v 102 x Fj(a)1566 1410 y Fh(0)1566 1469 y Fi(m)p Fh(\000)p Fi(l)p Fh(\000)p Fu(2)1809 1441 y Fr(\()1844 1339 y Fe(q)p 1935 1339 V 102 x Fj(a)1983 1410 y Fh(0)1983 1469 y Fi(m)p Fh(\000)p Fi(l)p Fh(\000)p Fu(1)2225 1441 y Fj(c)2264 1404 y Fh(0)2264 1464 y Fi(m)p Fh(\000)p Fi(l)p Fh(\000)p Fu(1)2527 1441 y Fr(+)g Fk(\001)15 b(\001)g(\001)21 b Fr(+)2835 1339 y Fe(q)p 2926 1339 215 4 v 102 x Fj(a)2974 1410 y Fh(0)2974 1469 y Fi(m)p Fh(\000)p Fi(k)3140 1441 y Fj(c)3179 1404 y Fh(0)3179 1464 y Fi(m)p Fh(\000)p Fi(k)3346 1441 y Fr(\))p Fj(:)71 b Fr(\(2)p Fj(:)p Fr(23\))50 1762 y(\(If)35 b(\(2.22\))i(holds)d(for)h Fj(i)f Fr(=)f(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(m)24 b Fk(\000)f Fr(2,)37 b(w)m(e)e(put)g Fj(l)g Fr(=)e Fj(m)23 b Fk(\000)h Fr(2.\))55 b(Then)35 b(dev)m(eloping)f Fj(L)3164 1729 y Fi(k)3164 1787 y(N)s(;m)3350 1762 y Fj(l)25 b Fk(\000)f Fj(k)i Fr(+)d(2)50 1908 y(times)30 b(as)g(in)f(\(2.19\),)k(w)m(e)e(obtain)221 2316 y Fj(L)283 2278 y Fi(k)283 2338 y(N)s(;m)459 2316 y Fk(\024)530 2236 y(p)p 605 2236 101 4 v 605 2316 a Fr(2)p Fj(\031)721 2192 y Fe(Z)772 2399 y Fi(D)831 2409 y Fm(m)p Ff(\000)p Fm(k)q Ff(\000)p Fn(2)1056 2399 y Fh(\\)p Fi(G)1162 2377 y Fm(\016)1193 2364 y Ff(00)1162 2423 y Fm(m;k)q Ff(\000)p Fn(1)1378 2316 y Fj(e)1431 2208 y Fn([)1449 2181 y Ff(p)p 1499 2181 71 3 v 27 x Fm(a)1534 2222 y Fn(1)1570 2208 y Fm(c)1600 2222 y Fn(1)1635 2208 y(\()p Fm(N)5 b Fn(\)+)p Ff(\001\001\001)n Fn(+)1894 2179 y Ff(p)p 1944 2179 96 3 v 29 x Fm(a)1979 2217 y(m)2039 2208 y(c)2069 2217 y(m)2129 2208 y Fn(\()p Fm(N)g Fn(\))p Ff(\000)2282 2181 y(p)p 2332 2181 71 3 v 27 x Fm(a)2367 2222 y Fn(1)2402 2208 y Fm(x)2438 2222 y Fn(1)2474 2208 y Ff(\000\001\001\001)o(\000)2634 2177 y Fh(p)p 2694 2177 261 4 v 31 x Fm(a)2729 2224 y(m)p Ff(\000)p Fm(k)q Ff(\000)p Fn(1)2955 2208 y Fm(x)2991 2224 y(m)p Ff(\000)p Fm(k)q Ff(\000)p Fn(1)3216 2208 y(])3234 2189 y(2)p 1431 2245 1839 4 v 2094 2283 a(2\()p Fm(a)2185 2299 y(m)p Ff(\000)p Fm(k)2330 2283 y Fn(+)p Ff(\001\001\001)o Fn(+)p Fm(a)2521 2292 y(m)2581 2283 y Fn(\))1823 2562 y Fk(\002)19 b Fj(e)1955 2525 y Fh(\000)p Fu(\()p Fi(x)2081 2497 y Fn(2)2081 2544 y Fm(m)p Ff(\000)p Fm(k)q Ff(\000)p Fn(1)2307 2525 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(x)2525 2497 y Fn(2)2525 2543 y(1)2560 2525 y Fu(\))p Fi(=)p Fu(2)2667 2562 y Fj(dx)2766 2576 y Fi(m)p Fh(\000)p Fi(k)r Fh(\000)p Fu(1)3040 2562 y Fj(:)c(:)g(:)i(dx)3261 2576 y Fu(1)550 2788 y Fr(+)652 2647 y Fe(p)p 743 2647 1211 4 v 79 x Fr(\(2)p Fj(\031)s Fr(\))913 2699 y Fi(l)p Fh(\000)p Fi(k)r Fu(+1)1131 2726 y Fr(\()p Fj(a)1214 2740 y Fi(m)p Fh(\000)p Fi(k)r Fu(+1)1492 2726 y Fr(+)j Fk(\001)15 b(\001)g(\001)22 b Fr(+)e Fj(a)1849 2740 y Fi(m)1918 2726 y Fr(\))p Fj(e)1995 2690 y Fh(\000)p Fi(c)2085 2663 y Ff(0)2109 2660 y Fn(2)2109 2708 y Fm(m)p Ff(\000)p Fm(k)q Fn(+1)2332 2690 y Fu(\()p Fi(N)7 b Fu(\))p Fi(=)p Fu(2)p 652 2767 1880 4 v 1467 2785 a Fk(p)p 1543 2785 83 4 v 77 x Fj(N)j(\016)1669 2836 y Fh(00)732 3108 y Fk(\002)823 2984 y Fe(Z)873 3191 y Fi(D)934 3169 y Ff(0)932 3215 y Fm(m)p Ff(\000)p Fm(l)p Ff(\000)p Fn(2)1164 3108 y Fj(e)1217 2968 y Fn([)1235 2916 y Fk(p)p 1311 2916 71 4 v 52 x Fm(a)1346 2953 y Ff(0)1346 2996 y Fn(1)1382 2968 y Fm(c)1412 2982 y Fn(1)1447 2968 y(\()p Fm(N)5 b Fn(\)+)p Ff(\001\001\001)n Fn(+)1706 2921 y Fk(p)p 1782 2921 181 4 v 47 x Fm(a)1817 2953 y Ff(0)1817 2999 y Fm(m)p Ff(\000)p Fm(k)1962 2968 y(c)1992 2950 y Ff(0)1992 2996 y Fm(m)p Ff(\000)p Fm(k)2136 2968 y Fn(\()p Fm(N)g Fn(\))p Ff(\000)2289 2916 y Fk(p)p 2364 2916 71 4 v 2364 2968 a Fm(a)2399 2953 y Ff(0)2399 2996 y Fn(1)2435 2968 y Fm(x)2471 2982 y Fn(1)2506 2968 y Ff(\000\001\001\001\000)2667 2921 y Fk(p)p 2743 2921 247 4 v 47 x Fm(a)2778 2953 y Ff(0)2778 2999 y Fm(m)p Ff(\000)p Fm(l)p Ff(\000)p Fn(2)2990 2968 y Fm(x)3026 2984 y(m)p Ff(\000)p Fm(l)p Ff(\000)p Fn(2)3237 2968 y(])3255 2950 y(2)p 1217 3015 2075 4 v 1923 3060 a(2\()p Fm(a)2014 3044 y Ff(0)2014 3090 y Fm(m)p Ff(\000)p Fm(l)p Ff(\000)p Fn(1)2225 3060 y(+)p Ff(\001\001\001)o Fn(+)p Fm(a)2416 3044 y Ff(0)2416 3090 y Fm(m)p Ff(\000)p Fm(k)2561 3060 y Fn(\))1823 3353 y Fk(\002)19 b Fj(e)1955 3316 y Fh(\000)p Fu(\()p Fi(x)2081 3288 y Fn(2)2081 3334 y Fm(m)p Ff(\000)p Fm(l)p Ff(\000)p Fn(2)2294 3316 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(x)2511 3288 y Fn(2)2511 3334 y(1)2547 3316 y Fu(\))p Fi(=)p Fu(2)2653 3353 y Fj(dx)2752 3367 y Fi(m)p Fh(\000)p Fi(l)p Fh(\000)p Fu(2)3010 3353 y Fj(:)c(:)g(:)h(dx)3230 3367 y Fu(1)3477 2795 y Fr(\(2)p Fj(:)p Fr(24\))50 3700 y(where)34 b Fj(G)389 3667 y Fi(\016)424 3640 y Ff(0)o(0)389 3726 y Fi(m;k)r Fh(\000)p Fu(1)644 3700 y Fr(=)d Fk(f)p Fr(\()p Fj(x)878 3714 y Fu(1)920 3700 y Fj(;)15 b(:)g(:)g(:)31 b(;)15 b(x)1188 3714 y Fi(m)p Fh(\000)p Fi(k)r Fh(\000)p Fu(1)1448 3700 y Fr(\))32 b(:)1572 3642 y Fk(p)p 1647 3642 89 4 v 1647 3700 a Fj(a)1695 3714 y Fu(1)1736 3700 y Fj(x)1788 3714 y Fu(1)1852 3700 y Fr(+)22 b Fk(\001)15 b(\001)g(\001)24 b Fr(+)2168 3646 y Fk(p)p 2243 3646 308 4 v 2243 3700 a Fj(a)2291 3714 y Fi(m)p Fh(\000)p Fi(k)r Fh(\000)p Fu(1)2551 3700 y Fj(x)2603 3714 y Fi(m)p Fh(\000)p Fi(k)r Fh(\000)p Fu(1)2893 3700 y Fj(<)2996 3642 y Fk(p)p 3071 3642 89 4 v 3071 3700 a Fj(a)3119 3714 y Fu(1)3160 3700 y Fj(c)3199 3714 y Fu(1)3240 3700 y Fr(\()p Fj(N)10 b Fr(\))24 b(+)e Fk(\001)15 b(\001)g(\001)24 b Fr(+)50 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4610 y Fj(a)2098 4624 y Fi(m)p Fh(\000)p Fi(i)p 1760 4650 490 4 v 1946 4734 a Fj(a)1994 4748 y Fi(m)2260 4671 y Fj(c)2299 4685 y Fi(m)2389 4671 y Fk(\000)2480 4612 y(p)p 2556 4612 291 4 v 59 x Fj(a)2604 4685 y Fi(m)p Fh(\000)p Fi(l)p Fh(\000)p Fu(1)2846 4671 y Fj(c)2885 4685 y Fi(m)p Fh(\000)p Fi(i)3037 4570 y Fe(\021)3117 4671 y Fr(=)25 b(0)p Fj(:)50 4954 y Fr(This)j(\014nishes)h(the)h(pro)s(of)g(of)g(\(2.20\))j(and)d (completes)g(the)h(pro)s(of)f(of)g(the)h(prop)s(osition.)38 b Fk(})50 5316 y Fa(Pro)s(of)44 b(of)g(Theorem)f(1.5:)120 b Fr(The)38 b(pro)s(of)f(of)h(Theorem)g(1.5)h(is)e(not)h(m)m(uc)m(h)g (more)g(than)g(a)g(corollary)50 5462 y(of)44 b(Theorem)f(1.3.)82 b(W)-8 b(e)45 b(ha)m(v)m(e)f(to)h(sho)m(w)e(that)i(for)e(an)m(y)h (collection)f(of)h(con)m(tin)m(uous,)j(lo)s(cal)c(functions)p eop %%Page: 27 27 27 26 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(27)50 513 y Fj(h)102 527 y Fu(1)143 513 y Fj(;)15 b(:)g(:)g(:)i(;)e(h)397 527 y Fi(j)436 513 y Fr(,)30 b Fj(j)h Fk(2)25 b Fg(N)7 b Fr(,)36 b(and)30 b(a)h(con)m(tin)m(uous)f(function)f Fj(G)c Fr(:)g Fg(R)2037 480 y Fi(j)2106 513 y Fk(!)g Fg(R)880 749 y(E)10 b Fj(G)1021 620 y Fe(\022)1094 625 y(Z)1200 749 y Fk(W)1297 711 y Fi(m)1290 771 y(\014)s(;N)1425 749 y Fr(\()p Fj(dw)r Fr(\))p Fj(h)1661 763 y Fu(1)1703 749 y Fr(\()p Fj(w)r Fr(\))p Fj(;)15 b(:)g(:)g(:)j(;)2043 625 y Fe(Z)2150 749 y Fk(W)2247 711 y Fi(m)2240 771 y(\014)s(;N)2375 749 y Fr(\()p Fj(dw)r Fr(\))p Fj(h)2611 763 y Fi(j)2651 749 y Fr(\()p Fj(w)r Fr(\))2788 620 y Fe(\023)905 1003 y Fk(!)25 b Fg(E)11 b Fj(F)1168 875 y Fe(\022)1235 879 y(Z)1341 1003 y Fk(W)1438 966 y Fi(m)1431 1026 y(\014)1507 1003 y Fr(\()p Fj(dw)r Fr(\))p Fj(h)1743 1017 y Fu(1)1786 1003 y Fr(\()p Fj(w)r Fr(\))p Fj(;)k(:)g(:)g(:)j(;)2126 879 y Fe(Z)2232 1003 y Fk(W)2329 966 y Fi(m)2322 1026 y(\014)2399 1003 y Fr(\()p Fj(dw)r Fr(\))p Fj(h)2635 1017 y Fi(j)2675 1003 y Fr(\()p Fj(w)r Fr(\))2812 875 y Fe(\023)3477 876 y Fr(\(2)p Fj(:)p Fr(26\))50 1231 y(No)m(w)31 b(clearly)80 1327 y Fe(Z)186 1450 y Fk(W)283 1413 y Fi(k)276 1473 y(\014)s(;N)411 1450 y Fr(\()p Fj(dw)r Fr(\))p Fj(h)647 1464 y Fi(i)679 1450 y Fr(\()p Fj(w)r Fr(\))90 1724 y(=)186 1600 y Fe(Z)292 1724 y Fk(P)362 1677 y Fu(\()p Fi(k)r Fu(\))355 1751 y Fi(N)464 1724 y Fr(\()p Fj(dy)s Fr(\))p Fj(h)681 1738 y Fi(i)728 1569 y Fe( )811 1588 y(R)887 1661 y Fk(P)957 1614 y Fu(\()p Fi(k)r Fu(\))950 1688 y Fi(N)1059 1661 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)1270 1628 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))1482 1661 y Fj(\016)s Fr(\()p Fj(x)1612 1675 y Fu(1)1674 1661 y Fk(\000)20 b Fj(y)1810 1675 y Fu(1)1851 1661 y Fr(\))p 811 1703 1075 4 v 1013 1736 a Fe(R)1089 1809 y Fk(P)1159 1762 y Fu(\()p Fi(k)r Fu(\))1152 1836 y Fi(N)1261 1809 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)1472 1783 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))1897 1724 y Fj(;)15 b(:)g(:)g(:)32 b(;)2125 1588 y Fe(R)2201 1661 y Fk(P)2271 1614 y Fu(\()p Fi(k)r Fu(\))2264 1688 y Fi(N)2373 1661 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)2584 1628 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))2796 1661 y Fj(\016)s Fr(\()p Fj(x)2926 1675 y Fu(1)2988 1661 y Fk(\000)20 b Fj(y)3124 1675 y Fu(1)3164 1661 y Fr(\))15 b Fj(:)g(:)g(:)i(\016)s Fr(\()p Fj(x)3466 1675 y Fi(k)3532 1661 y Fk(\000)j Fj(y)3668 1675 y Fi(k)3712 1661 y Fr(\))p 2125 1703 1623 4 v 2587 1736 a Fe(R)2662 1809 y Fk(P)2732 1762 y Fu(\()p Fi(k)r Fu(\))2725 1836 y Fi(N)2835 1809 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)3046 1783 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\)\))3759 1569 y Fe(!)3477 1915 y Fr(\(2)p Fj(:)p Fr(27\))50 2060 y(W)-8 b(e)24 b(will)c(use)j(the)g(fact)h(that)f(con)m(v)m(ergence)j (in)21 b(distribution)f(of)j(a)g(sequence)g Fk(P)2709 2013 y Fu(\()p Fi(k)r Fu(\))2702 2087 y Fi(N)2835 2060 y Fr(is)f(equiv)-5 b(alen)m(t)22 b(to)i(almost)50 2206 y(sure)i(con)m(v)m(ergence)i(of)f(a)g(sequence)g Fk(P)1352 2159 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))1345 2233 y Fi(N)1539 2206 y Fr(that)g(for)g(eac)m(h)g Fj(N)36 b Fr(ha)m(v)m(e)28 b(the)f(same)g(distribution)22 b(as)27 b Fk(P)3429 2159 y Fu(\()p Fi(k)r Fu(\))3422 2233 y Fi(N)3558 2206 y Fr(\(see)50 2351 y(e.g.)42 b([Ka,RE]\).)31 b(W)-8 b(e)32 b(can)e(of)h(course)g(replace)f Fk(P)1722 2304 y Fu(\()p Fi(k)r Fu(\))1715 2378 y Fi(N)1854 2351 y Fr(b)m(y)h Fk(P)2051 2304 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))2044 2378 y Fi(N)2242 2351 y Fr(in)e(\(2.26\))k(without)c(c)m(hanging)h(an)m (ything.)159 2570 y(W)-8 b(e)41 b(ha)m(v)m(e)f(already)f(sho)m(wn)g (that)h(the)g(denominators)e(in)g(the)i(argumen)m(ts)g(of)f Fj(h)3002 2584 y Fi(i)3072 2570 y Fr(con)m(v)m(erge)j(in)c(dis-)50 2715 y(tribution,)33 b(i.e.)631 2642 y Fe(R)707 2715 y Fk(P)777 2668 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))770 2742 y Fi(N)938 2715 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)1149 2682 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))1395 2715 y Fr(con)m(v)m(erges)j(a.s.)53 b(to)2105 2642 y Fe(R)2180 2715 y Fk(P)2250 2682 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))2411 2715 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)2622 2682 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))2834 2715 y Fr(.)f(The)34 b(same)g(argumen)m(t)50 2860 y(applies)28 b(to)j(the)g(n)m(umerators,)g(i.e.)40 b(for)30 b(\014xed)g Fj(x)g Fr(the)h(v)m(ector)95 2960 y Fe( )178 2979 y(R)254 3052 y Fk(P)324 3005 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))317 3079 y Fi(N)485 3052 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)696 3019 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))908 3052 y Fj(\016)s Fr(\()p Fj(x)1038 3066 y Fu(1)1100 3052 y Fk(\000)20 b Fj(y)1236 3066 y Fu(1)1276 3052 y Fr(\))p 178 3094 1134 4 v 380 3127 a Fe(R)456 3200 y Fk(P)526 3153 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))519 3227 y Fi(N)687 3200 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)898 3174 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))1323 3115 y Fj(;)15 b(:)g(:)g(:)h(;)1536 3053 y Fk(P)1606 3006 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))1599 3080 y Fi(N)1767 3053 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)1978 3020 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))2190 3053 y Fj(\016)s Fr(\()p Fj(x)2320 3067 y Fu(1)2382 3053 y Fk(\000)k Fj(y)2518 3067 y Fu(1)2558 3053 y Fr(\))15 b Fj(:)g(:)g(:)i(\016)s Fr(\()p 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Fu(\))2316 3363 y Fj(\016)s Fr(\()p Fj(x)2446 3377 y Fu(1)2508 3363 y Fk(\000)j Fj(y)2644 3377 y Fu(1)2684 3363 y Fr(\))15 b Fj(:)g(:)g(:)i(\016)s Fr(\()p Fj(x)2986 3377 y Fi(k)3052 3363 y Fk(\000)j Fj(y)3188 3377 y Fi(k)3233 3363 y Fr(\))p 1662 3404 1606 4 v 2100 3415 a Fe(R)2176 3488 y Fk(P)2246 3462 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))2407 3488 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)2618 3462 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))3279 3269 y Fe(!)3457 3424 y Fr(a.s.)3477 3615 y(\(2)p Fj(:)p Fr(28\))50 3761 y(Moreo)m(v)m(er,)33 b(this)c(v)m(ector)j(is)d(a)i(con)m(tin)m (uous)f(function)f(of)i Fj(x)p Fr(,)f(whic)m(h)f(in)g(turn)h(implies)d (that)161 4016 y Fj(h)213 4030 y Fi(i)259 3860 y Fe( )342 3879 y(R)417 3952 y Fk(P)487 3919 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))648 3952 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)859 3919 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))1072 3952 y Fj(\016)s Fr(\()p Fj(x)1202 3966 y Fu(1)1264 3952 y Fk(\000)20 b Fj(y)1400 3966 y Fu(1)1440 3952 y Fr(\))p 342 3995 1134 4 v 543 4006 a Fe(R)619 4080 y Fk(P)689 4053 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))850 4080 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)1061 4053 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))1486 4016 y Fj(;)15 b(:)g(:)g(:)i(;)1699 3954 y Fk(P)1769 3921 y Fu(\()p Fi(k)r Fh(\003)p Fu(\))1909 3954 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)2120 3921 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))2332 3954 y Fj(\016)s Fr(\()p Fj(x)2462 3968 y Fu(1)2524 3954 y Fk(\000)j Fj(y)2660 3968 y Fu(1)2700 3954 y Fr(\))15 b Fj(:)g(:)g(:)i(\016)s Fr(\()p Fj(x)3002 3968 y Fi(k)3068 3954 y Fk(\000)j Fj(y)3204 3968 y Fi(k)3248 3954 y Fr(\))p 1699 3995 1585 4 v 2137 4006 a Fe(R)2213 4080 y Fk(P)2283 4053 y Fu(\()p Fi(k)r Fh(\003)p Fu(\))2422 4080 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)2633 4053 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))3294 3860 y Fe(!)3477 4016 y Fr(\(2)p Fj(:)p Fr(29\))50 4271 y(is)29 b(a)i(con)m(tin)m(uous)f(function)f(of)i Fj(x)p Fr(,)f(and)g(th)m(us)80 4402 y Fe(Z)186 4525 y Fk(P)256 4482 y Fi(k)r(;)p Fh(\003)249 4552 y Fi(N)360 4525 y Fr(\()p Fj(dx)p Fr(\))p Fj(h)581 4539 y Fi(i)628 4370 y Fe( )711 4389 y(R)787 4462 y Fk(P)857 4429 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))1018 4462 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)1229 4429 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))1441 4462 y Fj(\016)s Fr(\()p Fj(x)1571 4476 y Fu(1)1633 4462 y Fk(\000)20 b Fj(y)1769 4476 y Fu(1)1809 4462 y Fr(\))p 711 4504 1134 4 v 913 4516 a Fe(R)989 4589 y Fk(P)1059 4563 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))1220 4589 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)1431 4563 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))1856 4525 y Fj(;)15 b(:)g(:)g(:)h(;)2069 4464 y Fk(P)2139 4431 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))2300 4464 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)2511 4431 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))2723 4464 y Fj(\016)s Fr(\()p Fj(x)2853 4478 y Fu(1)2915 4464 y Fk(\000)k Fj(y)3051 4478 y Fu(1)3091 4464 y Fr(\))15 b Fj(:)g(:)g(:)i(\016)s Fr(\()p Fj(x)3393 4478 y Fi(k)3459 4464 y Fk(\000)j Fj(y)3595 4478 y Fi(k)3639 4464 y Fr(\))p 2069 4504 1606 4 v 2507 4516 a Fe(R)2582 4589 y Fk(P)2652 4563 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))2813 4589 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)3024 4563 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))3685 4370 y Fe(!)90 4835 y Fk(!)206 4711 y Fe(Z)312 4835 y Fk(P)382 4797 y Fi(k)r(;)p Fh(\003)487 4835 y Fr(\()p Fj(x)p Fr(\))p Fj(h)661 4849 y Fi(i)707 4679 y Fe( )790 4698 y(R)866 4771 y Fk(P)936 4738 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))1097 4771 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)1308 4738 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))1520 4771 y Fj(\016)s Fr(\()p Fj(x)1650 4785 y Fu(1)1712 4771 y Fk(\000)g Fj(y)1848 4785 y Fu(1)1888 4771 y Fr(\))p 790 4814 1134 4 v 992 4825 a Fe(R)1068 4898 y Fk(P)1138 4872 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))1299 4898 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)1510 4872 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))1935 4835 y Fj(;)15 b(:)g(:)g(:)h(;)2148 4773 y Fk(P)2218 4740 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))2379 4773 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)2590 4740 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))2802 4773 y Fj(\016)s Fr(\()p Fj(x)2932 4787 y Fu(1)2994 4773 y Fk(\000)k Fj(y)3130 4787 y Fu(1)3170 4773 y Fr(\))15 b Fj(:)g(:)g(:)i(\016)s Fr(\()p Fj(x)3472 4787 y Fi(k)3538 4773 y Fk(\000)j Fj(y)3674 4787 y Fi(k)3718 4773 y Fr(\))p 2148 4814 1606 4 v 2586 4825 a Fe(R)2661 4898 y Fk(P)2731 4872 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))2892 4898 y Fr(\()p Fj(dx)p Fr(\))p Fj(e)3103 4872 y Fi(\014)s Fu(\()p Fi(\015)t(;x)p Fu(\))3764 4679 y Fe(!)3477 5025 y Fr(\(2)p Fj(:)p Fr(30\))50 5171 y(Th)m(us)37 b(all)f(argumen)m(ts)i(of)g Fj(G)f Fr(in)g(\(2.27\))j(con)m(v)m(erge)g (\(after)e(replacing)f Fk(P)2578 5124 y Fu(\()p Fi(k)r Fu(\))2571 5198 y Fi(N)2718 5171 y Fr(b)m(y)h Fk(P)2922 5124 y Fu(\()p Fi(k)r(;)p Fh(\003)p Fu(\))2915 5198 y Fi(N)3083 5171 y Fr(\))g(almost)g(surely)-8 b(.)50 5316 y(Since)27 b Fj(G)h Fr(is)f(b)s(ounded)f(and)i(con)m(tin)m(uous,)g (this)f(implies)f(con)m(v)m(ergence)k(of)f(the)f(mean)g(of)h Fj(G)p Fr(,)f(whic)m(h)f(pro)m(v)m(es)50 5462 y(the)j(theorem.)42 b Fk(})p eop %%Page: 28 28 28 27 bop 50 295 a Fo(28)3270 b Fl(Se)l(ction)27 b Fo(3)50 513 y Fv(3.)55 b(The)41 b(t)m(w)m(o-replica)f(distribution.)159 731 y Fr(W)-8 b(e)22 b(no)m(w)f(turn)e(to)j(the)f(analysis)e(of)i(the)g (o)m(v)m(erlap)g(distributions.)34 b(Let)21 b(us)f(\014rst)h(lo)s(ok)f (at)h(the)g(distribution)50 876 y(of)30 b(the)h(ultrametric)e(v)-5 b(aluation,)30 b Fj(f)1252 890 y Fi(\014)s(;N)1386 876 y Fr(,)h(de\014ned)e(in)g(\(1.20\).)159 1095 y(W)-8 b(e)21 b(need)f(some)h(notation.)38 b(F)-8 b(or)21 b(a)f(giv)m(en)h (con\014guration)e Fj(\033)2159 1109 y Fu(1)2200 1095 y Fj(\033)2252 1109 y Fu(2)2308 1095 y Fj(:)c(:)g(:)h(\033)2481 1109 y Fi(n)2551 1095 y Fr(w)m(e)21 b(denote)f(b)m(y)25 b(\026)-50 b Fj(\033)3125 1109 y Fi(l)3179 1095 y Fr(=)25 b Fj(\033)3327 1109 y Fi(J)3367 1119 y Fm(l)p Ff(\000)p Fn(1)3473 1109 y Fu(+1)3585 1095 y Fj(:)15 b(:)g(:)h(\033)3758 1109 y Fi(J)3798 1119 y Fm(l)50 1240 y Fr(its)29 b(pieces)h(for)f Fj(l)f Fr(=)d(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\))31 b(and)j(\026)-50 b Fj(\033)1404 1256 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)1651 1240 y Fr(=)25 b Fj(\033)1799 1254 y Fi(J)1839 1266 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))1953 1254 y Fu(+1)2064 1240 y Fj(:)15 b(:)g(:)i(\033)2238 1254 y Fi(n)2317 1240 y Fr(in)29 b(accordance)i(with)e(the)h(appropriate)50 1385 y(partition)e(\(1.12\)) k(and)d(\(1.13\).)42 b(Note)31 b(that)f(the)g(ev)m(en)m(t)h Fk(f)p Fj(d)2065 1399 y Fi(N)2135 1385 y Fr(\()p Fj(\033)n(;)15 b(\033)2315 1352 y Fh(0)2341 1385 y Fr(\))26 b Fk(\025)f Fj(q)2539 1399 y Fi(i)2569 1385 y Fk(g)30 b Fr(is)e(equiv)-5 b(alen)m(t)29 b(to)h(demanding)50 1531 y(that)h Fk(f)5 b Fr(\026)-50 b Fj(\033)344 1545 y Fu(1)400 1531 y Fj(:)15 b(:)g(:)21 b Fr(\026)-50 b Fj(\033)573 1545 y Fi(i)629 1531 y Fr(=)30 b(\026)-50 b Fj(\033)780 1498 y Fh(0)777 1553 y Fu(1)833 1531 y Fj(:)15 b(:)g(:)21 b Fr(\026)-50 b Fj(\033)1009 1498 y Fh(0)1006 1555 y Fi(i)1036 1531 y Fk(g)p Fr(.)41 b(Therefore)30 b(w)m(e)h(ha)m(v)m(e)h(that)216 1725 y(1)21 b Fk(\000)f Fj(f)418 1739 y Fi(\014)s(;N)552 1725 y Fr(\()p Fj(q)628 1739 y Fi(i)659 1725 y Fr(\))25 b(=)815 1638 y Fe(X)859 1829 y Fi(\033)962 1638 y Fe(X)993 1833 y Fi(\033)1037 1815 y Ff(0)1108 1725 y Fj(\026)1163 1739 y Fi(\014)s(;N)1298 1725 y Fr(\()p Fj(\033)s Fr(\))p Fj(\026)1478 1739 y Fi(\014)s(;N)1614 1725 y Fr(\()p Fj(\033)1704 1687 y Fh(0)1730 1725 y Fr(\)1)-23 b(I)1821 1741 y Fh(f)t Fu(\026)-40 b Fi(\033)1900 1750 y Fn(1)1935 1741 y Fi(:::)s Fu(\026)g Fi(\033)2042 1750 y Fm(i)2071 1741 y Fu(=)t(\026)g Fi(\033)2171 1719 y Ff(0)2169 1763 y Fn(1)2205 1741 y Fi(:::)r Fu(\026)g Fi(\033)2313 1719 y Ff(0)2311 1764 y Fm(i)2340 1741 y Fh(g)242 2003 y Fr(=)337 1879 y Fe(Z)444 2003 y Fk(R)521 1965 y Fi(i)521 2025 y(\014)s(;N)656 2003 y Fr(\()p Fj(dw)803 2017 y Fi(i)834 2003 y Fr(\))p Fj(w)936 1965 y Fu(2)934 2025 y Fi(i)1003 2003 y Fr(=)1099 1879 y Fe(Z)1205 2003 y Fk(W)1302 1965 y Fi(i)1295 2025 y(\014)s(;N)1429 2003 y Fr(\()p Fj(dw)1576 2017 y Fu(1)1618 2003 y Fj(;)15 b(:)g(:)g(:)i(;)e(dw)1932 2017 y Fi(i)1963 2003 y Fr(\))p Fj(w)2065 1965 y Fu(2)2063 2025 y Fi(i)2132 2003 y Fr(=)2228 1879 y Fe(Z)2334 2003 y Fk(W)2431 1965 y Fi(m)2424 2025 y(\014)s(;N)2558 2003 y Fr(\()p Fj(dw)2705 2017 y Fu(1)2747 2003 y Fj(;)g(:)g(:)g(:)i(;)e(dw) 3061 2017 y Fi(i)3092 2003 y Fr(\))p Fj(w)3192 2017 y Fi(i)3222 2003 y Fj(w)3287 2017 y Fi(m)3523 1878 y Fr(\(3)p Fj(:)p Fr(1\))50 2219 y(for)30 b(all)f Fj(m)c Fk(\025)g Fj(i)p Fr(.)159 2437 y(The)39 b(con)m(v)m(ergence)j(of)e(the)g(p)s(oin) m(t)e(pro)s(cesses)i Fk(W)1892 2404 y Fi(i)1885 2463 y(\014)s(;N)2059 2437 y Fr(for)f Fj(i)i Fk(\024)g Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))40 b(stated)h(in)d(Theorem)h(1.5)i(then)50 2583 y(immediately)28 b(implies)g(the)j(con)m(v)m(ergence)h(in)d (distribution)e(of)914 2799 y Fj(f)959 2813 y Fi(\014)s(;N)1093 2799 y Fr(\()p Fj(q)1169 2813 y Fi(i)1199 2799 y Fr(\))1276 2747 y Fh(D)1260 2799 y Fk(!)1376 2675 y Fe(Z)1493 2776 y(c)1482 2799 y Fk(W)1572 2813 y Fi(\014)1621 2799 y Fr(\()p Fj(dw)1768 2813 y Fu(1)1810 2799 y Fj(;)15 b(:)g(:)g(:)31 b(;)15 b(dw)2138 2815 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))2268 2799 y Fr(\))p Fj(w)2368 2815 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))2498 2799 y Fr(\(1)21 b Fk(\000)f Fj(w)2755 2813 y Fi(i)2785 2799 y Fr(\))p Fj(:)678 b Fr(\(3)p Fj(:)p Fr(2\))50 3020 y(for)42 b Fj(i)k Fr(=)f(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\).)79 b(T)-8 b(o)43 b(conclude)f(the)h(pro)s(of)e(of)i(Theorem)f(1.6,)47 b(w)m(e)c(m)m(ust)g(sho)m(w)f(that)h Fj(f)3439 3034 y Fi(\014)3530 3020 y Fr(is)f(in)50 3165 y(fact)36 b(atomic)g(with)e (atoms)i(only)f(at)h(the)f(v)-5 b(alues)35 b Fj(q)1821 3179 y Fi(i)1851 3165 y Fr(,)i Fj(i)d Fr(=)f(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\))37 b(and)d(that)i(the)g(limiting)c(o)m(v)m(erlap)50 3311 y(distribution)27 b(function)921 3287 y(~)902 3311 y Fj(f)947 3325 y Fi(\014)1025 3311 y Fr(do)s(es)j(the)h(same.)159 3529 y(F)-8 b(or)36 b(that)g(purp)s(ose)d(let)i(us)g(in)m(tro)s(duce)f (a)h(more)g(complicated)g(lo)s(oking)f(ob)5 b(ject.)56 b(Consider)33 b(the)j(seg-)50 3675 y(men)m(ts)30 b(\001)391 3689 y Fi(l)444 3675 y Fr(=)25 b([)p Fk(\000)15 b Fr(ln)23 b(\026)-54 b Fj(\013)800 3689 y Fi(l)828 3675 y Fj(=)15 b Fr(ln)g(2)p Fj(;)g Fr(ln)24 b(\026)-54 b Fj(\013)1215 3689 y Fi(l)1243 3675 y Fj(=)15 b Fr(ln)f(2])31 b(for)f Fj(l)e Fr(=)d(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\))31 b(and)50 3820 y(\001)126 3836 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)372 3820 y Fr(=)25 b([)p Fk(\000)15 b Fr(ln)o(\()p Fj(\013)748 3834 y Fi(J)788 3846 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))903 3834 y Fu(+1)1014 3820 y Fk(\001)g(\001)g(\001)h Fj(\013)1193 3834 y Fi(n)1243 3820 y Fr(\))p Fj(=)f Fr(ln)g(2)p Fj(;)g Fr(ln\()p Fj(\013)1685 3834 y Fi(J)1725 3846 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))1839 3834 y Fu(+1)1951 3820 y Fk(\001)g(\001)g(\001)h Fj(\013)2130 3834 y Fi(n)2180 3820 y Fr(\))p Fj(=)f Fr(ln)g(2].)50 3965 y(Clearly)-8 b(,)33 b(\()5 b(\026)-50 b Fj(\033)474 3979 y Fi(l)524 3965 y Fk(\001)28 b Fr(\026)-50 b Fj(\033)627 3932 y Fh(0)624 3991 y Fi(l)653 3965 y Fr(\))p Fj(=)-5 b(N)41 b Fk(2)29 b Fr(\001)1008 3979 y Fi(l)1036 3965 y Fr(,)34 b Fj(l)e Fr(=)e(1)p Fj(;)15 b(:)g(:)g(:)33 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\))23 b(+)f(1.)50 b(W)-8 b(e)34 b(in)m(tro)s(duce)f(a)g(random)g(measure)g Fj(f)3303 3918 y Fh(\012)p Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)3293 3993 y Fi(\014)s(;N)3614 3965 y Fr(on)50 4111 y(\001)126 4125 y Fu(1)177 4111 y Fk(\002)10 b Fr(\001)334 4125 y Fu(2)384 4111 y Fk(\002)g(\001)15 b(\001)g(\001)c(\002)f Fr(\001)738 4127 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)984 4111 y Fr(induced)24 b(b)m(y)h(the)h(o)m(v)m(erlaps)k(\026)-50 b Fj(\033)1989 4125 y Fi(l)2027 4111 y Fk(\001)15 b Fr(\026)-50 b Fj(\033)2117 4078 y Fh(0)2114 4137 y Fi(l)2169 4111 y Fr(of)25 b(all)g(the)g('lev)m(els')h(of)f(the)h(limiting)c(P)m (oisson)50 4256 y(cascade)32 b(as)e(follo)m(ws:)40 b(for)30 b(an)m(y)h(in)m(terv)-5 b(als)29 b Fj(I)1539 4270 y Fi(l)1592 4256 y Fk(2)c Fr(\001)1754 4270 y Fi(l)1782 4256 y Fr(,)31 b Fj(l)c Fr(=)e(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\))21 b(+)f(1,)31 b(set)485 4525 y Fj(f)540 4478 y Fh(\012)p Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)530 4552 y Fi(\014)s(;N)817 4525 y Fr(\()p Fj(I)892 4539 y Fu(1)954 4525 y Fk(\002)20 b(\001)15 b(\001)g(\001)21 b(\002)f Fj(I)1302 4541 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)1523 4525 y Fr(\))26 b Fk(\021)1690 4449 y Fg(E)1745 4463 y Fi(\033)n(;\033)1851 4444 y Ff(0)1902 4381 y Fe(Q)1988 4402 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)1988 4476 y Fi(l)p Fu(=1)2224 4449 y Fr(1)-23 b(I)2280 4465 y Fu(\()t(\026)-40 b Fi(\033)2350 4475 y Fm(l)2377 4465 y Fh(\001)t Fu(\026)g Fi(\033)2443 4443 y Ff(0)2441 4489 y Fm(l)2467 4465 y Fu(\))p Fi(=)l(N)7 b Fh(2)p Fi(I)2674 4475 y Fm(l)2705 4449 y Fj(e)2747 4416 y Fi(\014)2791 4362 y Fh(p)p 2852 4362 65 4 v 2852 4416 a Fi(N)f Fu(\()p Fi(X)3004 4425 y Fm(\033)3048 4416 y Fu(+)p Fi(X)3164 4433 y Fm(\033)3203 4420 y Ff(0)3231 4416 y Fu(\))p 1690 4504 1574 4 v 2379 4587 a Fj(Z)2448 4556 y Fu(2)2441 4615 y Fi(\014)s(;N)1584 4842 y Fr(=)24 b Fj(\026)1734 4803 y Fh(\012)p Fu(2)1734 4869 y Fi(\014)s(;N)1869 4741 y Fe(\020)1939 4720 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)1989 4755 y Fe(Y)1989 4951 y Fi(l)p Fu(=1)2171 4842 y Fr(1)-23 b(I)2226 4858 y Fu(\()t(\026)-40 b Fi(\033)2296 4868 y Fm(l)2323 4858 y Fh(\001)t Fu(\026)g Fi(\033)2389 4836 y Ff(0)2387 4882 y Fm(l)2414 4858 y Fu(\))p Fi(=)l(N)7 b Fh(2)p Fi(I)2621 4868 y Fm(l)2652 4741 y Fe(\021)2706 4842 y Fj(:)159 5171 y Fr(The)29 b(follo)m(wing)g(theorem)h(yields)e(then)i(all)f(desired)f(information) h(on)g(the)i(o)m(v)m(erlap)f(distribution)c(and)50 5316 y(on)36 b(the)h(ultrametric)f(distribution)d(functions)1747 5292 y(~)1727 5316 y Fj(f)1772 5330 y Fi(\014)1857 5316 y Fr(and)j Fj(f)2085 5330 y Fi(\014)2134 5316 y Fr(.)59 b(In)36 b(particular,)h(it)f(implies)e(Theorem)j(1.6)50 5462 y(readily)-8 b(.)p eop %%Page: 29 29 29 28 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(29)50 517 y Fd(Theorem)51 b(3.1:)141 b Fq(The)43 b(me)-5 b(asur)g(e)44 b Fj(f)1482 470 y Fh(\012)p Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)1472 544 y Fi(\014)s(;N)1802 517 y Fq(c)-5 b(onver)g(ges)44 b(in)e(distribution)i(to)f(the)h(fol)5 b(lowing)43 b(p)-5 b(oint)50 662 y(me)g(asur)g(e)34 b(on)f Fr(\001)608 676 y Fu(1)669 662 y Fk(\002)20 b Fr(\001)836 676 y Fu(2)896 662 y Fk(\002)g(\001)15 b(\001)g(\001)21 b(\002)f Fr(\001)1280 679 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)431 984 y Fj(f)486 937 y Fh(\012)p Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)476 1012 y Fi(\014)s(;N)805 933 y Fh(D)788 984 y Fk(!)26 b Fj(Q)977 998 y Fu(0)1017 984 y Fj(\016)1057 1001 y Fu(\(0)p Fi(;)p Fu(0)p Fi(;:::)12 b(;)p Fu(0\))1390 984 y Fr(+)1484 863 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1481 898 y Fe(X)1484 1092 y Fi(j)t Fu(=1)1627 984 y Fj(Q)1699 998 y Fi(j)1737 984 y Fj(\016)1777 1001 y Fu(\(ln)22 b(\026)-44 b Fi(\013)1926 1010 y Fn(1)1962 1001 y Fi(=)13 b Fu(ln)g(2)p Fi(;:::)e(;)p Fu(ln)20 b(\026)-43 b Fi(\013)2362 1010 y Fm(j)2396 1001 y Fi(=)13 b Fu(ln)g(2)p Fi(;)p Fu(0)p Fi(;:::)e(;)p Fu(0\))2988 984 y Fj(N)35 b Fk(!)26 b(1)p Fj(:)194 b Fr(\(3)p Fj(:)p Fr(3\))50 1288 y Fq(The)33 b(r)-5 b(andom)35 b(variables)e Fj(Q)1016 1302 y Fu(1)1057 1288 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(Q)1346 1304 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1508 1288 y Fq(ar)-5 b(e)33 b(given)f(as)241 1556 y Fj(Q)313 1570 y Fi(l)341 1556 y Fr(\()p Fj(\014)5 b Fr(\))26 b Fk(\021)589 1432 y Fe(Z)706 1533 y(c)695 1556 y Fk(W)785 1570 y Fi(\014)834 1556 y Fr(\()p Fj(dw)981 1570 y Fu(1)1022 1556 y Fj(;)15 b(:)g(:)g(:)i(;)e (dw)1336 1572 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1466 1556 y Fr(\))p Fj(w)1566 1572 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1696 1556 y Fr(\()p Fj(w)1796 1570 y Fi(j)1855 1556 y Fk(\000)20 b Fj(w)2011 1570 y Fi(j)t Fu(+1)2141 1556 y Fr(1)-23 b(I)2196 1572 y Fh(f)p Fi(j)t Fh(\024)p Fi(l)p Fu(\()p Fi(\014)s Fu(\))p Fh(\000)p Fu(1)p Fh(g)2583 1556 y Fr(\))p Fj(;)81 b(j)31 b Fr(=)25 b(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\))p Fj(:)193 b Fr(\(3)p Fj(:)p Fr(4\))50 2022 y Fq(Pr)-5 b(o)g(of.)65 b Fr(All)36 b(w)m(e)j(really)e(need)g(to)i(establish)e(is)g(the)h(supp)s(ort)e(of)i (the)h(measure)f Fj(f)2930 1975 y Fh(\012)p Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)2920 2049 y Fi(\014)s(;N)3245 2022 y Fr(in)f(the)h(limit)50 2167 y Fj(N)d Fk(")26 b(1)p Fr(.)40 b(The)30 b(assertion)g(\(3.4\))i(then)e(follo)m(w)g(from)g (Theorem)g(1.5)i(lik)m(e)d(\(3.2\).)159 2386 y(The)f(next)g(four)g (assertions)f(w)m(ould)g(pro)m(vide)h(us)f(the)i(supp)s(ort)d(of)j(the) f(measure)g Fj(f)2998 2338 y Fh(\012)p Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)2988 2413 y Fi(\014)s(;N)3276 2386 y Fr(.)39 b(W)-8 b(e)30 b(pro)m(v)m(e)50 2531 y(subsequen)m(tly)f(eac)m (h)i(of)g(them:)17 2749 y(\(i\))46 b(If)30 b Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))26 b(=)f(0,)31 b(then)f(for)g(an)m(y)h Fj(I)37 b Fr(suc)m(h)30 b(that)h(0)26 b Fk(62)f Fr(cl)o(\()p Fj(I)7 b Fr(\))1299 2997 y Fj(f)1354 2950 y Fh(\012)p Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)1344 3025 y Fi(\014)s(;N)1631 2997 y Fr(\()p Fj(I)g Fr(\))1790 2946 y Fh(D)1774 2997 y Fk(!)25 b Fr(0)92 b(as)30 b Fj(N)36 b Fk(!)25 b(1)p Fr(.)1044 b(\(3)p Fj(:)p Fr(5\))-8 3318 y(\(ii\))45 b(Let)d Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))44 b Fk(\025)g Fr(1.)74 b(Then)40 b(for)i(an)m(y)f Fj(l)46 b Fr(=)d(1)p Fj(;)15 b Fr(2)p Fj(;)g(:)g(:)g(:)33 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\))43 b(and)e(an)m(y)h Fj(I)2516 3332 y Fi(l)2587 3318 y Fk(2)i Fr(\001)2768 3332 y Fi(l)2837 3318 y Fr(suc)m(h)d(that)h (1)i Fk(62)f Fj(I)3494 3332 y Fi(l)3563 3318 y Fr(and)159 3463 y(0)26 b Fk(62)e Fr(cl)30 b(\()p Fj(I)486 3477 y Fi(l)515 3463 y Fr(\))771 3609 y Fj(f)826 3562 y Fh(\012)p Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)816 3636 y Fi(\014)s(;N)1103 3609 y Fr(\()p Fk(f)p Fr(1)p Fk(g)22 b(\002)e(\001)15 b(\001)g(\001)21 b(\002)f(f)p Fr(1)p Fk(g)i(\002)e Fj(I)1891 3623 y Fi(l)1939 3609 y Fk(\002)g Fr(\001)2106 3623 y Fi(l)p Fu(+1)2245 3609 y Fk(\002)g(\001)15 b(\001)g(\001)22 b(\002)e Fr(\001)2630 3625 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)2851 3609 y Fr(\))2927 3557 y Fh(D)2911 3609 y Fk(!)25 b Fr(0)p Fj(:)426 b Fr(\(3)p Fj(:)p Fr(6\))-33 3888 y(\(iii\))44 b(Let)g Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))47 b Fk(\025)f Fr(1.)80 b(Let)44 b Fj(I)1020 3902 y Fi(l)p Fh(\000)p Fu(1)1184 3888 y Fr(b)s(e)f(suc)m(h)f(that)i(1)j Fk(62)g Fj(I)1988 3902 y Fi(l)p Fh(\000)p Fu(1)2109 3888 y Fr(.)79 b(Then)42 b(for)h(an)m(y)h Fj(l)k Fr(=)e(2)p Fj(;)15 b Fr(3)p Fj(;)g(:)g(:)g(:)34 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\),)47 b(an)m(y)159 4033 y Fj(I)199 4047 y Fu(1)240 4033 y Fj(;)15 b(:)g(:)g(:)31 b(;)15 b(I)496 4047 y Fi(l)p Fh(\000)p Fu(2)648 4033 y Fr(and)30 b(an)m(y)h Fj(I)1037 4047 y Fi(l)1064 4033 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(I)1321 4050 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)1573 4033 y Fr(suc)m(h)30 b(that)h(\(0)p Fj(;)15 b Fr(0)p Fj(;)g(:)g(:)g(:)34 b(;)15 b Fr(0\))26 b Fk(62)f Fr(cl\()p Fj(I)2692 4047 y Fi(l)2740 4033 y Fk(\002)20 b(\001)15 b(\001)g(\001)21 b(\002)f Fj(I)3088 4050 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)3309 4033 y Fr(\))822 4281 y Fj(f)877 4234 y Fh(\012)p Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)867 4309 y Fi(\014)s(;N)1154 4281 y Fr(\()p Fj(I)1229 4295 y Fu(1)1290 4281 y Fk(\002)g(\001)15 b(\001)g(\001)21 b(\002)f Fj(I)1638 4295 y Fi(l)p Fh(\000)p Fu(2)1779 4281 y Fk(\002)g Fj(I)1910 4295 y Fi(l)p Fh(\000)p Fu(1)2051 4281 y Fk(\002)g Fj(I)2182 4295 y Fi(l)2230 4281 y Fk(\002)g(\001)15 b(\001)g(\001)22 b(\002)d Fj(I)2578 4298 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)2800 4281 y Fr(\))2876 4230 y Fh(D)2860 4281 y Fk(!)25 b Fr(0)p Fj(:)477 b Fr(\(3)p Fj(:)p Fr(7\))-30 4602 y(\(iv\))45 b(Let)31 b Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))26 b Fk(\025)f Fr(1.)41 b(Then)29 b(for)h(an)m(y)h Fj(I)1298 4616 y Fu(1)1339 4602 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(I)1596 4618 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1756 4602 y Fr(and)29 b Fj(I)1972 4618 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)2224 4602 y Fr(suc)m(h)h(that)h(0)26 b Fk(62)e Fr(cl\()p Fj(I)2923 4618 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)3145 4602 y Fr(\))1292 4850 y Fj(f)1347 4803 y Fh(\012)p Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)1337 4878 y Fi(\014)s(;N)1624 4850 y Fr(\()p Fj(I)1699 4864 y Fu(1)1760 4850 y Fk(\002)c(\001)15 b(\001)g(\001)21 b(\002)f Fj(I)2108 4867 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)2330 4850 y Fr(\))2406 4799 y Fh(D)2390 4850 y Fk(!)25 b Fr(0)p Fj(:)947 b Fr(\(3)p Fj(:)p Fr(8\))159 5171 y(\(cl)p Fj(I)39 b Fr(denotes)32 b(the)g(closure)f(if)g(the)h (subset)f Fj(I)7 b Fr(.\))45 b(W)-8 b(e)33 b(start)f(with)f(\(i\))g (where)h Fj(\014)2841 5138 y Fu(2)2882 5171 y Fj(A)2950 5185 y Fu(1)p Fi(;J)3048 5194 y Fn(1)3115 5171 y Fk(\024)c Fr(1.)45 b(Let)32 b(us)f(\014x)50 5316 y Fj(\017)25 b(>)g Fr(0)31 b(suc)m(h)f(that)872 5462 y(ln)o(\()p Fj(\013)1041 5476 y Fu(1)1097 5462 y Fj(:)15 b(:)g(:)h(\013)1276 5476 y Fi(j)1315 5462 y Fr(\))k Fk(\000)g Fr(\()p Fj(\014)1552 5424 y Fu(2)1614 5462 y Fk(\000)g Fr(\()p Fj(\014)26 b Fk(\000)19 b Fj(\017)p Fr(\))1979 5424 y Fu(2)2021 5462 y Fj(=)p Fr(2\)\()p Fj(a)2229 5476 y Fu(1)2291 5462 y Fr(+)h Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)2647 5476 y Fi(j)2686 5462 y Fr(\))25 b Fj(>)g Fr(0)636 b(\(3)p Fj(:)p Fr(9\))p eop %%Page: 30 30 30 29 bop 50 295 a Fo(30)3270 b Fl(Se)l(ction)27 b Fo(3)50 513 y Fr(for)j(all)f Fj(j)i Fr(=)25 b(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(n)31 b Fr(suc)m(h)f(that)h Fj(\014)1285 480 y Fu(2)1326 513 y Fj(A)1394 527 y Fu(1)p Fi(;j)1515 513 y Fj(<)25 b Fr(1.)41 b(In)m(tro)s(duce)30 b(the)h(ev)m(en)m(ts)437 767 y Fj(D)512 781 y Fi(\033)554 790 y Fn(1)590 781 y Fi(:::)o(\033)697 790 y Fm(j)760 767 y Fr(=)25 b Fk(f)p Fj(!)k Fr(:)1037 704 y Fk(p)p 1113 704 89 4 v 63 x Fj(a)1161 781 y Fu(1)1202 767 y Fj(X)1277 781 y Fi(\033)1319 790 y Fn(1)1380 767 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)1688 710 y Fk(p)p 1764 710 87 4 v 57 x Fj(a)1812 781 y Fi(j)1850 767 y Fj(X)1925 781 y Fi(\033)1967 790 y Fn(1)2003 781 y Fi(:::)o(\033)2110 790 y Fm(j)2173 767 y Fj(<)k Fr(\()p Fj(\014)h Fr(+)20 b Fj(\017)p Fr(\)\()p Fj(a)2627 781 y Fu(1)2689 767 y Fr(+)g Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)3045 781 y Fi(j)3083 767 y Fr(\))3118 686 y Fk(p)p 3194 686 83 4 v 81 x Fj(N)10 b Fk(g)50 1021 y Fr(if)32 b Fj(\014)192 988 y Fu(2)233 1021 y Fj(A)301 1035 y Fu(1)p Fi(;j)428 1021 y Fj(<)e Fr(1)j(\(whic)m(h)g(is)f(alw)m(a) m(ys)i(true)f(for)g(all)f Fj(j)k(>)30 b(J)1984 1035 y Fu(1)2025 1021 y Fr(,)35 b(and)d(if)h Fj(\014)5 b(A)2475 1035 y Fu(1)p Fi(;J)2573 1044 y Fn(1)2643 1021 y Fj(<)30 b Fr(1,)35 b(then)e(for)g(all)f Fj(j)k Fk(\024)30 b Fj(J)3554 1035 y Fu(1)3628 1021 y Fr(as)50 1167 y(w)m(ell)f(\))i(and)341 1421 y Fj(D)416 1435 y Fi(\033)458 1444 y Fn(1)494 1435 y Fi(:::)o(\033)601 1444 y Fm(j)665 1421 y Fr(=)25 b Fk(f)p Fj(!)j Fr(:)942 1358 y Fk(p)p 1017 1358 89 4 v 1017 1421 a Fj(a)1065 1435 y Fu(1)1106 1421 y Fj(X)1181 1435 y Fi(\033)1223 1444 y Fn(1)1284 1421 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)1592 1364 y Fk(p)p 1668 1364 87 4 v 57 x Fj(a)1716 1435 y Fi(j)1754 1421 y Fj(X)1829 1435 y Fi(\033)1871 1444 y Fn(1)1907 1435 y Fi(:::)o(\033)2014 1444 y Fm(j)2078 1421 y Fj(<)k(u)2226 1437 y Fu(ln\()p Fi(\013)2361 1446 y Fn(1)2398 1437 y Fh(\001\001\001)n Fi(\013)2509 1446 y Fm(j)2544 1437 y Fu(\))p Fi(;N)2663 1421 y Fr(\(0\))2778 1348 y Fe(p)p 2870 1348 504 4 v 2870 1421 a Fj(a)2918 1435 y Fu(1)2979 1421 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(a)3335 1435 y Fi(j)3373 1421 y Fk(g)50 1675 y Fr(if)f Fj(\014)179 1642 y Fu(2)220 1675 y Fj(A)288 1689 y Fu(1)p Fi(;j)410 1675 y Fr(=)25 b(1)20 b(\(that)i(can)f(happ)s(en)d(only)i(if) f Fj(\014)1580 1642 y Fu(2)1621 1675 y Fj(A)1689 1689 y Fu(1)p Fi(;J)1787 1698 y Fn(1)1853 1675 y Fr(=)25 b(1)20 b(and)g(only)g(for)g Fj(j)31 b Fk(\024)25 b Fj(J)2714 1689 y Fu(1)2775 1675 y Fr(suc)m(h)20 b(that)h Fj(A)3225 1689 y Fu(1)p Fi(;j)3347 1675 y Fr(=)k Fj(A)3511 1689 y Fu(1)p Fi(;J)3609 1698 y Fn(1)3649 1675 y Fr(\).)50 1821 y(W)-8 b(e)31 b(decomp)s(ose)204 2108 y Fj(f)259 2060 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)249 2135 y Fi(\014)s(;N)480 2108 y Fr(\()p Fj(I)7 b Fr(\))26 b(=)693 2007 y Fe(h)736 2108 y Fr(2)781 2070 y Fh(\000)p Fu(2)p Fi(N)944 2108 y Fj(e)986 2070 y Fh(\000)p Fi(\014)1087 2043 y Fn(2)1123 2070 y Fi(N)1328 2021 y Fe(X)1207 2216 y Fi(\033)n(;\033)1313 2198 y Ff(0)1338 2216 y Fu(:)p Fi(\033)r Fh(\001)p Fi(\033)1469 2198 y Ff(0)1494 2216 y Fh(2)p Fi(I)1595 2108 y Fj(e)1637 2070 y Fi(\014)1681 2016 y Fh(p)p 1741 2016 65 4 v 54 x Fi(N)7 b Fu(\()p Fi(X)1894 2079 y Fm(\033)1937 2070 y Fu(+)p Fi(X)2053 2087 y Fm(\033)2092 2074 y Ff(0)2120 2070 y Fu(\))2153 2108 y Fr(1)-23 b(I)2209 2124 y Fh(f\\)2295 2102 y Fm(n)2295 2147 y(j)s Fn(=1)2407 2124 y Fu(\()p Fi(D)2494 2133 y Fm(\033)2531 2147 y Fn(1)2567 2133 y Fm(:::\033)2667 2148 y(j)2705 2124 y Fh(\\)p Fi(D)2813 2143 y Fm(\033)2852 2127 y Ff(0)2850 2171 y Fn(1)2886 2143 y Fm(:::\033)2988 2127 y Ff(0)2986 2172 y Fm(j)3024 2124 y Fu(\))p Fh(g)3094 2007 y Fe(i)3207 2046 y Fj(e)3249 2013 y Fi(\014)3293 1986 y Fn(2)3329 2013 y Fi(N)p 3148 2087 309 4 v 3148 2170 a Fr(\()p Fj(Z)3245 2184 y Fi(\014)s(;N)3381 2170 y Fr(\))3416 2144 y Fu(2)623 2425 y Fr(+)693 2324 y Fe(h)736 2425 y Fr(2)781 2388 y Fh(\000)p Fi(N)907 2425 y Fj(e)949 2388 y Fh(\000)p Fi(\014)1050 2360 y Fn(2)1087 2388 y Fi(N)q(=)p Fu(2)1223 2425 y Fj(Z)1292 2386 y Fh(\000)p Fu(1)1285 2453 y Fi(\014)s(;N)1465 2339 y Fe(X)1345 2534 y Fi(\033)n(;\033)1451 2516 y Ff(0)1476 2534 y Fu(:)p Fi(\033)r Fh(\001)p Fi(\033)1607 2516 y Ff(0)1631 2534 y Fh(2)p Fi(I)1732 2425 y Fj(e)1774 2388 y Fi(\014)1818 2333 y Fh(p)p 1879 2333 65 4 v 1879 2388 a Fi(N)6 b Fu(\()p Fi(X)2031 2397 y Fm(\033)2075 2388 y Fu(+)p Fi(X)2191 2405 y Fm(\033)2230 2392 y Ff(0)2258 2388 y Fu(\))2291 2425 y Fr(1)-23 b(I)2346 2447 y Fh(f[)2432 2425 y Fm(n)2432 2470 y(j)s Fn(=1)2544 2447 y Fu(\()2589 2431 y(\026)2572 2447 y Fi(D)2631 2456 y Fm(\033)2668 2470 y Fn(1)2705 2456 y Fm(:::)o(\033)2804 2471 y(j)2843 2447 y Fh([)2908 2431 y Fu(\026)2892 2447 y Fi(D)2951 2466 y Fm(\033)2990 2450 y Ff(0)2988 2494 y Fn(1)3024 2466 y Fm(:::)o(\033)3125 2450 y Ff(0)3123 2495 y Fm(j)3162 2447 y Fu(\))p Fh(g)3232 2324 y Fe(i)3286 2364 y Fj(e)3328 2331 y Fi(\014)3372 2303 y Fn(2)3408 2331 y Fi(N)q(=)p Fu(2)p 3286 2404 259 4 v 3317 2488 a Fj(Z)3379 2502 y Fi(\014)s(;N)623 2670 y Fk(\021)o Fj(L)755 2633 y Fu(1)755 2693 y Fi(N)844 2670 y Fr(+)20 b Fj(L)997 2633 y Fu(2)997 2693 y Fi(N)1066 2670 y Fj(:)50 2894 y Fr(The)h(second)g(factors)i(in)d Fj(L)959 2861 y Fu(1)959 2919 y Fi(N)1049 2894 y Fr(and)h Fj(L)1279 2861 y Fu(2)1279 2919 y Fi(N)1369 2894 y Fr(con)m(v)m(erge)j (in)c(probabilit)m(y)f(to)j(the)g(constan)m(ts)h Fj(C)3001 2861 y Fh(\000)p Fu(2)3098 2894 y Fr(\()p Fj(\014)5 b Fr(\))22 b(and)f Fj(C)3486 2861 y Fh(\000)p Fu(1)3583 2894 y Fr(\()p Fj(\014)5 b Fr(\))50 3040 y(resp)s(ectiv)m(elly)34 b(b)m(y)h(Theorem)g(1.6.)56 b(The)35 b(exp)s(ectation)h(of)f(the)h (\014rst)f(factor)h(in)e(square)h(brac)m(k)m(ets)i(of)e Fj(L)3641 3007 y Fu(1)3641 3065 y Fi(N)50 3185 y Fr(can)30 b(b)s(e)g(estimated)h(b)m(y)f(\(2.14\))j(It)d(do)s(es)g(not)h(exceed) 132 3476 y(2)177 3438 y Fh(\000)p Fu(2)p Fi(N)723 3389 y Fe(X)355 3589 y Fi(\033)n(;\033)461 3571 y Ff(0)486 3589 y Fu(:)p Fi(d)545 3598 y Fm(u)587 3589 y Fu(\()p Fi(\033)n(;\033)721 3571 y Ff(0)747 3589 y Fu(\))p Fi()g Fr(0)30 b(one)50 5024 y(can)d(\014nd)d Fj(\016)30 b(>)25 b Fr(0)h(suc)m(h)g(that)i Fj(\036)p Fr(\()p Fj(m)p Fr(\))d Fj(>)g(\016)31 b Fr(for)26 b(all)f Fk(j)p Fj(m)p Fk(j)h Fj(>)f(\017)p Fr(.)39 b(Then,)26 b(since)g(0)g Fk(62)f Fr(cl)p Fj(I)7 b Fr(,)27 b(the)g(\014rst)e(term)i (of)f(\(3.10\))50 5169 y(b)s(eing)19 b(the)j(sum)e(of)h(these)h(n)m(um) m(b)s(ers)d(with)h Fk(j)p Fj(m)p Fk(j)26 b Fj(>)f(\017)p Fr(,)e(is)d(exp)s(onen)m(tially)f(small.)36 b(It)21 b(follo)m(ws)g (that)g Fj(L)3357 5136 y Fu(1)3357 5194 y Fi(N)3451 5169 y Fk(!)26 b Fr(0)21 b(in)50 5315 y(probabilit)m(y)-8 b(.)38 b(The)28 b(\014rst)f(factor)j(in)d Fj(L)1337 5282 y Fu(2)1337 5340 y Fi(N)1434 5315 y Fr(is)g(not)i(bigger)f(than)g(2) 2209 5282 y Fh(\000)p Fi(N)2335 5315 y Fj(e)2377 5282 y Fh(\000)p Fi(\014)2478 5254 y Fn(2)2514 5282 y Fi(N)q(=)p Fu(2)2666 5246 y Fe(P)2762 5342 y Fi(\033)2826 5315 y Fj(e)2868 5282 y Fi(\014)2912 5227 y Fh(p)p 2973 5227 65 4 v 2973 5282 a Fi(N)6 b(X)3097 5291 y Fm(\033)3145 5315 y Fr(1)-23 b(I)3200 5336 y Fh(f[)3286 5314 y Fm(n)3286 5359 y(j)s Fn(=1)3414 5320 y Fu(\026)3398 5336 y Fi(D)3457 5345 y Fm(\033)3494 5359 y Fn(1)3530 5345 y Fm(:::)o(\033)3629 5360 y(j)3668 5336 y Fh(g)50 5460 y Fr(whic)m(h)34 b(is)h(exactly)749 5437 y(\026)730 5460 y Fj(Z)799 5427 y Fi(T)792 5486 y(\014)s(;N)927 5460 y Fj(=)p Fg(E)11 b Fj(Z)1096 5427 y Fi(T)1089 5486 y(\014)t(;N)1266 5460 y Fr(in)34 b(terms)i(of)f(the)h (notation)g(of)g(Lemma)f(2.1.)58 b(In)35 b(the)h(pro)s(of)e(of)i(that)p eop %%Page: 31 31 31 30 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(31)50 513 y Fr(lemma)31 b(it)g(has)g(already)g(b)s(een)f(sho)m(wn)h(that)h (this)e(term)i(con)m(v)m(erges)h(to)f(zero)g(in)e(probabilit)m(y)f (\(see)j(\(2.10\))50 658 y(and)d(the)g(discussion)e(follo)m(wing)h (it\).)40 b(Hence,)31 b Fj(L)1707 625 y Fu(2)1707 683 y Fi(N)1801 658 y Fk(!)25 b Fr(0)30 b(in)e(probabilit)m(y)f(as)j(w)m (ell,)f(\014nishing)d(the)k(pro)s(of)f(of)50 804 y(the)h(assertion)g (\(i\).)159 1022 y(Let)h(us)e(turn)h(to)h(\(ii\).)40 b(W)-8 b(e)31 b(in)m(tro)s(duce)f(the)g(ev)m(en)m(ts)138 1246 y Fj(E)209 1260 y Fu(\026)-40 b Fi(\033)247 1269 y Fn(1)283 1260 y Fi(:::)s Fu(\026)f Fi(\033)389 1269 y Fm(i)448 1246 y Fr(=)25 b Fk(f)p Fj(!)k Fr(:)c Fj(\015)772 1260 y Fu(1)813 1246 y Fj(u)865 1207 y Fh(\000)p Fu(1)865 1273 y(ln)c(\026)-44 b Fi(\013)985 1282 y Fn(1)1021 1273 y Fi(;N)1112 1246 y Fr(\()1173 1223 y(\026)1147 1246 y Fj(X)1222 1260 y Fi(\033)1264 1269 y Fn(1)1300 1260 y Fh(\001\001\001)o Fi(\033)1407 1269 y Fm(J)1442 1283 y Fn(1)1486 1246 y Fr(\))21 b(+)f Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(\015)1897 1260 y Fi(i)1927 1246 y Fj(u)1979 1207 y Fh(\000)p Fu(1)1979 1273 y(ln)h(\026)-43 b Fi(\013)2100 1282 y Fm(i)2129 1273 y Fi(;N)2219 1246 y Fr(\()2281 1223 y(\026)2254 1246 y Fj(X)2336 1183 y Fi(\033)2378 1192 y Fn(1)2415 1183 y Fi(:::)n(\033)2521 1192 y Fm(J)2556 1207 y(i)p Ff(\000)p Fn(1)2329 1259 y Fi(\033)2371 1268 y Fm(J)2406 1283 y(i)p Ff(\000)p Fn(1)2516 1268 y(+1)2598 1259 y Fh(\001\001\001)o Fi(\033)2705 1268 y Fm(J)2740 1283 y(i)2778 1246 y Fr(\))26 b Fk(\024)f Fr(\()p Fj(\015)3017 1260 y Fu(1)3078 1246 y Fr(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fr(+)f Fj(\015)3433 1260 y Fi(i)3464 1246 y Fr(\))p Fj(x)p Fk(g)p Fj(:)50 1470 y Fr(Then)215 1653 y Fj(f)270 1606 y Fh(\012)p Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)260 1680 y Fi(\014)s(N)548 1653 y Fr(\()p Fk(f)p Fr(1)p Fk(g)i(\002)d(\001) c(\001)g(\001)22 b(\002)e(f)p Fr(1)p Fk(g)h(\002)f Fj(I)1335 1667 y Fi(l)1383 1653 y Fk(\002)g Fr(\001)1550 1667 y Fi(l)p Fu(+1)1690 1653 y Fk(\002)g(\001)15 b(\001)g(\001)21 b(\002)f Fr(\001)2074 1669 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)2295 1653 y Fr(\))241 1896 y(=)k Fj(U)408 1857 y Fh(\000)p Fu(2)398 1923 y Fi(N)529 1783 y(l)p Fh(\000)p Fu(1)522 1810 y Fe(X)521 2005 y Fi(k)r Fu(=1)733 1810 y Fe(X)669 2001 y Fi(\033)711 2010 y Fn(1)746 2001 y Fi(:::)o(\033)853 2010 y Fm(J)888 2024 y Fn(1)943 1896 y Fj(e)985 1848 y Fu(2)p Fi(\014)s(\015)1103 1857 y Fn(1)1140 1848 y Fi(u)1183 1819 y Ff(\000)p Fn(1)1183 1872 y(ln)18 b(\026)-38 b Fm(\013)1289 1886 y Fn(1)1324 1872 y Fm(;N)1403 1848 y Fu(\()1451 1832 y(\026)1431 1848 y Fi(X)1491 1857 y Fm(\033)1528 1871 y Fn(1)1563 1857 y Ff(\001\001\001)p Fm(\033)1663 1873 y(J)1698 1887 y Fn(1)1743 1848 y Fu(\))1791 1896 y Fk(\001)15 b(\001)g(\001)2079 1810 y Fe(X)1912 2001 y Fi(\033)1954 2010 y Fm(J)1989 2026 y(k)q Ff(\000)p Fn(1)2109 2010 y(+1)2191 2001 y Fi(:::)o(\033)2298 2010 y Fm(J)2333 2026 y(k)2393 1896 y Fj(e)2435 1835 y Fu(2)p Fi(\014)s(\015)2553 1845 y Fm(k)2594 1835 y Fi(u)2637 1806 y Ff(\000)p Fn(1)2637 1859 y(ln)i(\026)-38 b Fm(\013)2742 1875 y(k)2782 1859 y(;N)2861 1835 y Fu(\()2909 1819 y(\026)2889 1835 y Fi(X)2954 1774 y Fm(\033)2991 1788 y Fn(1)3026 1774 y Fm(:::\033)3126 1790 y(J)3161 1806 y(k)q Ff(\000)p Fn(1)2949 1847 y Fm(\033)2986 1863 y(J)3021 1879 y(k)q Ff(\000)p Fn(1)3141 1863 y(+1)3223 1847 y Fm(:::\033)3323 1863 y(J)3358 1879 y(k)3407 1835 y Fu(\))1023 2156 y Fk(\002)20 b Fr(1)-23 b(I)1170 2186 y Fh(f\\)1256 2156 y Fm(k)q Ff(\000)p Fn(1)1256 2209 y Fm(j)s Fn(=1)1376 2186 y Fi(E)1432 2164 y Fm(N)1433 2205 y Fn(\026)-35 b Fm(\033)1466 2219 y Fn(1)1502 2205 y Fm(:::)s Fn(\026)g Fm(\033)1601 2220 y(j)1640 2186 y Fh(\\)1705 2170 y Fu(\026)1689 2186 y Fi(E)1745 2164 y Fm(N)1746 2205 y Fn(\026)g Fm(\033)1779 2219 y Fn(1)1815 2205 y Fm(:::)s Fn(\026)g Fm(\033)1914 2221 y(k)1959 2186 y Fh(g)2000 2156 y Fj(V)2057 2170 y Fu(\026)-40 b Fi(\033)2095 2179 y Fn(1)2131 2170 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)2294 2180 y Fm(k)235 2406 y Fr(+)20 b Fj(U)398 2367 y Fh(\000)p Fu(2)388 2433 y Fi(N)575 2320 y Fe(X)511 2511 y Fi(\033)553 2520 y Fn(1)589 2511 y Fi(:::)o(\033)696 2520 y Fm(J)731 2534 y Fn(1)786 2406 y Fj(e)828 2358 y Fu(2)p Fi(\014)s(\015)946 2367 y Fn(1)983 2358 y Fi(u)1026 2329 y Ff(\000)p Fn(1)1026 2382 y(ln)d(\026)-38 b Fm(\013)1131 2396 y Fn(1)1166 2382 y Fm(;N)1245 2358 y Fu(\()1294 2342 y(\026)1273 2358 y Fi(X)1333 2367 y Fm(\033)1370 2381 y Fn(1)1406 2367 y Ff(\001\001\001)o Fm(\033)1505 2383 y(J)1540 2397 y Fn(1)1585 2358 y Fu(\))1633 2406 y Fk(\001)15 b(\001)g(\001)1948 2320 y Fe(X)1755 2511 y Fi(\033)1797 2520 y Fm(J)1832 2536 y(l)p Ff(\000)p Fn(2)1938 2520 y(+1)2021 2511 y Fi(:::)n(\033)2127 2520 y Fm(J)2162 2536 y(l)p Ff(\000)p Fn(1)2289 2406 y Fj(e)2331 2345 y Fu(2)p Fi(\014)s(\015)2449 2355 y Fm(l)p Ff(\000)p Fn(1)2557 2345 y Fi(u)2600 2316 y Ff(\000)p Fn(1)2600 2369 y(ln)i(\026)-38 b Fm(\013)2705 2385 y(l)p Ff(\000)p Fn(1)2811 2369 y Fm(;N)2890 2345 y Fu(\()2939 2329 y(\026)2918 2345 y Fi(X)2983 2284 y Fm(\033)3020 2298 y Fn(1)3056 2284 y Fm(:::\033)3156 2300 y(J)3191 2316 y(l)p Ff(\000)p Fn(2)2978 2357 y Fm(\033)3015 2373 y(J)3050 2389 y(l)p Ff(\000)p Fn(2)3157 2373 y(+1)3239 2357 y Fm(:::\033)3339 2373 y(J)3374 2389 y(l)p Ff(\000)p Fn(1)3490 2345 y Fu(\))235 2684 y Fk(\002)20 b Fr(1)-23 b(I)382 2714 y Fh(f\\)468 2684 y Fm(l)p Ff(\000)p Fn(1)468 2737 y Fm(j)s Fn(=1)580 2714 y Fi(E)636 2692 y Fm(N)637 2733 y Fn(\026)-35 b Fm(\033)670 2747 y Fn(1)706 2733 y Fm(:::)s Fn(\026)g Fm(\033)805 2748 y(j)844 2714 y Fh(g)1121 2598 y Fe(X)987 2776 y Fn(\026)g Fm(\033)1020 2792 y(l)1046 2776 y(;:::)11 b(;)t Fn(\026)-35 b Fm(\033)1199 2794 y(l)p Fn(\()p Fm(\014)r Fn(\)+1)855 2857 y(\026)g Fm(\033)890 2842 y Ff(0)888 2888 y Fm(l)914 2857 y(;:::)11 b(;)t Fn(\026)-35 b Fm(\033)1069 2842 y Ff(0)1067 2890 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)1258 2857 y(:)t(\026)g Fm(\033)1313 2873 y(l)1340 2857 y Ff(\001)t Fn(\026)g Fm(\033)1400 2842 y Ff(0)1398 2888 y Fm(l)1424 2857 y Ff(2)p Fm(I)1495 2873 y(l)1488 2684 y Fj(S)1548 2698 y Fu(\026)-40 b Fi(\033)1586 2707 y Fn(1)1621 2698 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)1784 2708 y Fm(l)p Ff(\000)p Fn(1)1895 2684 y Fr(\()5 b(\026)-50 b Fj(\033)1982 2698 y Fi(l)2010 2684 y Fj(;)15 b(:)g(:)g(:)32 b(;)20 b Fr(\026)-50 b Fj(\033)2279 2701 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)2500 2684 y Fr(;)20 b(\026)-50 b Fj(\033)2595 2647 y Fh(0)2592 2707 y Fi(l)2621 2684 y Fj(;)15 b(:)g(:)g(:)32 b(;)20 b Fr(\026)-50 b Fj(\033)2893 2647 y Fh(0)2890 2709 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)3112 2684 y Fr(\))1599 3024 y Fk(\002)20 b Fr(1)-23 b(I)1746 3060 y Fh(f[)1832 3027 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))1832 3085 y Fm(k)q Fn(=)p Fm(l)1946 3060 y Fh([)1995 3027 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))1995 3085 y Fm(p)p Fn(=)p Fm(l)2109 3060 y Fu(\()2153 3044 y(\026)2137 3060 y Fi(E)2194 3069 y Fn(\026)-35 b Fm(\033)2227 3083 y Fn(1)2263 3069 y Fm(:::)s Fn(\026)g Fm(\033)2362 3085 y(l)p Ff(\000)p Fn(1)2473 3069 y(\026)g Fm(\033)2506 3085 y(l)2532 3069 y(:::)t Fn(\026)g Fm(\033)2632 3085 y(k)2676 3060 y Fh(\\)2741 3044 y Fu(\026)2725 3060 y Fi(E)2782 3079 y Fn(\026)g Fm(\033)2815 3093 y Fn(1)2851 3079 y Fm(:::)t Fn(\026)g Fm(\033)2951 3095 y(l)p Ff(\000)p Fn(1)3061 3079 y(\026)g Fm(\033)3096 3064 y Ff(0)3094 3110 y Fm(l)3121 3079 y(:::)s Fn(\026)g Fm(\033)3222 3064 y Ff(0)3220 3098 y Fm(p)3263 3060 y Fu(\))p Fh(g)235 3288 y Fr(+)20 b Fj(U)398 3249 y Fh(\000)p Fu(2)388 3314 y Fi(N)586 3201 y Fe(X)511 3392 y Fi(\033)553 3401 y Fn(1)589 3392 y Fi(;:::)n(\033)717 3401 y Fm(J)752 3415 y Fn(1)808 3288 y Fj(e)850 3240 y Fu(2)p Fi(\014)s(\015)968 3249 y Fn(1)1004 3240 y Fi(u)1047 3210 y Ff(\000)p Fn(1)1047 3264 y(ln)e(\026)-38 b Fm(\013)1153 3278 y Fn(1)1188 3264 y Fm(;N)1267 3240 y Fu(\()1315 3224 y(\026)1295 3240 y Fi(X)1355 3249 y Fm(\033)1392 3263 y Fn(1)1428 3249 y Ff(\001\001\001)o Fm(\033)1527 3264 y(J)1562 3278 y Fn(1)1607 3240 y Fu(\))1655 3288 y Fk(\001)15 b(\001)g(\001)1970 3201 y Fe(X)1776 3392 y Fi(\033)1818 3401 y Fm(J)1853 3417 y(l)p Ff(\000)p Fn(2)1960 3401 y(+1)2042 3392 y Fi(:::)o(\033)2149 3401 y Fm(J)2184 3417 y(l)p Ff(\000)p Fn(1)2310 3288 y Fj(e)2352 3227 y Fu(2)p Fi(\014)s(\015)2470 3237 y Fm(l)p Ff(\000)p Fn(1)2578 3227 y Fi(u)2621 3197 y Ff(\000)p Fn(1)2621 3251 y(ln)j(\026)-38 b Fm(\013)2727 3267 y(l)p Ff(\000)p Fn(1)2833 3251 y Fm(;N)2912 3227 y Fu(\()2961 3211 y(\026)2940 3227 y Fi(X)3005 3166 y Fm(\033)3042 3180 y Fn(1)3078 3166 y Fm(:::)o(\033)3177 3181 y(J)3212 3197 y(l)p Ff(\000)p Fn(2)3000 3239 y Fm(\033)3037 3254 y(J)3072 3270 y(l)p Ff(\000)p Fn(2)3179 3254 y(+1)3261 3239 y Fm(:::\033)3361 3254 y(J)3396 3270 y(l)p Ff(\000)p Fn(1)3511 3227 y Fu(\))235 3566 y Fk(\002)20 b Fr(1)-23 b(I)382 3595 y Fh(f\\)468 3566 y Fm(l)p Ff(\000)p Fn(1)468 3618 y Fm(j)s Fn(=1)580 3595 y Fi(E)636 3573 y Fm(N)637 3614 y Fn(\026)-35 b Fm(\033)670 3628 y Fn(1)706 3614 y Fm(:::)s Fn(\026)g Fm(\033)805 3629 y(j)844 3595 y Fh(g)1136 3479 y Fe(X)1002 3657 y Fn(\026)g Fm(\033)1035 3673 y(l)1061 3657 y(;:::)11 b(;)t Fn(\026)-35 b Fm(\033)1214 3676 y(l)p Fn(\()p Fm(\014)r Fn(\)+1)870 3739 y(\026)g Fm(\033)905 3723 y Ff(0)903 3769 y Fm(l)930 3739 y(;:::)10 b(;)t Fn(\026)-35 b Fm(\033)1084 3723 y Ff(0)1082 3772 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)1274 3739 y(:)t(\026)g Fm(\033)1329 3755 y(l)1355 3739 y Ff(\001)t Fn(\026)g Fm(\033)1415 3723 y Ff(0)1413 3769 y Fm(l)1440 3739 y Ff(2)p Fm(I)1511 3755 y(l)1503 3566 y Fj(S)1563 3580 y Fu(\026)-40 b Fi(\033)1601 3589 y Fn(1)1636 3580 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)1799 3590 y Fm(l)p Ff(\000)p Fn(1)1910 3566 y Fr(\()5 b(\026)-50 b Fj(\033)1997 3580 y Fi(l)2025 3566 y Fj(;)15 b(:)g(:)g(:)32 b(;)20 b Fr(\026)-50 b Fj(\033)2294 3582 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)2515 3566 y Fr(;)21 b(\026)-51 b Fj(\033)2610 3528 y Fh(0)2607 3588 y Fi(l)2636 3566 y Fj(;)15 b(:)g(:)g(:)32 b(;)20 b Fr(\026)-50 b Fj(\033)2908 3528 y Fh(0)2905 3590 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)3127 3566 y Fr(\))1569 3906 y Fk(\002)20 b Fr(1)-23 b(I)1715 3942 y Fh(f)p Fu(\()p Fh(\\)1829 3909 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))1829 3966 y Fm(k)q Fn(=)p Fm(l)1944 3942 y Fi(E)2001 3951 y Fn(\026)-35 b Fm(\033)2034 3965 y Fn(1)2070 3951 y Fm(:::)s Fn(\026)g Fm(\033)2169 3967 y(l)p Ff(\000)p Fn(1)2280 3951 y(\026)g Fm(\033)2313 3967 y(l)2339 3951 y(:::)t Fn(\026)g Fm(\033)2439 3967 y(k)2483 3942 y Fu(\))p Fh([)p Fu(\()p Fh(\\)2637 3909 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2637 3966 y Fm(p)p Fn(=)p Fm(l)2752 3942 y Fi(E)2809 3961 y Fn(\026)g Fm(\033)2842 3975 y Fn(1)2878 3961 y Fm(:::)s Fn(\026)g Fm(\033)2977 3977 y(l)p Ff(\000)p Fn(1)3088 3961 y(\026)g Fm(\033)3123 3945 y Ff(0)3121 3991 y Fm(l)3147 3961 y(:::)t Fn(\026)g Fm(\033)3249 3945 y Ff(0)3247 3979 y Fm(p)3290 3942 y Fu(\))p Fh(g)241 4108 y Fk(\021)24 b Fj(U)408 4069 y Fh(\000)p Fu(2)398 4134 y Fi(N)506 4108 y Fr(\()p Fj(M)639 4070 y Fu(1)629 4130 y Fi(N)719 4108 y Fr(+)c Fj(M)908 4070 y Fu(2)898 4130 y Fi(N)987 4108 y Fr(+)g Fj(M)1176 4070 y Fu(3)1166 4130 y Fi(N)1235 4108 y Fr(\))3477 4217 y(\(3)p Fj(:)p Fr(11\))50 4362 y(with)29 b(the)h(notations)215 4527 y Fj(S)275 4541 y Fu(\026)-40 b Fi(\033)313 4550 y Fn(1)348 4541 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)511 4551 y Fm(l)p Ff(\000)p Fn(1)622 4527 y Fr(\()5 b(\026)-50 b Fj(\033)709 4541 y Fi(l)737 4527 y Fj(;)15 b(:)g(:)g(:)32 b(;)20 b Fr(\026)-50 b Fj(\033)1006 4544 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)1228 4527 y Fr(;)20 b(\026)-50 b Fj(\033)1323 4490 y Fh(0)1320 4550 y Fi(l)1349 4527 y Fj(;)15 b(:)g(:)g(:)31 b(;)21 b Fr(\026)-51 b Fj(\033)1620 4490 y Fh(0)1617 4552 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)1839 4527 y Fr(\))240 4782 y Fk(\021)25 b Fj(e)378 4685 y Fi(\014)s(\015)460 4695 y Fm(l)488 4685 y Fi(u)531 4655 y Ff(\000)p Fn(1)531 4709 y(ln)17 b(\026)-38 b Fm(\013)636 4725 y(l)662 4709 y(;N)741 4685 y Fu(\()790 4669 y(\026)769 4685 y Fi(X)834 4624 y Fm(\033)871 4638 y Fn(1)907 4624 y Fm(:::)o(\033)1006 4639 y(J)1041 4655 y(l)p Ff(\000)p Fn(1)829 4697 y Fm(\033)866 4712 y(J)901 4728 y(l)p Ff(\000)p Fn(1)1008 4712 y(+1)1090 4697 y Fm(:::\033)1190 4712 y(J)1225 4728 y(l)1260 4685 y Fu(\)+)p Fi(\014)s(\015)1426 4695 y Fm(l)1454 4685 y Fi(u)1497 4655 y Ff(\000)p Fn(1)1497 4709 y(ln)17 b(\026)-38 b Fm(\013)1602 4725 y(l)1628 4709 y(;N)1707 4685 y Fu(\()1756 4669 y(\026)1735 4685 y Fi(X)1800 4624 y Fm(\033)1837 4638 y Fn(1)1873 4624 y Fm(:::)o(\033)1972 4639 y(J)2007 4655 y(l)p Ff(\000)p Fn(1)1795 4718 y Fm(\033)1834 4703 y Ff(0)1832 4748 y Fm(J)1867 4764 y(l)p Ff(\000)p Fn(1)1974 4748 y(+1)2056 4718 y Fm(:::\033)2158 4703 y Ff(0)2156 4748 y Fm(J)2191 4764 y(l)2226 4685 y Fu(\))296 5075 y Fk(\002)20 b Fj(e)429 4972 y Fi(\014)s(\015)511 4984 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))626 4972 y Fi(u)669 4943 y Ff(\000)p Fn(1)669 4997 y(ln)d(\026)-38 b Fm(\013)774 5015 y(l)p Fn(\()p Fm(\014)r Fn(\))888 4997 y Fm(;N)967 4972 y Fu(\()1015 4956 y(\026)995 4972 y Fi(X)1060 4905 y Fm(\033)1097 4919 y Fn(1)1132 4905 y Fm(:::\033)1232 4921 y(J)1267 4939 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)1055 4985 y Fm(\033)1092 5000 y(J)1127 5018 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)1321 5000 y(+1)1403 4985 y Fm(:::\033)1503 5000 y(J)1538 5018 y(l)p Fn(\()p Fm(\014)r Fn(\))1661 4972 y Fu(\)+)p Fi(\014)s(\015)1827 4984 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))1942 4972 y Fi(u)1985 4943 y Ff(\000)p Fn(1)1985 4997 y(ln)17 b(\026)-38 b Fm(\013)2090 5015 y(l)p Fn(\()p Fm(\014)r Fn(\))2204 4997 y Fm(;N)2283 4972 y Fu(\()2331 4956 y(\026)2311 4972 y Fi(X)2376 4893 y Fm(\033)2413 4907 y Fn(1)2448 4893 y Fm(:::\033)2550 4875 y Ff(0)2548 4921 y Fm(J)2583 4939 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)2371 5006 y Fm(\033)2410 4990 y Ff(0)2408 5036 y Fm(J)2443 5054 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)2637 5036 y(+1)2719 5006 y Fm(:::\033)2821 4990 y Ff(0)2819 5036 y Fm(J)2854 5054 y(l)p Fn(\()p Fm(\014)r Fn(\))2977 4972 y Fu(\))296 5293 y Fk(\002)20 b Fj(e)429 5204 y Fi(\014)473 5150 y Fh(p)p 533 5150 65 4 v 54 x Fi(N)7 b Fu(\()626 5164 y Fk(p)p 702 5164 271 4 v 40 x Fi(a)741 5213 y Fm(J)776 5232 y(l)p Fn(\()p Fm(\014)r Fn(\))890 5213 y(+1)973 5204 y Fi(X)1033 5213 y Fm(\033)1070 5227 y Fn(1)1105 5213 y Fm(:::)o(\033)1204 5229 y(J)1239 5247 y(l)p Fn(\()p Fm(\014)r Fn(\))1353 5229 y(+1)1440 5204 y Fu(+)p Fh(\001\001\001)o Fu(+)1617 5162 y Fh(p)p 1676 5162 85 4 v 1676 5204 a Fi(a)1715 5213 y Fm(n)1760 5204 y Fi(X)1820 5213 y Fm(\033)1857 5227 y Fn(1)1892 5213 y Fm(:::\033)1992 5222 y(n)2041 5204 y Fu(+)2097 5164 y Fk(p)p 2173 5164 271 4 v 40 x Fi(a)2212 5213 y Fm(J)2247 5232 y(l)p Fn(\()p Fm(\014)r Fn(\))2361 5213 y(+1)2443 5204 y Fi(X)2503 5223 y Fm(\033)2540 5237 y Fn(1)2575 5223 y Fm(:::\033)2677 5207 y Ff(0)2675 5253 y Fm(J)2710 5272 y(l)p Fn(\()p Fm(\014)r Fn(\))2824 5253 y(+1)2911 5204 y Fu(+)p Fh(\001\001\001)n Fu(+)3087 5162 y Fh(p)p 3147 5162 85 4 v 42 x Fi(a)3186 5213 y Fm(n)3231 5204 y Fi(X)3291 5223 y Fm(\033)3328 5237 y Fn(1)3363 5223 y Fm(:::)o(\033)3464 5207 y Ff(0)3462 5242 y Fm(n)3512 5204 y Fu(\))296 5466 y Fk(\002)20 b Fj(e)429 5418 y Fh(\000)p Fi(\014)530 5390 y Fn(2)566 5418 y Fi(N)7 b Fu(\()p Fi(a)698 5427 y Fm(J)733 5445 y(l)p Fn(\()p Fm(\014)r Fn(\))847 5427 y(+1)929 5418 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(a)1145 5427 y Fm(n)1190 5418 y Fu(\))p Fh(\000)p Fu(2)p Fi(N)19 b Fu(ln)q(\()p Fi(\013)1524 5427 y Fm(J)1559 5445 y(l)p Fn(\()p Fm(\014)r Fn(\)+1)1756 5418 y Fh(\001\001\001)n Fi(\013)1867 5427 y Fm(n)1912 5418 y Fu(\))p eop %%Page: 32 32 32 31 bop 50 295 a Fo(32)3270 b Fl(Se)l(ction)27 b Fo(3)97 689 y Fj(V)154 703 y Fu(\026)-40 b Fi(\033)192 712 y Fn(1)227 703 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)390 713 y Fm(k)122 875 y Fk(\021)218 774 y Fe(h)436 789 y(X)276 980 y Fi(\033)318 989 y Fm(J)353 1005 y(k)393 989 y Fn(+1)475 980 y Fi(:::)o(\033)582 989 y Fm(J)617 1005 y(l)p Ff(\000)p Fn(1)743 875 y Fj(e)785 817 y Fu(2)p Fi(\014)865 763 y Fh(p)p 926 763 65 4 v 926 817 a Fi(N)7 b Fu(\()1019 784 y Fh(p)p 1079 784 197 4 v 33 x Fi(a)1118 826 y Fm(J)1153 842 y(k)1193 826 y Fn(+1)1275 817 y Fi(X)1335 826 y Fm(\033)1372 840 y Fn(1)1407 826 y Fm(:::\033)1507 842 y(J)1542 858 y(k)1582 842 y Fn(+1)1669 817 y Fu(+)p Fh(\001\001\001)n Fu(+)1845 774 y Fk(p)p 1921 774 186 4 v 43 x Fi(a)1960 826 y Fm(J)1995 842 y(l)p Ff(\000)p Fn(1)2106 817 y Fi(X)2166 826 y Fm(\033)2203 840 y Fn(1)2238 826 y Fm(:::\033)2338 842 y(J)2373 858 y(l)p Ff(\000)p Fn(1)2489 817 y Fu(\))342 1114 y Fe(X)179 1285 y Fm(\033)216 1301 y(J)251 1317 y(l)p Ff(\000)p Fn(1)358 1301 y(+1)440 1285 y Fm(;:::)j(;\033)592 1294 y(n)32 1376 y(\033)71 1360 y Ff(0)69 1406 y Fm(J)104 1422 y(l)p Ff(\000)p Fn(1)210 1406 y(+1)293 1376 y Fm(;:::)g(;\033)447 1360 y Ff(0)445 1394 y Fm(n)490 1376 y Fn(:)34 b(\026)-35 b Fm(\033)575 1392 y(l)602 1376 y Ff(\001)t Fn(\026)g Fm(\033)662 1360 y Ff(0)660 1406 y Fm(l)686 1376 y Ff(2)p Fm(I)757 1392 y(l)492 1200 y Fj(e)534 1118 y Fi(\014)578 1064 y Fh(p)p 638 1064 65 4 v 54 x Fi(N)7 b Fu(\()731 1075 y Fk(p)p 807 1075 264 4 v 43 x Fi(a)846 1127 y Fm(J)881 1143 y(l)p Ff(\000)p Fn(1)988 1127 y(+1)1070 1118 y Fi(X)1130 1127 y Fm(\033)1167 1141 y Fn(1)1202 1127 y Fm(:::\033)1302 1143 y(J)1337 1159 y(l)p Ff(\000)p Fn(1)1443 1143 y(+1)1530 1118 y Fu(+)p Fh(\001\001\001)o Fu(+)1707 1077 y Fh(p)p 1766 1077 85 4 v 1766 1118 a Fi(a)1805 1127 y Fm(n)1850 1118 y Fi(X)1910 1127 y Fm(\033)1947 1141 y Fn(1)1982 1127 y Fm(:::\033)2082 1136 y(n)2131 1118 y Fu(+)2187 1075 y Fk(p)p 2263 1075 264 4 v 43 x Fi(a)2302 1127 y Fm(J)2337 1143 y(l)p Ff(\000)p Fn(1)2444 1127 y(+1)2526 1118 y Fi(X)2586 1137 y Fm(\033)2623 1151 y Fn(1)2658 1137 y Fm(:::\033)2760 1122 y Ff(0)2758 1167 y Fm(J)2793 1183 y(l)p Ff(\000)p Fn(1)2899 1167 y(+1)2986 1118 y Fu(+)p Fh(\001\001\001)o Fu(+)3163 1077 y Fh(p)p 3222 1077 85 4 v 3222 1118 a Fi(a)3261 1127 y Fm(n)3306 1118 y Fi(X)3366 1137 y Fm(\033)3403 1151 y Fn(1)3438 1137 y Fm(:::\033)3540 1122 y Ff(0)3538 1156 y Fm(n)3587 1118 y Fu(\))3620 1100 y Fe(i)117 1598 y Fk(\002)208 1497 y Fe(h)375 1512 y(X)266 1702 y Fi(\033)308 1711 y Fm(J)343 1728 y(k)383 1711 y Fn(+1)465 1702 y Fi(:::)o(\033)572 1711 y Fm(n)632 1598 y Fj(e)674 1548 y Fi(\014)718 1493 y Fh(p)p 778 1493 65 4 v 55 x Fi(N)g Fu(\()871 1514 y Fh(p)p 931 1514 197 4 v 34 x Fi(a)970 1557 y Fm(J)1005 1573 y(k)1045 1557 y Fn(+1)1127 1548 y Fi(X)1187 1557 y Fm(\033)1224 1571 y Fn(1)1259 1557 y Fm(:::\033)1359 1572 y(J)1394 1588 y(k)1434 1572 y Fn(+1)1521 1548 y Fu(+)p Fh(\001\001\001)n Fu(+)1697 1506 y Fh(p)p 1757 1506 85 4 v 42 x Fi(a)1796 1557 y Fm(n)1841 1548 y Fi(X)1901 1557 y Fm(\033)1938 1571 y Fn(1)1973 1557 y Fm(:::)o(\033)2072 1566 y(n)2121 1548 y Fu(\))2154 1497 y Fe(i)2197 1516 y Fh(\000)p Fu(2)117 1927 y Fk(\002)208 1826 y Fe(h)430 1841 y(X)266 2031 y Fi(\033)308 2040 y Fm(J)343 2056 y(k)383 2040 y Fn(+1)465 2031 y Fi(:::)o(\033)572 2040 y Fm(J)607 2059 y(l)p Fn(\()p Fm(\014)r Fn(\))741 1927 y Fj(e)783 1860 y Fi(\014)s(\015)865 1870 y Fm(k)q Fn(+1)983 1860 y Fi(u)1026 1830 y Ff(\000)p Fn(1)1026 1884 y(ln)17 b(\026)-37 b Fm(\013)1132 1900 y(k)q Fn(+1)1248 1884 y Fm(;N)1328 1860 y Fu(\()1376 1844 y(\026)1356 1860 y Fi(X)1421 1806 y Fm(\033)1458 1820 y Fn(1)1493 1806 y Fm(:::\033)1593 1822 y(J)1628 1838 y(k)1416 1872 y(\033)1453 1887 y(J)1488 1903 y(k)1528 1887 y Fn(+1)1610 1872 y Fm(:::)o(\033)1709 1887 y(J)1744 1903 y(k)q Fn(+1)1871 1860 y Fu(\)+)p Fh(\001\001\001)o Fu(+)p Fi(\014)s(\015)2158 1872 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2272 1860 y Fi(u)2315 1830 y Ff(\000)p Fn(1)2315 1884 y(ln)18 b(\026)-38 b Fm(\013)2421 1902 y(l)p Fn(\()p Fm(\014)r Fn(\))2534 1884 y Fm(;N)2614 1860 y Fu(\()2662 1844 y(\026)2642 1860 y Fi(X)2707 1793 y Fm(\033)2744 1807 y Fn(1)2779 1793 y Fm(:::\033)2879 1808 y(J)2914 1826 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)2702 1872 y Fm(\033)2739 1887 y(J)2774 1906 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)2968 1887 y(+1)3050 1872 y Fm(:::\033)3150 1887 y(J)3185 1906 y(l)p Fn(\()p Fm(\014)r Fn(\))3308 1860 y Fu(\))211 2154 y Fe(X)36 2345 y Fi(\033)78 2354 y Fm(J)113 2373 y(l)p Fn(\()p Fm(\014)r Fn(\))227 2354 y(+1)309 2345 y Fi(;:::)11 b(;\033)472 2354 y Fm(n)350 2240 y Fj(e)392 2176 y Fi(\014)436 2122 y Fh(p)p 496 2122 65 4 v 54 x Fi(N)c Fu(\()589 2136 y Fk(p)p 665 2136 271 4 v 40 x Fi(a)704 2185 y Fm(J)739 2204 y(l)p Fn(\()p Fm(\014)r Fn(\))853 2185 y(+1)936 2176 y Fi(X)996 2185 y Fm(\033)1033 2199 y Fn(1)1068 2185 y Fm(:::)o(\033)1167 2201 y(J)1202 2219 y(l)p Fn(\()p Fm(\014)r Fn(\))1316 2201 y(+1)1391 2176 y Fu(+)p Fh(\001\001\001)n Fu(+)1567 2135 y Fh(p)p 1627 2135 85 4 v 41 x Fi(a)1666 2185 y Fm(n)1711 2176 y Fi(X)1771 2185 y Fm(\033)1808 2199 y Fn(1)1843 2185 y Fm(:::)o(\033)1942 2194 y(n)1992 2176 y Fu(\))p Fh(\000)p Fi(\014)2121 2149 y Fn(2)2157 2176 y Fi(N)g Fu(\()p Fi(a)2289 2185 y Fm(J)2324 2204 y(l)p Fn(\()p Fm(\014)r Fn(\))2438 2185 y(+1)2520 2176 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(a)2736 2185 y Fm(n)2781 2176 y Fu(\))p Fi(=)p Fu(2)p Fh(\000)p Fi(N)19 b Fu(ln)q(\()p Fi(\013)3152 2185 y Fm(J)3187 2204 y(l)p Fn(\()p Fm(\014)r Fn(\)+1)3359 2176 y Fh(\001\001\001)o Fi(\013)3471 2185 y Fm(n)3516 2176 y Fu(\))3549 2140 y Fe(i)3591 2158 y Fu(2)50 2555 y Fr(and)130 2807 y Fj(U)192 2821 y Fi(N)286 2807 y Fr(=)447 2721 y Fe(X)382 2912 y Fi(\033)424 2921 y Fn(1)460 2912 y Fi(:::)o(\033)567 2921 y Fm(J)602 2935 y Fn(1)657 2807 y Fj(e)699 2759 y Fi(\014)s(\015)781 2768 y Fn(1)818 2759 y Fi(u)861 2730 y Ff(\000)p Fn(1)861 2784 y(ln)e(\026)-38 b Fm(\013)966 2798 y Fn(1)1001 2784 y Fm(;N)1080 2759 y Fu(\()1129 2743 y(\026)1108 2759 y Fi(X)1168 2768 y Fm(\033)1205 2782 y Fn(1)1241 2768 y Ff(\001\001\001)p Fm(\033)1341 2784 y(J)1376 2798 y Fn(1)1420 2759 y Fu(\))1468 2807 y Fk(\001)15 b(\001)g(\001)1831 2721 y Fe(X)1590 2912 y Fi(\033)1632 2921 y Fm(J)1667 2940 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)1861 2921 y(+1)1943 2912 y Fi(:::)o(\033)2050 2921 y Fm(J)2085 2940 y(l)p Fn(\()p Fm(\014)r Fn(\))2219 2807 y Fj(e)2261 2740 y Fi(\014)s(\015)2343 2752 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2458 2740 y Fi(u)2501 2711 y Ff(\000)p Fn(1)2501 2764 y(ln)i(\026)-38 b Fm(\013)2606 2783 y(l)p Fn(\()p Fm(\014)r Fn(\))2720 2764 y Fm(;N)2799 2740 y Fu(\()2847 2724 y(\026)2827 2740 y Fi(X)2892 2673 y Fm(\033)2929 2687 y Fn(1)2964 2673 y Fm(:::\033)3064 2688 y(J)3099 2707 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)2887 2752 y Fm(\033)2924 2768 y(J)2959 2786 y(l)p Fn(\()p Fm(\014)r Fn(\))p Ff(\000)p Fn(1)3153 2768 y(+1)3235 2752 y Fm(:::\033)3335 2768 y(J)3370 2786 y(l)p Fn(\()p Fm(\014)r Fn(\))3493 2740 y Fu(\))229 3035 y Fe(X)54 3225 y Fi(\033)96 3234 y Fm(J)131 3253 y(l)p Fn(\()p Fm(\014)r Fn(\))246 3234 y(+1)328 3225 y Fi(;:::)11 b(;\033)491 3234 y Fm(n)368 3121 y Fj(e)410 3057 y Fi(\014)454 3003 y Fh(p)p 515 3003 65 4 v 515 3057 a Fi(N)6 b Fu(\()607 3017 y Fk(p)p 683 3017 271 4 v 40 x Fi(a)722 3066 y Fm(J)757 3084 y(l)p Fn(\()p Fm(\014)r Fn(\))872 3066 y(+1)954 3057 y Fi(X)1014 3066 y Fm(\033)1051 3080 y Fn(1)1086 3066 y Fm(:::\033)1186 3081 y(J)1221 3100 y(l)p Fn(\()p Fm(\014)r Fn(\))1335 3081 y(+1)1421 3057 y Fu(+)p Fh(\001\001\001)o Fu(+)1598 3015 y Fh(p)p 1658 3015 85 4 v 42 x Fi(a)1697 3066 y Fm(n)1742 3057 y Fi(X)1802 3066 y Fm(\033)1839 3080 y Fn(1)1874 3066 y Fm(:::)o(\033)1973 3075 y(n)2022 3057 y Fu(\))p Fh(\000)p Fi(\014)2151 3029 y Fn(2)2188 3057 y Fi(N)h Fu(\()p Fi(a)2320 3066 y Fm(J)2355 3084 y(l)p Fn(\()p Fm(\014)r Fn(\))2469 3066 y(+1)2551 3057 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(a)2767 3066 y Fm(n)2811 3057 y Fu(\))p Fi(=)p Fu(2)p Fh(\000)p Fi(N)20 b Fu(ln)q(\()p Fi(\013)3183 3066 y Fm(J)3218 3084 y(l)p Fn(\()p Fm(\014)r Fn(\)+1)3415 3057 y Fh(\001\001\001)n Fi(\013)3526 3066 y Fm(n)3571 3057 y Fu(\))3604 3121 y Fj(:)159 3529 y Fr(The)32 b(third)f(term)i(in)e(\(3.11\))k Fj(U)1243 3490 y Fh(\000)p Fu(2)1233 3555 y Fi(N)1340 3529 y Fj(M)1438 3496 y Fu(3)1428 3554 y Fi(N)1530 3529 y Fr(is)d(not)h(bigger)f(than) 2311 3506 y(~)2277 3529 y Fj(M)2375 3496 y Fu(3)2365 3554 y Fi(N)2434 3529 y Fj(=U)2541 3543 y Fi(N)2643 3529 y Fr(where)2942 3506 y(~)2909 3529 y Fj(M)3007 3496 y Fu(3)2997 3554 y Fi(N)3098 3529 y Fr(is)g(the)h(sum)e(o)m(v)m(er)50 3674 y(the)h(same)g(terms)f(as)h Fj(U)866 3688 y Fi(N)967 3674 y Fr(but)f(truncated)g(b)m(y)h(1)-23 b(I)p Fk(f\\)1839 3627 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1839 3699 y Fi(j)t Fu(=1)1970 3674 y Fj(E)2041 3688 y Fu(\026)-40 b Fi(\033)2079 3697 y Fn(1)2114 3688 y Fi(:::)s Fu(\026)g Fi(\033)2221 3697 y Fm(j)2260 3674 y Fk(g)p Fr(.)45 b(By)32 b(Theorem)f(1.3)i(the)f (denominator)50 3820 y Fj(U)112 3834 y Fi(N)214 3820 y Fr(con)m(v)m(erges)k(in)c(la)m(w)i(to)g(the)g(a.s.)51 b(\014nite)32 b(in)m(tegral)i(o)m(v)m(er)g(the)g(pro)s(cess)f Fk(P)2675 3787 y Fu(\()p Fi(l)p Fu(\()p Fi(\014)s Fu(\)\))2862 3820 y Fr(.)51 b(Then)32 b(for)i(an)m(y)f Fj(\017)e(>)f Fr(0)50 3965 y(one)35 b(can)g(\014nd)e(a)i(small)f Fj(\016)s Fr(\()p Fj(\017)p Fr(\))g Fj(>)e Fr(0)j(suc)m(h)f(that)i Fg(P)p Fr(\()p Fj(U)1831 3979 y Fi(N)1933 3965 y Fj(<)d(\016)s Fr(\()p Fj(\017)p Fr(\)\))h Fj(<)e(\017)j Fr(for)f(all)g(su\016cien)m (tly)g(large)g Fj(N)10 b Fr(.)54 b(The)50 4123 y(exp)s(ectation)36 b(of)g(the)g(n)m(umerator)1297 4100 y(~)1263 4123 y Fj(M)1361 4090 y Fu(3)1351 4148 y Fi(N)1456 4123 y Fr(is)f(b)s(ounded)e(b)m(y)j Fj(e)2103 4070 y Fi(x)2157 4008 y Fe(P)2253 4031 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2253 4104 y Fm(j)s Fn(=1)2367 4070 y Fu(\()p Fi(\014)s(\015)2477 4079 y Fm(j)2512 4070 y Fh(\000)p Fu(1\))2673 4123 y Fr(for)g(all)e Fj(N)46 b Fr(large)35 b(enough)h(b)m(y)50 4268 y(Lemma)k(2.2.)71 b(Then)39 b(for)h(an)m(y)g(giv)m(en)g(pair)f(of)h Fj(\017;)16 b Fr(~)-46 b Fj(\017)42 b(>)f Fr(0)f(one)g(can)h(\014nd)d Fj(x)k(<)f Fr(0)f(su\016cien)m(tly)f(large)h(b)m(y)50 4413 y(absolute)30 b(v)-5 b(alue)30 b(suc)m(h)g(that)501 4596 y Fg(P)p Fr(\()p Fj(M)689 4558 y Fu(3)679 4618 y Fi(N)749 4596 y Fj(=U)866 4558 y Fu(2)856 4618 y Fi(N)951 4596 y Fj(>)25 b Fr(~)-45 b Fj(\017)p Fr(\))25 b Fk(\024)g Fg(P)p Fr(\()1365 4573 y(~)1330 4596 y Fj(M)1428 4558 y Fu(3)1418 4618 y Fi(N)1488 4596 y Fj(=U)1595 4610 y Fi(N)1690 4596 y Fj(>)h Fr(~)-46 b Fj(\017)p Fr(\))25 b Fk(\024)g Fg(P)p Fr(\()2104 4573 y(~)2069 4596 y Fj(M)2167 4558 y Fu(3)2157 4618 y Fi(N)2227 4596 y Fj(=\016)s Fr(\()p Fj(\017)p Fr(\))i Fj(>)f Fr(~)-46 b Fj(\017)p Fr(\))21 b(+)f Fg(P)p Fr(\()p Fj(U)2881 4610 y Fi(N)2976 4596 y Fj(>)25 b(\016)s Fr(\()p Fj(\017)p Fr(\)\))617 4851 y Fj(<)724 4790 y(e)766 4737 y Fi(x)820 4676 y Fe(P)916 4698 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))916 4771 y Fm(j)s Fn(=1)1030 4737 y Fu(\()p Fi(\014)s(\015)1140 4746 y Fm(j)1175 4737 y Fh(\000)p Fu(1\))p 724 4830 577 4 v 919 4914 a Fr(~)-46 b Fj(\017\016)s Fr(\()p Fj(\017)p Fr(\))1332 4851 y(+)20 b Fj(\017)25 b(<)g Fr(2)p Fj(\017)50 5098 y Fr(for)30 b(all)f Fj(N)40 b Fr(starting)31 b(from)f(some)g(n)m (um)m(b)s(er)f Fj(N)1614 5112 y Fu(0)1655 5098 y Fr(.)159 5316 y(Giv)m(en)k(~)-45 b Fj(\017)o(;)15 b(\017)30 b(>)f Fr(0,)34 b(let)f(us)f(then)g(\014x)h(an)f(appropriate)g Fj(x)d(<)g Fr(0)k(and)g(consider)e(the)i(terms)g Fj(M)3295 5283 y Fu(1)3285 5341 y Fi(N)3354 5316 y Fj(=U)3471 5283 y Fu(2)3461 5341 y Fi(N)3563 5316 y Fr(and)50 5462 y Fj(M)148 5429 y Fu(2)138 5487 y Fi(N)207 5462 y Fj(=U)324 5429 y Fu(2)314 5487 y Fi(N)420 5462 y Fr(with)i(this)g(c)m(hoice.)59 b(T)-8 b(o)37 b(treat)h Fj(M)1606 5429 y Fu(2)1596 5487 y Fi(N)1665 5462 y Fj(=U)1782 5429 y Fu(2)1772 5487 y Fi(N)1841 5462 y Fr(,)g(it)e(su\016ces)g(to)h(sho)m(w)f(that)h Fj(M)2965 5429 y Fu(2)2955 5487 y Fi(N)3061 5462 y Fr(is)e(non-zero)i (with)p eop %%Page: 33 33 33 32 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(33)50 513 y Fr(v)-5 b(anishing)28 b(probabilit)m(y)g(as)i Fj(N)41 b Fr(tends)30 b(to)h(in\014nit)m(y)-8 b(.)39 b(This)28 b(means)i(that)h(the)g(random)f(v)-5 b(ariable)171 747 y Fj(\021)216 761 y Fi(N)311 747 y Fr(=)514 660 y Fe(X)411 851 y Fu(\026)-40 b Fi(\033)449 860 y Fn(1)484 851 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)647 861 y Fm(l)p Ff(\000)p Fn(1)769 747 y Fr(1)-23 b(I)824 777 y Fh(f\\)910 747 y Fm(l)p Ff(\000)p Fn(1)910 799 y Fm(j)s Fn(=1)1022 777 y Fi(E)1078 755 y Fm(N)1079 796 y Fn(\026)-35 b Fm(\033)1112 810 y Fn(1)1148 796 y Fm(:::)s Fn(\026)g Fm(\033)1247 811 y(j)1286 777 y Fh(g)1518 660 y Fe(X)1384 838 y Fn(\026)g Fm(\033)1417 854 y(l)1443 838 y(;:::)11 b(;)t Fn(\026)-35 b Fm(\033)1596 857 y(l)p Fn(\()p Fm(\014)r Fn(\)+1)1252 920 y(\026)g Fm(\033)1287 904 y Ff(0)1285 950 y Fm(l)1311 920 y(;:::)11 b(;)t Fn(\026)-35 b Fm(\033)1466 904 y Ff(0)1464 953 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)1655 920 y(:)t(\026)g Fm(\033)1710 936 y(l)1737 920 y Ff(\001)t Fn(\026)g Fm(\033)1797 904 y Ff(0)1795 950 y Fm(l)1821 920 y Ff(2)p Fm(I)1892 936 y(l)1945 747 y Fr(1)-23 b(I)2001 783 y Fh(f[)2087 750 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2087 807 y Fm(k)q Fn(=)p Fm(l)2201 783 y Fh([)2250 750 y Fm(l)p Fn(\()p Fm(\014)r Fn(\))2250 807 y Fm(p)p Fn(=)p Fm(l)2364 783 y Fu(\()2408 767 y(\026)2392 783 y Fi(E)2449 792 y Fn(\026)-35 b Fm(\033)2482 806 y Fn(1)2518 792 y Fm(:::)s Fn(\026)g Fm(\033)2617 808 y(l)p Ff(\000)p Fn(1)2728 792 y(\026)g Fm(\033)2761 808 y(l)2787 792 y(:::)t Fn(\026)g Fm(\033)2887 808 y(k)2931 783 y Fh(\\)2996 767 y Fu(\026)2980 783 y Fi(E)3037 802 y Fn(\026)g Fm(\033)3070 816 y Fn(1)3106 802 y Fm(:::)t Fn(\026)g Fm(\033)3206 818 y(l)p Ff(\000)p Fn(1)3316 802 y(\026)g Fm(\033)3351 786 y Ff(0)3349 832 y Fm(l)3376 802 y(:::)s Fn(\026)g Fm(\033)3477 786 y Ff(0)3475 820 y Fm(p)3518 783 y Fu(\))p Fh(g)50 1166 y Fr(do)s(es)28 b(not)h(equal)f(zero)h(with)f(probabilit)m(y)e(con)m(v) m(erging)j(to)h(zero.)40 b(Let)30 b(us)d(decomp)s(ose)i Fj(\021)3073 1180 y Fi(N)3168 1166 y Fr(=)3264 1098 y Fe(P)3359 1119 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))3359 1194 y Fi(k)r(;p)p Fu(=)p Fi(l)3558 1166 y Fj(\021)3606 1122 y Fi(p;k)3603 1193 y(N)50 1312 y Fr(where)124 1505 y Fj(\021)172 1461 y Fi(p;k)169 1531 y(N)302 1505 y Fr(=)472 1418 y Fe(X)402 1609 y Fu(\026)-40 b Fi(\033)440 1618 y Fn(1)475 1609 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)638 1619 y Fm(k)693 1505 y Fr(1)-23 b(I)748 1534 y Fh(f\\)834 1505 y Fm(k)q Ff(\000)p Fn(1)834 1557 y Fm(j)s Fn(=1)955 1534 y Fi(E)1012 1543 y Fn(\026)-35 b Fm(\033)1045 1557 y Fn(1)1080 1543 y Fm(;:::)11 b(;)t Fn(\026)-35 b Fm(\033)1233 1558 y(j)1271 1534 y Fh(\\)1336 1518 y Fu(\026)1320 1534 y Fi(E)1377 1543 y Fn(\026)g Fm(\033)1410 1557 y Fn(1)1446 1543 y Fm(;:::)11 b(;)t Fn(\026)-35 b Fm(\033)1599 1559 y(k)1643 1534 y Fh(g)1807 1418 y Fe(X)1598 1617 y Fu(\026)-40 b Fi(\033)1638 1595 y Ff(0)1636 1641 y Fm(l)1663 1617 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)1828 1599 y Ff(0)1826 1635 y Fm(p)1863 1617 y Fu(:)t(\026)g Fi(\033)1926 1627 y Fm(l)1953 1617 y Fh(\001)t Fu(\026)g Fi(\033)2019 1595 y Ff(0)2017 1641 y Fm(l)2043 1617 y Fh(2)p Fi(I)2124 1627 y Fm(l)2060 1505 y Fr(1)-23 b(I)2115 1534 y Fh(f\\)2201 1504 y Fm(p)p Ff(\000)p Fn(1)2201 1557 y Fm(i)p Fn(=1)2320 1534 y Fi(E)2377 1553 y Fn(\026)-35 b Fm(\033)2410 1567 y Fn(1)2446 1553 y Fm(;:::)10 b(;)t Fn(\026)-35 b Fm(\033)2598 1569 y(l)p Ff(\000)p Fn(1)2709 1553 y(\026)g Fm(\033)2744 1538 y Ff(0)2742 1584 y Fm(l)2768 1553 y(:::)t Fn(\026)g Fm(\033)2870 1538 y Ff(0)2868 1583 y Fm(i)2901 1534 y Fh(\\)2966 1518 y Fu(\026)2950 1534 y Fi(E)3007 1553 y Fn(\026)g Fm(\033)3040 1567 y Fn(1)3076 1553 y Fm(;:::)10 b(;)t Fn(\026)-35 b Fm(\033)3228 1569 y(l)p Ff(\000)p Fn(1)3339 1553 y(\026)g Fm(\033)3374 1538 y Ff(0)3372 1584 y Fm(l)3398 1553 y(;:::)11 b(;)t Fn(\026)-35 b Fm(\033)3553 1538 y Ff(0)3551 1572 y Fm(p)3593 1534 y Fh(g)50 1819 y Fj(p;)15 b(k)28 b Fr(=)d Fj(l)r(;)15 b(:)g(:)g(:)32 b(;)15 b(l)r Fr(\()p Fj(\014)5 b Fr(\))32 b(and)d(estimate)i(for)g Fj(p)25 b Fk(\025)g Fj(k)j Fk(\025)d Fj(l)124 2035 y(\021)172 1991 y Fi(p;k)169 2062 y(N)301 2035 y Fk(\024)32 b Fr(~)-52 b Fj(\021)445 1991 y Fi(p;k)442 2062 y(N)575 2035 y Fr(=)745 1949 y Fe(X)675 2140 y Fu(\026)-41 b Fi(\033)712 2149 y Fn(1)748 2140 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)911 2150 y Fm(k)965 2035 y Fr(1)-23 b(I)1021 2056 y Fh(f)1074 2040 y Fu(\026)1058 2056 y Fi(E)1115 2065 y Fn(\026)-35 b Fm(\033)1148 2079 y Fn(1)1184 2065 y Fm(;:::)11 b(;)t Fn(\026)-35 b Fm(\033)1337 2081 y(k)1381 2056 y Fh(g)352 2207 y Fe(X)143 2405 y Fu(\026)-40 b Fi(\033)183 2383 y Ff(0)181 2429 y Fm(l)208 2405 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)373 2387 y Ff(0)371 2423 y Fm(p)408 2405 y Fu(:)t(\026)g Fi(\033)471 2415 y Fm(l)498 2405 y Fh(\001)t Fu(\026)g Fi(\033)564 2383 y Ff(0)562 2429 y Fm(l)588 2405 y Fh(2)p Fi(I)669 2415 y Fm(l)711 2293 y Fr(1)-23 b(I)766 2378 y Fh(f)p Fi(\015)841 2388 y Fm(l)868 2378 y Fi(u)911 2348 y Ff(\000)p Fn(1)911 2402 y(ln)11 b Fm(\013)1017 2418 y(l)1043 2402 y(;N)1122 2378 y Fu(\()1170 2362 y(\026)1150 2378 y Fi(X)1215 2317 y Fm(\033)1252 2331 y Fn(1)1287 2317 y Fm(;:::)g(;\033)1440 2332 y(J)1475 2348 y(l)p Ff(\000)p Fn(1)1210 2411 y Fm(\033)1249 2396 y Ff(0)1247 2441 y Fm(J)1282 2457 y(l)p Ff(\000)p Fn(1)1389 2441 y(+1)1471 2411 y Fm(;:::)f(;\033)1625 2396 y Ff(0)1623 2441 y Fm(J)1658 2457 y(l)1694 2378 y Fu(\)+)p Fh(\001\001\001)o Fu(+)p Fi(\015)1937 2387 y Fm(p)1975 2378 y Fi(u)2018 2348 y Ff(\000)p Fn(1)2018 2402 y(ln)h Fm(\013)2124 2411 y(p)2161 2402 y(;N)2240 2378 y Fu(\()2289 2362 y(\026)2268 2378 y Fi(X)2333 2302 y Fm(\033)2370 2316 y Fn(1)2406 2302 y Fm(;:::)f(;\033)2560 2287 y Ff(0)2558 2332 y Fm(J)2593 2347 y(p)p Ff(\000)p Fn(1)2328 2411 y Fm(\033)2367 2396 y Ff(0)2365 2441 y Fm(J)2400 2456 y(p)p Ff(\000)p Fn(1)2519 2441 y(+1)2601 2411 y Fm(;:::)h(;\033)2756 2396 y Ff(0)2754 2441 y Fm(J)2789 2450 y(p)2836 2378 y Fu(\))p Fi(>)p Fu(\()p Fi(\015)2987 2388 y Fm(l)3014 2378 y Fu(+)p Fh(\001\001\001)o Fu(+)p Fi(\015)3229 2387 y Fm(p)3267 2378 y Fu(\))p Fi(x)p Fh(g)3378 2293 y Fj(:)3477 2225 y Fr(\(3)p Fj(:)p Fr(12\))50 2650 y(W)-8 b(e)35 b(will)d(sho)m(w)i Fg(P)p Fr(\()7 b(~)-52 b Fj(\021)753 2607 y Fi(p;k)750 2677 y(N)891 2650 y Fk(6)p Fr(=)31 b(0\))i Fk(!)f Fr(0)j(as)f Fj(N)42 b Fk(!)32 b Fr(0.)53 b(First)34 b(of)h(all,)f(note)h(that)g(for)f(an)m (y)h Fj(\017)d(>)g Fr(0)i(there)h(exists)50 2796 y Fj(K)7 b Fr(\()p Fj(\017)p Fr(\))39 b(suc)m(h)f(that)i Fg(P)p Fr(\()789 2728 y Fe(P)890 2823 y Fu(\026)-40 b Fi(\033)928 2832 y Fn(1)963 2823 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)1126 2833 y Fm(k)1185 2796 y Fr(1)-23 b(I)1241 2817 y Fh(f)1294 2801 y Fu(\026)1278 2817 y Fi(E)1335 2826 y Fn(\026)-35 b Fm(\033)1368 2840 y Fn(1)1404 2826 y Fm(;:::)10 b(;)t Fn(\026)-35 b Fm(\033)1556 2842 y(k)1601 2817 y Fh(g)1682 2796 y Fj(>)39 b(K)7 b Fr(\()p Fj(\017)p Fr(\)\))40 b Fj(<)f(\017)f Fr(for)h(all)f Fj(N)49 b Fr(large)38 b(enough.)66 b(Namely)-8 b(,)42 b(b)m(y)50 2941 y(Theorem)36 b(1.1)i(this)e(probabilit)m(y)f(con)m(v)m(erges)k(to)e(the)g (probabilit)m(y)e(that)i(the)g(pro)s(cess)g Fk(P)3175 2908 y Fu(\()p Fi(k)r Fu(\))3314 2941 y Fr(in)f Fj(R)3497 2908 y Fi(k)3578 2941 y Fr(has)50 3087 y(more)27 b(than)g Fj(K)34 b Fr(p)s(oin)m(ts)26 b(in)g(the)h(domain)f Fj(\015)1487 3101 y Fu(1)1528 3087 y Fj(x)1580 3101 y Fu(1)1634 3087 y Fr(+)14 b Fk(\001)h(\001)g(\001)f Fr(+)g Fj(\015)1970 3101 y Fi(k)2015 3087 y Fj(x)2067 3101 y Fi(k)2137 3087 y Fj(>)25 b Fr(\()p Fj(\015)2315 3101 y Fu(1)2370 3087 y Fr(+)14 b Fk(\001)h(\001)g(\001)f Fr(+)g Fj(\015)2706 3101 y Fi(k)2750 3087 y Fr(\))p Fj(x)p Fr(.)40 b(This)25 b(last)i(probabilit)m(y)50 3232 y(tends)k(to)h(zero)g(as)g Fj(K)h Fk(!)27 b(1)k Fr(since)g(b)m(y)g(Prop)s(osition)f(1.4)i(\(i\))f Fk(P)2248 3199 y Fu(\()p Fi(k)r Fu(\))2382 3232 y Fr(has)g(a.s.)44 b(a)32 b(\014nite)e(n)m(um)m(b)s(er)g(of)i(p)s(oin)m(ts)50 3378 y(in)42 b(this)h(domain.)79 b(Second,)47 b(observ)m(e)d(that)h (the)f(sums)e(of)i(the)g(second)g(lev)m(el)2881 3309 y Fe(P)2981 3405 y Fu(\026)-40 b Fi(\033)3021 3383 y Ff(0)3019 3429 y Fm(l)3046 3405 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)3211 3387 y Ff(0)3209 3423 y Fm(p)3247 3405 y Fu(:)t(\026)g Fi(\033)3310 3415 y Fm(l)3336 3405 y Fh(\001)t Fu(\026)g Fi(\033)3402 3383 y Ff(0)3400 3429 y Fm(l)3426 3405 y Fh(2)p Fi(I)3507 3415 y Fm(l)3554 3378 y Fr(1)-23 b(I)3609 3394 y Fh(f\001g)50 3523 y Fr(are)36 b Fq(indep)-5 b(endent)37 b Fr(from)e(the)h(indicators)f(in)f(fron)m(t) i(of)g(them)f(in)g(the)g(\014rst)g(lev)m(el)h(due)f(to)h(the)g (condition)55 3669 y(\026)-50 b Fj(\033)102 3683 y Fi(l)140 3669 y Fk(\001)15 b Fr(\026)-50 b Fj(\033)230 3636 y Fh(0)227 3694 y Fi(l)282 3669 y Fk(2)24 b Fj(I)407 3683 y Fi(l)461 3669 y Fr(with)g(1)i Fk(62)e Fj(I)859 3683 y Fi(l)887 3669 y Fr(.)39 b(Third,)24 b(for)i(an)m(y)f(giv)m(en)31 b(\026)-50 b Fj(\033)1816 3683 y Fi(l)1869 3669 y Fr(let)26 b(us)e(compute)i(the)g(n)m(um)m(b)s(er)e(of)h(con\014gurations)30 b(\026)-50 b Fj(\033)3684 3636 y Fh(0)3681 3694 y Fi(l)50 3814 y Fr(suc)m(h)29 b(that)34 b(\026)-50 b Fj(\033)501 3828 y Fi(l)547 3814 y Fk(\001)23 b Fr(\026)-50 b Fj(\033)645 3781 y Fh(0)642 3840 y Fi(l)696 3814 y Fk(2)25 b Fj(I)822 3828 y Fi(l)879 3814 y Fr(and)j(estimate)i(it)f(b)m(y)g(Stirling's)d (form)m(ula)i(as)i(it)e(w)m(as)i(done)f(in)f(\(i\).)40 b(This)27 b(n)m(um)m(b)s(er)59 3959 y(\026)-54 b Fj(\013)108 3927 y Fi(N)108 3985 y(l)177 3959 y Fr(\(1)19 b(+)e Fj(o)p Fr(\(1\)\))573 3891 y Fe(P)670 3987 y Fi(m)p Fh(2)p Fi(I)816 3997 y Fm(l)847 3959 y Fr(\(2)p Fj(\031)s(N)10 b Fr(\(1)19 b(+)e Fj(m)p Fr(\)\(1)i Fk(\000)e Fj(m)p Fr(\)\))1704 3927 y Fh(\000)p Fu(1)p Fi(=)p Fu(2)1876 3959 y Fj(e)1918 3927 y Fh(\000)p Fi(N)7 b(\036)p Fu(\()p Fi(m)p Fu(\))2238 3959 y Fr(turns)28 b(out)h(to)h(b)s(e)e(exp)s(onen)m(tially)f(smaller) 50 4105 y(than)45 b(\026)-54 b Fj(\013)326 4072 y Fi(N)326 4131 y(l)433 4105 y Fr(b)m(y)36 b(virtue)g(of)h(the)g(condition)f(0)g Fk(6)g Fr(cl)g Fk(2)f Fj(I)1871 4119 y Fi(l)1899 4105 y Fr(.)60 b(In)36 b(other)h(w)m(ords,)i(it)d(is)g(b)s(ounded)e(b)m(y)j (some)44 b Fe(b)-58 b Fj(\013)3640 4072 y Fi(N)3640 4131 y(l)50 4250 y Fr(where)36 b Fe(b)-57 b Fj(\013)371 4264 y Fi(l)424 4250 y Fj(<)34 b Fr(\026)-54 b Fj(\013)578 4264 y Fi(l)606 4250 y Fr(.)41 b(These)30 b(argumen)m(ts)g(com)m(bined) g(together)i(imply)284 4485 y Fg(P)p Fr(\()7 b(~)-52 b Fj(\021)422 4441 y Fi(p;k)419 4511 y(N)553 4485 y Fk(6)p Fr(=)25 b(0\))h Fk(\024)f Fj(K)7 b Fr(\()p Fj(\017)p Fr(\))p Fg(P)p Fr(\(max)1195 4539 y Fi(\033)1341 4462 y Fe(b)1317 4485 y Fj(X)1392 4499 y Fi(\033)1467 4485 y Fj(>)1563 4415 y Fk(p)p 1638 4415 76 4 v 70 x Fr(\026)-46 b Fj(a)1686 4499 y Fi(l)1714 4485 y Fj(u)1766 4499 y Fu(ln)21 b(\026)-43 b Fi(\013)1887 4509 y Fm(l)1913 4499 y Fi(;N)2004 4485 y Fr(\()p Fj(x)p Fr(\))21 b(+)f Fk(\001)15 b(\001)g(\001)21 b Fr(+)2455 4412 y Fe(p)p 2546 4412 91 4 v 2547 4485 a Fr(\026)-46 b Fj(a)2594 4499 y Fi(p)2636 4485 y Fj(u)2688 4499 y Fu(ln)20 b(\026)-43 b Fi(\013)2808 4508 y Fm(p)2846 4499 y Fi(;N)2937 4485 y Fr(\()p Fj(x)p Fr(\)\))21 b(+)f Fj(\017)234 b Fr(\(3)p Fj(:)p Fr(13\))50 4735 y(where)332 4712 y Fe(b)308 4735 y Fj(X)383 4749 y Fi(\033)459 4735 y Fr(are)26 b(Gaussian)f(random)h(v)-5 b(ariables)24 b(with)h(zero)i(mean)f(and)f(v)-5 b(ariance)28 b(\026)-47 b Fj(a)2895 4749 y Fi(l)2935 4735 y Fr(+)12 b Fk(\001)j(\001)g(\001)d Fr(+)h(\026)-46 b Fj(a)3266 4749 y Fi(p)3333 4735 y Fr(sub)5 b(jected)50 4880 y(to)28 b(the)g(GREM)f(of)h Fj(p)14 b Fk(\000)g Fj(l)i Fr(+)e(1)28 b(lev)m(els)f(with)f(parametes)35 b Fe(b)-58 b Fj(\013)2003 4894 y Fi(l)2031 4880 y Fj(;)25 b Fr(\026)-55 b Fj(\013)2129 4894 y Fi(l)p Fu(+1)2250 4880 y Fj(;)15 b(:)g(:)g(:)32 b(;)24 b Fr(\026)-54 b Fj(\013)2525 4894 y Fi(p)2595 4880 y Fr(and)28 b(\026)-47 b Fj(a)2816 4894 y Fi(l)2844 4880 y Fj(;)15 b(:)g(:)g(:)32 b(;)17 b Fr(\026)-47 b Fj(a)3109 4894 y Fi(p)3152 4880 y Fr(.)39 b(By)28 b(Theorem)50 5025 y(1.5)34 b(\(iii\))e(from)g([BK1])j(max)1030 5039 y Fi(\033)1094 5025 y Fj(X)1169 5039 y Fi(\033)1218 5025 y Fj(=)1263 4948 y Fk(p)p 1339 4948 83 4 v 77 x Fj(N)40 b Fk(!)1573 4953 y(p)p 1649 4953 269 4 v 1650 5025 a Fr(\026)-46 b Fj(a)1697 5039 y Fi(l)1740 5025 y Fr(ln)21 b Fe(b)-58 b Fj(\013)1889 5039 y Fi(l)1939 5025 y Fr(+)22 b Fk(\001)15 b(\001)g(\001)23 b Fr(+)2253 4952 y Fe(p)p 2344 4952 297 4 v 2345 5025 a Fr(\026)-46 b Fj(a)2392 5039 y Fi(p)2449 5025 y Fr(ln)23 b(\026)-54 b Fj(\013)2598 5039 y Fi(p)2673 5025 y Fr(a.s.)50 b(whic)m(h)32 b(is)g(smaller)g(than) 50 5171 y(the)d(v)-5 b(alue)28 b(in)f(\(3.13\))k(as)k Fe(b)-57 b Fj(\013)971 5185 y Fi(l)1024 5171 y Fj(<)34 b Fr(\026)-54 b Fj(\013)1178 5185 y Fi(l)1206 5171 y Fr(.)40 b(Then)28 b(the)h(\014rst)f(term)h(in)e(the)i(r.h.s.)39 b(of)29 b(\(3.13\))i(con)m(v)m(erges)g(to)e(zero)g(as)50 5316 y Fj(N)40 b Fk(!)31 b(1)p Fr(.)50 b(This)32 b(pro)m(v)m(es)i Fg(P)p Fr(\()7 b(~)-52 b Fj(\021)1084 5273 y Fi(p;k)1081 5343 y(N)1220 5316 y Fk(6)p Fr(=)30 b(0\))i Fk(!)e Fr(0)k(for)g(all)e Fj(p)f Fk(\025)f Fj(k)k Fk(\025)c Fj(l)r Fr(.)50 b(The)34 b(case)g Fj(k)g Fk(\025)c Fj(p)h Fk(\025)f Fj(l)36 b Fr(is)c(symmetric.)50 5462 y(Then)d Fg(P)p Fr(\()p Fj(M)475 5429 y Fu(2)465 5487 y Fi(N)560 5462 y Fk(6)p Fr(=)c(0\))i Fk(!)e Fr(0)30 b(and)g Fj(M)1229 5429 y Fu(2)1219 5487 y Fi(N)1288 5462 y Fj(=U)1405 5429 y Fu(2)1395 5487 y Fi(N)1490 5462 y Fk(!)25 b Fr(0)31 b(in)e(probabilit)m(y)-8 b(.)p eop %%Page: 34 34 34 33 bop 50 295 a Fo(34)3270 b Fl(Se)l(ction)27 b Fo(3)159 513 y Fr(If)44 b Fj(l)52 b Fr(=)d(1,)g(then)c(the)g(term)g Fj(M)1305 480 y Fu(1)1295 538 y Fi(N)1364 513 y Fj(=U)1481 480 y Fu(2)1471 538 y Fi(N)1586 513 y Fr(in)e(\(3.11\))48 b(is)c(absen)m(t)h(and)f(the)h(assertion)g(\(ii\))f(is)g(pro)m(v)m(en.) 50 658 y(Otherwise)d(w)m(e)i(pro)s(ceed)f(b)m(y)g(induction.)75 b(As)42 b(in)f(the)i(analysis)d(of)j(the)g(third)d(term,)46 b(for)c(an)m(y)h(giv)m(en)50 804 y Fj(\017)25 b(>)g Fr(0)j(w)m(e)g(c)m (ho)s(ose)h Fj(\016)s Fr(\()p Fj(\017)p Fr(\))g(to)f(mak)m(e)h(the)f (denominator)e Fj(U)1973 771 y Fu(2)1963 829 y Fi(N)2060 804 y Fr(smaller)g(than)i Fj(\016)s Fr(\()p Fj(\017)p Fr(\))h(with)d(probabilit)m(y)f(smaller)50 949 y(than)32 b Fj(\017)g Fr(and)f(after)i(that)f(concen)m(trate)j(on)d(the)g(n)m (umerator)g Fj(M)2244 916 y Fu(1)2234 974 y Fi(N)2303 949 y Fr(.)46 b(Assume)31 b(that)i(\(ii\))e(has)h(b)s(een)f(pro)m(v)m (en)50 1095 y(up)39 b(to)i Fj(l)29 b Fk(\000)e Fr(1)40 b(and)g(consider)g(the)g Fj(k)s Fr(th)h(term)f(of)h Fj(M)1885 1062 y Fu(1)1875 1120 y Fi(N)1944 1095 y Fr(.)70 b(F)-8 b(or)42 b(an)m(y)e Fj(\017)i(>)g Fr(0)f(one)f(can)h(\014nd)e(suc)m(h)h (a)h(large)50 1240 y Fj(K)7 b Fr(\()p Fj(\017)p Fr(\))36 b(that)g Fg(P)p Fr(\()569 1172 y Fe(P)670 1267 y Fu(\026)-41 b Fi(\033)707 1276 y Fn(1)743 1267 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)906 1277 y Fm(k)965 1240 y Fr(1)-23 b(I)1021 1262 y Fh(f)1074 1246 y Fu(\026)1058 1262 y Fi(E)1114 1240 y Fm(N)1115 1281 y Fn(\026)-35 b Fm(\033)1148 1295 y Fn(1)1184 1281 y Fm(:::)s Fn(\026)g Fm(\033)1283 1297 y(k)1328 1262 y Fh(g)1403 1240 y Fj(>)34 b(K)7 b Fr(\))33 b Fj(<)h(\017)p Fr(.)56 b(Once)35 b(again,)i(b)m(y)f(Theorem)f (1.1,)j(this)c(probabilit)m(y)50 1386 y(con)m(v)m(erges)43 b(to)g(the)e(probabilit)m(y)e(that)k(the)e(pro)s(cess)g Fk(P)2008 1353 y Fu(\()p Fi(k)r Fu(\))2152 1386 y Fr(in)f Fg(R)2329 1353 y Fi(k)2421 1386 y Fr(has)i(more)f(than)h Fj(K)48 b Fr(p)s(oin)m(ts)40 b(in)h(the)50 1532 y(domain)35 b Fj(\015)426 1546 y Fu(1)467 1532 y Fj(x)519 1546 y Fu(1)584 1532 y Fr(+)25 b Fk(\001)15 b(\001)g(\001)25 b Fr(+)f Fj(\015)952 1546 y Fi(k)997 1532 y Fj(x)1049 1546 y Fi(k)1130 1532 y Fj(>)35 b Fr(\()p Fj(\015)1318 1546 y Fu(1)1384 1532 y Fr(+)24 b Fk(\001)15 b(\001)g(\001)26 b Fr(+)e Fj(\015)1752 1546 y Fi(k)1797 1532 y Fr(\))p Fj(x)37 b Fr(and)f(this)f(last)i(probabilit)m(y)d(tends)j(to)g(zero)h (when)50 1677 y Fj(K)46 b Fk(!)39 b(1)g Fr(b)m(y)f(Prop)s(osition)f (1.4)j(\(i\).)66 b(Th)m(us,)40 b(with)e(a)h(large)g(enough)f (probabilit)m(y)f(the)i Fj(k)s Fr(th)g(term)f(is)g(a)50 1823 y(linear)25 b(com)m(bination)i(of)g(not)g(more)h(than)e Fj(K)34 b Fr(terms)27 b(of)h(t)m(yp)s(e)f Fj(V)2225 1837 y Fu(\026)-40 b Fi(\033)2263 1846 y Fn(1)2298 1837 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)2461 1847 y Fm(k)2505 1823 y Fr(.)40 b(Eac)m(h)28 b(of)f(these)g(terms)g Fj(V)3429 1837 y Fu(\026)-40 b Fi(\033)3467 1846 y Fn(1)3503 1837 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)3666 1847 y Fm(k)50 1968 y Fr(con)m(v)m(erges)32 b(in)d(probabilit)m(y)e(to)k (zero,)h(since)d(the)h(pro)s(duct)f(of)i(the)f(\014rst)f(t)m(w)m(o)j (factors)f([)p Fk(\001)p Fr(][)p Fk(\001)p Fr(])3140 1935 y Fh(\000)p Fu(2)3269 1968 y Fr(of)f Fj(V)3429 1982 y Fu(\026)-40 b Fi(\033)3467 1991 y Fn(1)3503 1982 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)3666 1992 y Fm(k)50 2114 y Fr(do)32 b(so)h(b)m(y)f(induction)e(assumption,)i(while)f(the)h (third)f(factor)i([)p Fk(\001)p Fr(])2300 2081 y Fu(2)2374 2114 y Fr(con)m(v)m(erges)i(to)e(the)g(square)f(of)h(an)f(a.s.)50 2259 y(\014nite)38 b(in)m(tegral)h(o)m(v)m(er)i(the)e(pro)s(cess)g Fk(P)1396 2226 y Fu(\()p Fi(l)p Fu(\()p Fi(\014)s Fu(\))p Fh(\000)p Fi(k)r Fu(\))1719 2259 y Fr(b)m(y)g(Theorem)g(1.3.)69 b(Th)m(us)38 b(for)h(an)m(y)g Fj(k)44 b Fr(=)c(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(l)28 b Fk(\000)e Fr(1,)50 2405 y(the)38 b Fj(k)s Fr(th)f(term)h(of)g Fj(M)821 2372 y Fu(1)811 2430 y Fi(N)917 2405 y Fr(con)m(v)m(erges)i(in)c(probabilit)m (y)f(to)k(zero)f(and)f(the)h(pro)s(of)f(of)h(the)g(assertion)f(\(ii\))g (is)50 2550 y(terminated.)159 2768 y(The)24 b(pro)s(of)g(of)h(the)g (assertion)f(\(iii\))f(go)s(es)i(b)m(y)g(induction)d(along)j(the)f (same)i(lines)c(as)j(for)g(\(ii\))e(using)h(trun-)50 2914 y(cation)e(b)m(y)f Fj(E)502 2928 y Fu(\026)-40 b Fi(\033)540 2937 y Fn(1)576 2928 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)739 2938 y Fm(k)783 2914 y Fr(,)23 b(the)f(p)s(oin)m(t)e(pro) s(cess)h(con)m(v)m(ergence)j(of)e(Theorem)f(1.1)i(together)f(with)f (the)g(prop)s(erties)50 3059 y(of)28 b(the)h(pro)s(cess)f Fk(P)691 3026 y Fu(\()p Fi(m)p Fu(\))846 3059 y Fr(of)h(Prop)s(osition) d(1.4,)k(Theorem)e(1.3)i(and)e(the)g(estimate)i(of)e(the)h(Lemma)f (2.2.)42 b(W)-8 b(e)50 3205 y(omit)22 b(similar)e(details)i(to)h (\(ii\))f(and)g(only)g(emphasize)g(the)h(di\013erences.)37 b(Observ)m(e)23 b(that)g(to)g(obtain)f(\(3.13\))j(it)50 3350 y(w)m(as)f(crucial)e(that)i(the)g(sums)f(of)g(the)h(second)g(lev)m (el)f(of)h(\(3.12\))i(w)m(ere)e(indep)s(enden)m(t)d(from)i(the)h (indicators)e(in)50 3495 y(fron)m(t)g(of)g(them.)38 b(This)20 b(prop)s(ert)m(y)h(w)m(as)h(ac)m(heiv)m(ed)h(b)m(y)e(the)h(assumption)f (1)k Fk(62)g Fj(I)2644 3509 y Fi(l)2672 3495 y Fr(.)38 b(In)21 b(\(iii\))f(this)h(indep)s(endence)50 3641 y(prop)s(ert)m(y)36 b(is)g(already)g(pro)m(vided)f(b)m(y)i(the)g(assumption)e(1)h Fk(62)f Fj(I)2233 3655 y Fi(l)p Fh(\000)p Fu(1)2354 3641 y Fr(,)k(since)d(then)g(the)h(random)f(v)-5 b(ariables)50 3786 y Fj(X)125 3800 y Fi(\033)167 3809 y Fn(1)203 3800 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)366 3810 y Fm(l)p Ff(\000)p Fn(1)476 3800 y Fu(\026)g Fi(\033)514 3810 y Fm(l)540 3800 y Fi(;:::)654 3786 y Fr(and)23 b Fj(X)899 3802 y Fi(\033)941 3811 y Fn(1)977 3802 y Fi(;:::)10 b(;)t Fu(\026)-40 b Fi(\033)1141 3780 y Ff(0)1139 3826 y Fm(l)p Ff(\000)p Fn(1)1250 3802 y Fu(\026)g Fi(\033)1290 3780 y Ff(0)1288 3826 y Fm(l)1315 3802 y Fi(:::)1407 3786 y Fr(with)22 b(\()5 b(\026)-50 b Fj(\033)1694 3800 y Fi(l)p Fh(\000)p Fu(1)1821 3786 y Fk(\001)10 b Fr(\026)-49 b Fj(\033)1907 3753 y Fh(0)1904 3812 y Fi(l)p Fh(\000)p Fu(1)2024 3786 y Fr(\))p Fj(=)-5 b(N)36 b Fk(2)25 b Fj(I)2334 3800 y Fi(l)2385 3786 y Fr(are)e(indep)s(enden)m(t.)36 b(Therefore)23 b(under)50 3932 y(\(iii\))36 b(the)i(con)m(tribution)f (of)h(an)m(y)g Fj(I)1254 3946 y Fi(l)1320 3932 y Fr(suc)m(h)f(that)i(0) f Fk(62)f Fr(cl)p Fj(I)2024 3946 y Fi(l)2090 3932 y Fr(is)g (negligible.)60 b(F)-8 b(urthermore,)40 b(if)d(ev)-5 b(aluating)50 4077 y(\(3.12\),)27 b(w)m(e)d(ha)m(v)m(e)h(0)g Fk(2)g Fr(cl)p Fj(I)924 4091 y Fi(l)952 4077 y Fr(,)g(but)d(for)h(some) h Fj(j)5 b Fr(,)26 b Fj(l)h(<)e(j)31 b Fk(\024)25 b Fj(p)p Fr(,)f(0)i Fk(62)f Fr(cl)p Fj(I)2279 4091 y Fi(p)2321 4077 y Fr(,)g(then)e(an)m(yw)m(a)m(y)h Fg(P)p Fr(\()7 b(~)-52 b Fj(\021)3027 4034 y Fi(p;k)3024 4104 y(N)3158 4077 y Fk(6)p Fr(=)25 b(0\))h Fk(!)f Fr(0:)38 b(W)-8 b(e)50 4223 y(w)m(ould)25 b(get)i(\(3.13\))h(where)998 4200 y Fe(b)975 4223 y Fj(X)1050 4237 y Fi(\033)1125 4223 y Fr(are)e(sub)5 b(jected)26 b(to)h(the)f(GREM)g(of)g Fj(p)11 b Fk(\000)g Fj(l)i Fr(+)e(1)28 b(lev)m(els)d(with)g(the)h (parameters)59 4368 y(\026)-54 b Fj(\013)108 4382 y Fi(l)136 4368 y Fj(;)15 b(:)g(:)g(:)23 b Fe(b)-58 b Fj(\013)355 4382 y Fi(j)394 4368 y Fj(;)15 b(:)g(:)g(:)32 b(;)24 b Fr(\026)-54 b Fj(\013)669 4382 y Fi(p)711 4368 y Fr(,)47 b Fe(b)-58 b Fj(\013)834 4382 y Fi(j)911 4368 y Fj(<)46 b Fr(\026)-54 b Fj(\013)1077 4382 y Fi(j)1116 4368 y Fr(,)40 b(and)d(then)h(with)f(the)h(maxim)m(um)f(smaller)f(than)i(the)g (v)-5 b(alue)38 b(in)e(\(3.13\).)50 4514 y(Finally)-8 b(,)28 b(if)h(cl)o Fj(I)564 4528 y Fi(l)592 4514 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b Fr(cl)p Fj(I)915 4530 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))1074 4514 y Fr(all)28 b(con)m(tain)i(zero)g (but)f Fj(I)1916 4530 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)2167 4514 y Fr(do)s(es)g(not,)h(then)g(truncating)f(b)m(y)g Fj(E)3404 4528 y Fu(\026)-40 b Fi(\033)3442 4537 y Fn(1)3478 4528 y Fi(;:::)11 b(;)t Fu(\026)-40 b Fi(\033)3641 4538 y Fm(k)3685 4514 y Fr(,)50 4659 y(applying)22 b(induction)f(and)j(the)g (same)h(argumen)m(ts)f(as)g(in)f(\(ii\))g(w)m(e)i(reduce)e(the)i (problem)d(to)j(ev)-5 b(aluating)24 b(the)50 4829 y(sum)240 4761 y Fe(P)340 4857 y Fu(\026)-40 b Fi(\033)378 4869 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)570 4857 y Fi(;)t Fu(\026)g Fi(\033)636 4835 y Ff(0)634 4883 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)825 4857 y Fu(:\()t(\026)g Fi(\033)916 4869 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)1108 4857 y Fh(\001)t Fu(\026)g Fi(\033)1174 4835 y Ff(0)1172 4883 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)1363 4857 y Fu(\))p Fi(=)l(N)7 b Fh(2)p Fi(I)1570 4869 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)1781 4829 y Fj(e)1823 4756 y Fi(\014)1867 4698 y Fk(p)p 1944 4698 621 4 v 1944 4756 a Fi(N)g Fu(\()p Fi(a)2076 4768 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)2268 4756 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(\013)2491 4765 y Fm(n)2536 4756 y Fu(\))q(\()p Fi(X)2657 4765 y Fn(\026)-35 b Fm(\033)2690 4784 y(l)p Fn(\()p Fm(\014)r Fn(\)+1)2886 4756 y Fu(+)p Fi(X)3006 4775 y Fn(\026)g Fm(\033)3041 4760 y Ff(0)3039 4808 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)3234 4756 y Fu(\))3295 4829 y Fr(divided)26 b(b)m(y)50 4994 y Fj(e)92 4960 y Fh(\000)p Fi(\014)193 4933 y Fn(2)229 4960 y Fi(N)7 b Fu(\()p Fi(a)361 4972 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)553 4960 y Fu(+)p Fh(\001\001\001)n Fu(+)p Fi(\013)776 4969 y Fm(n)821 4960 y Fu(\))p Fh(\000)p Fi(N)19 b Fu(ln)q(\()p Fi(a)1111 4972 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)1303 4960 y Fh(\001\001\001)o Fi(\013)1415 4969 y Fm(n)1460 4960 y Fu(\))1521 4994 y Fr(where)28 b Fj(X)1861 5008 y Fu(\026)-40 b Fi(\033)1899 5020 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)2123 4994 y Fr(and)28 b Fj(X)2377 5010 y Fu(\026)-40 b Fi(\033)2417 4988 y Ff(0)2415 5036 y Fm(l)p Fn(\()p Fm(\014)r Fn(\)+1)2640 4994 y Fr(are)28 b(indep)s(enden)m(t)e(b)s (ecause)j(of)50 5139 y(1)d Fk(62)e Fj(I)246 5153 y Fi(l)p Fh(\000)p Fu(1)367 5139 y Fr(.)41 b(The)30 b(exp)s(ectation)h(of)f (this)g(term)g(con)m(v)m(erges)i(to)f(zero)h(if)d(only)g(0)d Fk(62)f Fr(cl)p Fj(I)2864 5156 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)3085 5139 y Fr(.)159 5357 y(It)36 b(is)f(easy)i(to)f(deriv)m(e) g(from)f(\(i\),)j(\(ii\))d(and)g(\(iii\))g(that)h(the)g(measure)g Fj(f)2636 5310 y Fh(\012)p Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)2626 5385 y Fi(\014)s(;N)2950 5357 y Fr(in)e(the)i(limit)e(has)i(an)p eop %%Page: 35 35 35 34 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(35)50 513 y Fr(atomic)31 b(structure)e(with)g(atoms)i(\()p Fk(f)p Fr(1)p Fk(g)21 b(\002)f(\001)15 b(\001)g(\001)21 b(\002)e(f)p Fr(1)p Fk(g)i(\002)f(f)p Fr(0)p Fk(g)15 b(\001)g(\001)g(\001)22 b(\002)d(f)p Fr(0)p Fk(g)i(\002)f(f)p Fr(0)p Fk(g)p Fr(\))32 b(and)d(it)h(remains)f(to)i(study)50 658 y(its)d(asymptotic)i(b)s(eha)m(viour)e(on)h(the)g(set)h(\()p Fk(f)p Fr(1)p Fk(g)19 b(\002)f(\001)d(\001)g(\001)j(\002)g(f)p Fr(1)p Fk(g)h(\002)e Fr(\001)2292 675 y Fi(l)p Fu(\()p Fi(\014)s Fu(\)+1)2513 658 y Fr(\))30 b(and)e(sho)m(w)h(that)h(in)e (the)h(limit)e(it)50 804 y(is)33 b(concen)m(trated)j(on)e(the)g(unique) e(atom)j(of)g(this)e(set)h(\()p Fk(f)p Fr(1)p Fk(g)25 b(\002)d(\001)15 b(\001)g(\001)24 b(\002)e(f)p Fr(1)p Fk(g)i(\002)f(f)p Fr(0)p Fk(g)p Fr(\).)53 b(This)33 b(is)g(the)h(sub)5 b(ject)50 949 y(of)31 b(\(iv\).)43 b(Its)31 b(pro)s(of)f(is)g(carried)g (again)h(b)m(y)g(induction)e(as)i(in)f(\(ii\))g(and)h(using)e (truncatons)i(b)m(y)g(1)-23 b(I)3336 965 y Fh(f)p Fi(E)3430 974 y Fn(\026)-35 b Fm(\033)3463 988 y Fn(1)3499 974 y Fm(:::)s Fn(\026)g Fm(\033)3598 990 y(k)3643 965 y Fh(g)3685 949 y Fr(.)50 1095 y(W)-8 b(e)35 b(outline)e(only)g(that)i (the)f(basis)f(of)h(the)h(induction)d(is)h(the)h(statemen)m(t)i(\(i\))e (for)g Fj(l)r Fr(\()p Fj(\014)5 b Fr(\))32 b(=)f(0.)53 b(All)32 b(other)50 1240 y(argumen)m(ts)e(are)h(the)g(same)g(as)f(in)f (\(ii\),)h(therefore)h(w)m(e)g(omit)f(further)f(details)g(of)i(the)f (pro)s(of.)40 b Fk(})50 1469 y Fv(4.)55 b(Pro)s(of)41 b(of)g(the)g(Ghirlanda-Guerra)f(relations.)159 1687 y Fr(In)c(this)f(section)i(w)m(e)g(\014nd)e(the)i(mean)g(of)g(the)g(o)m (v)m(erlap)g(distribution)c(function)i(stated)j(Prop)s(osition)50 1833 y(1.7)g(and)f(Ghirlanda-Guerra)e(relations)h(stated)i(in)e(Prop)s (osition)f(1.8.)62 b(Throughout)36 b(this)g(section)h(w)m(e)50 1978 y(assume)30 b(that)h(the)f(mo)s(del)g(parameters)g(are)h(suc)m(h)f (that)h(the)g(inequalities)d(\(1.9\))k(are)e(all)g(strict.)159 2196 y(The)g(follo)m(wing)f(simple)f(observ)-5 b(ation)30 b(is)f(the)i(k)m(ey)g(to)m(w)m(ards)g(b)s(oth)f(of)g(these)h(results.) 50 2415 y Fd(Prop)s(osition)38 b(4.1:)p Fq(F)-7 b(or)34 b(any)f(value)f(of)h Fj(\014)5 b Fq(,)33 b(and)g(any)h Fj(i)25 b Fr(=)g(1)p Fj(;)15 b(:)g(:)g(:)i(;)e(n)p Fq(,)972 2660 y Fg(E)1121 2598 y Fj(d)p 1038 2639 202 4 v 1038 2722 a(d)1085 2657 y Fk(p)p 1167 2657 49 4 v 1167 2722 a Fj(a)1215 2749 y Fi(i)1256 2660 y Fj(F)1314 2674 y Fi(\014)s(;N)1475 2660 y Fr(=)25 b Fk(\000)p Fj(\014)1698 2622 y Fu(2)1739 2597 y Fk(p)p 1814 2597 79 4 v 1814 2660 a Fj(a)1862 2674 y Fi(i)1893 2660 y Fg(E)10 b Fj(f)1992 2674 y Fi(\014)s(;N)2133 2559 y Fe(\020)2255 2546 y Fi(i)2202 2573 y Fe(X)2205 2767 y Fi(j)t Fu(=1)2349 2660 y Fr(ln)k Fj(\013)2498 2674 y Fi(j)2536 2660 y Fj(=)h Fr(ln)g(2)2733 2559 y Fe(\021)3523 2660 y Fr(\(4)p Fj(:)p Fr(1\))50 3075 y Fa(Pro)s(of)36 b(of)f(Prop)s(osition)h(4.1:)87 b Fr(Ob)m(viously)-8 b(,)946 3309 y Fg(E)1094 3247 y Fj(d)p 1011 3288 202 4 v 1011 3371 a(d)1059 3312 y Fk(p)p 1140 3312 79 4 v 1140 3371 a Fj(a)1188 3385 y Fi(i)1230 3309 y Fj(F)1288 3323 y Fi(\014)s(;N)1448 3309 y Fr(=)25 b Fk(\000)p Fj(N)1698 3271 y Fh(\000)p Fu(1)1795 3309 y Fj(\014)5 b Fg(E)1917 3247 y(E)1978 3261 y Fi(\033)2033 3170 y Fk(p)p 2109 3170 83 4 v 77 x Fj(N)k(X)2266 3261 y Fi(\033)2308 3270 y Fn(1)2344 3261 y Fi(:::)o(\033)2451 3270 y Fm(i)2485 3247 y Fj(e)2527 3214 y Fi(\014)2571 3160 y Fh(p)p 2631 3160 65 4 v 54 x Fi(N)e(X)2756 3223 y Fm(\033)p 1923 3288 880 4 v 2149 3385 a Fg(E)2204 3399 y Fi(\033)2259 3385 y Fj(e)2301 3359 y Fi(\014)2345 3305 y Fh(p)p 2405 3305 65 4 v 54 x Fi(N)g(X)2530 3368 y Fm(\033)3523 3309 y Fr(\(4)p Fj(:)p Fr(2\))50 3524 y(No)m(w)39 b(if)e Fj(X)46 b Fr(is)37 b(standard)h(normal)f(v)-5 b(ariable,)40 b(and)d Fj(g)42 b Fr(an)m(y)d(function)e(of)i(at)g(most)f(p)s (olynomial)e(gro)m(wth,)50 3669 y(then)1480 3815 y Fg(E)10 b Fr([)p Fj(X)e(g)t Fr(\()p Fj(X)f Fr(\))q(])31 b(=)25 b Fg(E)11 b Fj(g)2095 3777 y Fh(0)2127 3815 y Fr(\()p Fj(X)c Fr(\))1244 b(\(4)p Fj(:)p Fr(3\))50 3998 y(Using)33 b(this)f(iden)m(tit)m(y)h(in)g(the)h(righ)m(t)f(hand)g(side)f(of)i (\(4.2\))i(with)c(resp)s(ect)i(to)g(the)g(a)m(v)m(erage)i(o)m(v)m(er)f Fj(X)3466 4012 y Fi(\033)3508 4021 y Fn(1)3544 4012 y Fi(:::)o(\033)3651 4021 y Fm(i)3685 3998 y Fr(,)50 4144 y(w)m(e)c(get)g(immediately)e(that)240 4389 y Fg(E)306 4328 y(E)366 4342 y Fi(\033)421 4251 y Fk(p)p 497 4251 83 4 v 77 x Fj(N)10 b(X)655 4342 y Fi(\033)697 4351 y Fn(1)733 4342 y Fi(:::)o(\033)840 4351 y Fm(i)873 4328 y Fj(e)915 4295 y Fi(\014)959 4240 y Fh(p)p 1020 4240 65 4 v 1020 4295 a Fi(N)c(X)1144 4304 y Fm(\033)p 312 4368 880 4 v 538 4466 a Fg(E)592 4480 y Fi(\033)647 4466 y Fj(e)689 4440 y Fi(\014)733 4385 y Fh(p)p 794 4385 65 4 v 794 4440 a Fi(N)g(X)918 4449 y Fm(\033)1228 4389 y Fr(=)25 b Fj(N)10 b(\014)1463 4326 y Fk(p)p 1539 4326 79 4 v 63 x Fj(a)1587 4403 y Fi(i)1617 4389 y Fg(E)1687 4234 y Fe( )1765 4389 y Fr(1)20 b Fk(\000)1932 4316 y Fg(E)1987 4330 y Fi(\033)2042 4316 y Fg(E)2096 4330 y Fi(\033)2141 4311 y Ff(0)2176 4316 y Fr(1)-23 b(I)2231 4332 y Fi(\033)2273 4341 y Fn(1)2309 4332 y Fi(:::)o(\033)2416 4341 y Fm(i)2445 4332 y Fu(=)p Fi(\033)2545 4310 y Ff(0)2543 4354 y Fn(1)2578 4332 y Fi(:::)o(\033)2687 4310 y Ff(0)2685 4354 y Fn(2)2725 4316 y Fj(\017)2762 4283 y Fi(\014)2806 4229 y Fh(p)p 2866 4229 65 4 v 54 x Fi(N)7 b Fu(\()p Fi(X)3019 4292 y Fm(\033)3062 4283 y Fu(+)p Fi(X)3178 4300 y Fm(\033)3217 4287 y Ff(0)p 1932 4368 1318 4 v 2321 4466 a Fr(\()p Fg(E)2411 4480 y Fi(\033)2466 4466 y Fj(e)2508 4440 y Fi(\014)2552 4385 y Fh(p)p 2613 4385 65 4 v 2613 4440 a Fi(N)f(X)2737 4449 y Fm(\033)2784 4466 y Fr(\))2819 4440 y Fu(2)3261 4234 y Fe(!)1228 4644 y Fr(=)25 b Fj(N)10 b(\014)1463 4581 y Fk(p)p 1539 4581 79 4 v 63 x Fj(a)1587 4658 y Fi(i)1617 4644 y Fg(E)1687 4543 y Fe(\020)1747 4644 y Fr(1)21 b Fk(\000)f Fj(\026)1959 4605 y Fh(\012)p Fu(2)1959 4671 y Fi(\014)s(;N)2109 4543 y Fe(\020)2163 4644 y Fr(1)-23 b(I)2218 4660 y Fi(\033)2260 4669 y Fn(1)2296 4660 y Fi(:::)o(\033)2403 4669 y Fm(i)2432 4660 y Fu(=)p Fi(\033)2532 4638 y Ff(0)2530 4681 y Fn(1)2565 4660 y Fi(:::)o(\033)2674 4638 y Ff(0)2672 4682 y Fm(i)2705 4543 y Fe(\021)q(\021)3523 4485 y Fr(\(4)p Fj(:)p Fr(4\))50 4832 y(whic)m(h)29 b(is)g(ob)m(viously)g(\(4.1\).)43 b Fk(})50 5050 y Fa(Pro)s(of)d(of)g(Prop)s(osition)h(1.7.:)106 b Fr(Due)35 b(to)g(the)g(con)m(v)m(exit)m(y)h(of)f(the)g(function)e Fj(F)2900 5064 y Fi(\014)s(;N)3070 5050 y Fr(and)h(Prop)s(osition)50 5195 y(4.1)783 5369 y Fg(E)11 b Fj(f)883 5383 y Fi(\014)937 5269 y Fe(\020)1059 5256 y Fi(i)1007 5283 y Fe(X)1009 5477 y Fi(j)t Fu(=1)1153 5369 y Fr(ln)j Fj(\013)1302 5383 y Fi(j)1340 5369 y Fj(=)h Fr(ln)g(2)1537 5269 y Fe(\021)1617 5369 y Fr(=)25 b Fk(\000)p Fj(\014)1840 5332 y Fh(\000)p Fu(2)1937 5369 y Fr(\()1972 5306 y Fk(p)p 2048 5306 V 63 x Fj(a)2096 5383 y Fi(i)2127 5369 y Fr(\))2162 5332 y Fh(\000)p Fu(1)2348 5308 y Fj(d)p 2271 5349 202 4 v 2271 5432 a(d)2318 5373 y Fk(p)p 2394 5373 79 4 v 59 x Fj(a)2442 5446 y Fi(i)2540 5369 y Fr(lim)2498 5428 y Fi(N)7 b Fh(!1)2723 5369 y Fg(E)j Fj(F)2836 5383 y Fi(\014)s(;N)p eop %%Page: 36 36 36 35 bop 50 295 a Fo(36)3270 b Fl(Se)l(ction)27 b Fo(4)50 513 y Fr(where)35 b(the)g(last)g(limit)e(is)h(found)g(in)g(Theorem)h (1.2.)56 b(T)-8 b(aking)35 b(the)h(deriv)-5 b(ativ)m(e)34 b(of)i(the)f(r.h.s.)55 b(of)35 b(\(1.14\),)50 658 y(the)30 b(result)g(follo)m(ws.)40 b Fk(})p Fr(.)159 876 y(W)-8 b(e)31 b(no)m(w)g(turn)e(to)i(the)g(pro)s(of)e(Prop)s(osition)g(1.8.)50 1095 y Fa(Pro)s(of)g(of)g(Prop)s(osition)h(1.8:)82 b Fr(Let)25 b(us)g(\014rst)f(note)h(that)h(w)m(e)f(need)g(to)h(pro)m(v)m (e)f(\(1.30\))j(only)c(for)g Fj(i)i Fk(\024)f Fj(J)3555 1111 y Fi(l)p Fu(\()p Fi(\014)s Fu(\))3685 1095 y Fr(,)50 1240 y(since)30 b(for)g(large)g(v)-5 b(alues)30 b(of)g Fj(i)h Fr(,)g(b)s(oth)e(sides)g(are)i(trivially)d(equal)i(to)h(zero.) 159 1458 y(The)f(starting)g(p)s(oin)m(t)f(is)h(the)g(follo)m(wing)f(ob) m(vious)h(generalization)g(of)g(\(4.4\):)50 1676 y Fd(Lemma)38 b(4.2:)51 b Fq(L)-5 b(et)33 b Fj(h)25 b Fr(:)h Fk(S)1032 1643 y Fi(n)1025 1701 y(N)1119 1676 y Fk(!)f Fg(R)41 b Fq(b)-5 b(e)33 b(any)g(b)-5 b(ounde)g(d)34 b(function)f(of)g Fj(n)f Fq(spins.)42 b(Then)299 1921 y Fr(1)p 242 1962 159 4 v 242 1980 a Fk(p)p 318 1980 83 4 v 77 x Fj(N)412 1983 y Fg(E)11 b Fj(\026)522 1944 y Fh(\012)p Fi(n)522 2010 y(\014)s(;N)678 1882 y Fe(\020)732 1983 y Fj(X)807 2005 y Fi(\033)851 1983 y Fm(k)849 2027 y Fn(1)891 2005 y Fi(:::)o(\033)1000 1983 y Fm(k)998 2028 y(i)1045 1983 y Fj(h)p Fr(\()p Fj(\033)1187 1945 y Fu(1)1229 1983 y Fj(;)k(:)g(:)g(:)h(;)f(\033)1485 1945 y Fi(n)1535 1983 y Fr(\))1570 1882 y Fe(\021)257 2263 y Fr(=)25 b Fj(\014)409 2199 y Fk(p)p 485 2199 79 4 v 64 x Fj(a)533 2277 y Fi(i)563 2263 y Fg(E)11 b Fj(\026)673 2224 y Fh(\012)p Fi(n)p Fu(+1)673 2290 y Fi(\014)s(;N)892 2107 y Fe( )964 2263 y Fj(h)p Fr(\()p Fj(\033)1106 2225 y Fu(1)1148 2263 y Fj(;)k(:)g(:)g(:)h(;)f(\033)1404 2225 y Fi(n)1454 2263 y Fr(\))1504 2107 y Fe( )1620 2149 y Fi(n)1577 2176 y Fe(X)1585 2372 y Fi(l)p Fu(=1)1723 2263 y Fr(1)-23 b(I)1778 2285 y Fi(\033)1822 2263 y Fm(k)1820 2307 y Fn(1)1862 2285 y Fi(:::)o(\033)1971 2263 y Fm(k)1969 2308 y(i)2011 2285 y Fu(=)p Fi(\033)2111 2263 y Fm(l)2109 2307 y Fn(1)2145 2285 y Fi(:::)o(\033)2254 2263 y Fm(l)2252 2308 y(i)2305 2263 y Fk(\000)20 b Fj(n)p Fr(1)-23 b(I)2506 2291 y Fi(\033)2550 2269 y Fm(k)2548 2313 y Fn(1)2590 2291 y Fi(:::)o(\033)2699 2269 y Fm(k)2697 2314 y(i)2739 2291 y Fu(=)p Fi(\033)2839 2262 y Fm(n)p Fn(+1)2837 2313 y(1)2961 2291 y Fi(:::)o(\033)3070 2262 y Fm(n)p Fn(+1)3068 2314 y Fm(i)3197 2107 y Fe(!!)3523 2144 y Fr(\(4)p Fj(:)p Fr(5\))50 2760 y Fa(Pro)s(of:)57 b Fr(It)35 b(is)f(completely)g(analogous)h(to)h(the)f(pro)s(of)f(of)g (Prop)s(osition)f(4.1)j(using)d(the)i(in)m(tegration)g(b)m(y)50 2906 y(parts)30 b(\(4.3\).)42 b Fk(})159 3124 y Fr(The)31 b(strength)h(of)g(Lemma)g(4.2)h(comes)f(out)g(when)f(com)m(bined)g (with)f(a)j(factorization)f(result)f(that)h(in)50 3269 y(turn)d(is)h(a)g(consequence)h(of)g(self-a)m(v)m(eraging.)50 3487 y Fd(Lemma)h(4.3:)48 b Fq(L)-5 b(et)28 b Fj(h)g Fq(b)-5 b(e)29 b(as)f(in)g(the)h(pr)-5 b(evious)29 b(lemma.)41 b(Then)29 b(under)g(the)f(hyp)-5 b(othesis)31 b(of)d(the)h(the)-5 b(or)g(em,)147 3786 y Fr(lim)123 3846 y Fi(N)7 b Fh("1)380 3725 y Fr(1)p 323 3765 159 4 v 323 3784 a Fk(p)p 399 3784 83 4 v 77 x Fj(N)508 3682 y Fe(\014)508 3736 y(\014)508 3791 y(\014)538 3786 y Fg(E)k Fj(\026)648 3747 y Fh(\012)p Fi(n)648 3814 y(\014)s(;N)804 3685 y Fe(\020)858 3786 y Fj(X)933 3808 y Fi(\033)977 3787 y Fm(k)975 3830 y Fn(1)1018 3808 y Fi(:::)o(\033)1127 3787 y Fm(k)1125 3831 y(i)1171 3786 y Fj(h)p Fr(\()p Fj(\033)1313 3749 y Fu(1)1355 3786 y Fj(;)k(:)g(:)g(:)i(;)e(\033)1612 3749 y Fi(n)1662 3786 y Fr(\))1697 3685 y Fe(\021)1772 3786 y Fk(\000)k Fg(E)11 b Fj(\026)1972 3800 y Fi(\014)s(;N)2128 3685 y Fe(\020)2182 3786 y Fj(X)2257 3808 y Fi(\033)2301 3787 y Fm(k)2299 3830 y Fn(1)2342 3808 y Fi(:::)o(\033)2451 3787 y Fm(k)2449 3831 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Fm(k)2240 4641 y(i)2286 4596 y Fr(\))p Fg(E)g Fj(\026)2431 4557 y Fh(\012)p Fi(n)2431 4623 y(\014)s(;N)2572 4596 y Fr(\()p Fj(h)p Fr(\()p Fj(\033)2749 4558 y Fu(1)2791 4596 y Fj(;)k(:)g(:)g(:)i (;)e(\033)3048 4558 y Fi(n)3098 4596 y Fr(\)\))3168 4495 y Fe(\021)3223 4514 y Fu(2)334 4814 y Fr(=)430 4714 y Fe(\020)485 4814 y Fg(E)10 b Fj(\026)594 4776 y Fh(\012)p Fi(n)594 4842 y(\014)t(;N)750 4714 y Fe(h\020)848 4814 y Fj(X)923 4837 y Fi(\033)967 4815 y Fm(k)965 4859 y Fn(1)1007 4837 y Fi(:::)o(\033)1116 4815 y Fm(k)1114 4860 y(i)1181 4814 y Fk(\000)20 b Fg(E)10 b Fj(\026)1381 4776 y Fh(\012)p Fi(n)1381 4842 y(\014)s(;N)1522 4814 y Fr(\()p Fj(X)1632 4837 y Fi(\033)1676 4815 y Fm(k)1674 4859 y Fn(1)1717 4837 y Fi(:::)o(\033)1826 4815 y Fm(k)1824 4860 y(i)1870 4814 y Fr(\))1905 4714 y Fe(\021)1960 4814 y Fj(h)p Fr(\()p Fj(\033)2102 4777 y Fu(1)2144 4814 y Fj(;)15 b(:)g(:)g(:)h(;)f(\033)2400 4777 y Fi(n)2450 4814 y Fr(\))2485 4714 y Fe(i)q(\021)2583 4732 y Fu(2)334 5033 y Fk(\024)25 b Fg(E)11 b Fj(\026)540 4994 y Fh(\012)p Fi(n)540 5060 y(\014)s(;N)681 4932 y Fe(\020)735 5033 y Fj(X)810 5055 y Fi(\033)854 5033 y Fm(k)852 5077 y Fn(1)895 5055 y Fi(:::)n(\033)1003 5033 y Fm(k)1001 5078 y(i)1068 5033 y Fk(\000)20 b Fg(E)11 b Fj(\026)1269 4994 y Fh(\012)p Fi(n)1269 5060 y(\014)s(;N)1410 5033 y Fr(\()p Fj(X)1520 5055 y Fi(\033)1564 5033 y Fm(k)1562 5077 y Fn(1)1604 5055 y Fi(:::)o(\033)1713 5033 y Fm(k)1711 5078 y(i)1758 5033 y Fr(\))1793 4932 y Fe(\021)1848 4951 y Fu(2)1888 5033 y Fg(E)g Fj(\026)1998 4994 y Fh(\012)p Fi(n)1998 5060 y(\014)s(;N)2154 4959 y Fe(\000)2196 5033 y Fj(h)p Fr(\()p Fj(\033)2338 4996 y Fu(1)2379 5033 y Fj(;)k(:)g(:)g(:)i(;)e(\033)2636 4996 y Fi(n)2686 5033 y Fr(\))2721 4959 y Fe(\001)2763 4977 y Fu(2)3523 4805 y Fr(\(4)p Fj(:)p Fr(7\))50 5316 y(where)32 b(the)h(last)f(inequalit)m (y)f(is)h(the)h(Cauc)m(h)m(y{Sc)m(h)m(w)m(arz)h(inequalit)m(y)d (applied)g(to)i(the)g(join)m(t)f(exp)s(ectation)50 5462 y(with)j(resp)s(ect)h(to)h(the)g(Gibbs)d(measure)j(and)e(the)i (disorder.)57 b(Ob)m(viously)34 b(the)i(\014rst)g(factor)h(in)e(the)i (last)p eop %%Page: 37 37 37 36 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(37)50 513 y Fr(line)29 b(is)g(equal)h(to)646 780 y Fg(E)11 b Fj(\026)756 794 y Fi(\014)s(;N)897 679 y Fe(\020)951 780 y Fj(X)1026 802 y Fi(\033)1070 780 y Fm(k)1068 824 y Fn(1)1110 802 y Fi(:::)o(\033)1219 780 y Fm(k)1217 825 y(i)1284 780 y Fk(\000)20 b Fj(\026)1430 794 y Fi(\014)s(;N)1565 780 y Fr(\()p Fj(X)1675 802 y Fi(\033)1719 780 y Fm(k)1717 824 y Fn(1)1760 802 y Fi(:::)n(\033)1868 780 y Fm(k)1866 825 y(i)1913 780 y Fr(\))1948 679 y Fe(\021)2003 698 y Fu(2)757 998 y Fr(+)g Fg(E)903 898 y Fe(h)952 998 y Fj(\026)1007 1012 y Fi(\014)s(;N)1142 998 y Fr(\()p Fj(X)1252 1021 y Fi(\033)1296 999 y Fm(k)1294 1043 y Fn(1)1336 1021 y Fi(:::)o(\033)1445 999 y Fm(k)1443 1044 y(i)1490 998 y Fr(\))g Fk(\000)g Fg(E)11 b Fj(\026)1746 1012 y Fi(\014)s(;N)1887 998 y Fr(\()p Fj(X)1997 1021 y Fi(\033)2041 999 y Fm(k)2039 1043 y Fn(1)2082 1021 y Fi(:::)n(\033)2190 999 y Fm(k)2188 1044 y(i)2235 998 y Fr(\))2270 898 y Fe(i)2313 916 y Fu(2)671 1244 y Fr(=)25 b Fk(\000)p Fj(\014)894 1207 y Fh(\000)p Fu(2)992 1244 y Fg(E)1140 1183 y Fj(d)1187 1150 y Fu(2)p 1063 1223 243 4 v 1063 1319 a Fj(d)1110 1260 y Fk(p)p 1187 1260 79 4 v 1187 1319 a Fj(a)1235 1333 y Fi(i)1265 1275 y Fu(2)1317 1244 y Fj(F)1375 1258 y Fi(\014)s(;N)1530 1244 y Fr(+)20 b Fj(N)10 b(\014)1760 1207 y Fh(\000)p Fu(2)1858 1244 y Fg(E)1928 1116 y Fe(\022)2089 1183 y Fj(d)p 2012 1223 202 4 v 2012 1307 a(d)2059 1248 y Fk(p)p 2135 1248 79 4 v 59 x Fj(a)2183 1321 y Fi(i)2224 1244 y Fj(F)2282 1258 y Fi(\014)s(;N)2438 1244 y Fk(\000)19 b Fg(E)2677 1183 y Fj(d)p 2594 1223 202 4 v 2594 1307 a(d)2641 1248 y Fk(p)p 2723 1248 79 4 v 2723 1307 a Fj(a)2771 1321 y Fi(i)2812 1244 y Fj(F)2870 1258 y Fi(\014)s(;N)3006 1116 y Fe(\023)3073 1135 y Fu(2)3523 1025 y Fr(\(4)p Fj(:)p Fr(8\))50 1537 y(W)-8 b(e)41 b(kno)m(w)f(that)h(b)m(y)f(Theorem) g(1.2)h Fj(F)1414 1551 y Fi(\014)s(;N)1590 1537 y Fr(con)m(v)m(erges)h (as)e Fj(N)52 b Fk(")42 b(1)e Fr(and)g(that)g(the)h(limit)d(is)h (in\014nitely)50 1682 y(di\013eren)m(tiable)24 b(as)i(a)h(function)e (of)h(the)g(parameters)g Fj(a)1887 1696 y Fu(1)1928 1682 y Fj(;)15 b(:)g(:)g(:)i(;)e(a)2178 1699 y Fi(J)6 b Fu(\()p Fi(l)p Fu(\()p Fi(\014)s Fu(\))2382 1682 y Fr(,)27 b(except)g(at)g(the) f(set)h(of)f(v)-5 b(alues)25 b(where)50 1828 y(some)37 b(of)g(the)f(conditions)g(\(1.9\))i(hold)d(with)h(equalit)m(y;)j(moreo) m(v)m(er,)h Fk(\000)p Fj(F)2577 1842 y Fi(\014)s(;N)2749 1828 y Fr(is)35 b(con)m(v)m(ex)k(in)c(eac)m(h)j(of)f(the)50 1973 y(v)-5 b(ariables)425 1915 y Fk(p)p 501 1915 V 58 x Fj(a)549 1987 y Fi(i)579 1973 y Fr(.)40 b(Then)28 b(standard)g (results)f(of)i(con)m(v)m(ex)i(analysis)c(\(see)j([Ro],)g(Theorem)e (25.7\))j(imply)26 b(that)954 2264 y(lim)15 b(sup)1006 2341 y Fi(N)7 b Fh("1)1232 2264 y Fr(\()p Fk(\000)p Fg(E)1487 2202 y Fj(d)1534 2169 y Fu(2)p 1410 2243 243 4 v 1410 2338 a Fj(d)1457 2280 y Fk(p)p 1533 2280 79 4 v 58 x Fj(a)1581 2352 y Fi(i)1611 2295 y Fu(2)1663 2264 y Fj(F)1721 2278 y Fi(\014)s(;N)1856 2264 y Fr(\))26 b(=)f Fk(\000)2171 2202 y Fj(d)2218 2169 y Fu(2)p 2094 2243 243 4 v 2094 2338 a Fj(d)2141 2280 y Fk(p)p 2217 2280 79 4 v 58 x Fj(a)2265 2352 y Fi(i)2296 2295 y Fu(2)2386 2264 y Fr(lim)2363 2323 y Fi(N)7 b Fh("1)2552 2264 y Fg(E)j Fj(F)2665 2278 y Fi(\014)s(;N)3523 2264 y Fr(\(4)p Fj(:)p Fr(9\))50 2562 y(whic)m(h)37 b(is)g(\014nite.)63 b(Th)m(us,)40 b(the)e(\014rst)g(term)g(in)f(\(4.8\))j(will)c(v)-5 b(anish)36 b(when)i(divided)d(b)m(y)j Fj(N)10 b Fr(.)65 b(T)-8 b(o)38 b(see)h(that)50 2707 y(the)f(co)s(e\016cien)m(t)h(of)f Fj(N)48 b Fr(of)38 b(the)h(second)f(term)g(giv)m(es)g(a)h(v)-5 b(anishing)35 b(con)m(tribution,)k(w)m(e)g(use)e(the)i(general)50 2852 y(fact)32 b(that)g(if)e(the)h(v)-5 b(ariance)31 b(of)h(family)d(of)j(a)f(con)m(v)m(ex)i(\(or)f(conca)m(v)m(e\))i (functions)29 b(tends)i(to)h(zero,)h(then)e(the)50 2998 y(same)k(is)f(true)h(for)f(its)g(deriv)-5 b(ativ)m(e,)36 b(pro)m(vided)e(that)h(the)g(second)g(deriv)-5 b(ativ)m(e)34 b(is)g(b)s(ounded)f(\(see)j(Lemma)50 3143 y(8.9)d(in)d([BG])j(or)e (Lemma)h(4.3)h(in)d([T1].\))45 b(In)31 b(Theorem)h(1.3)g(w)m(e)g(ha)m (v)m(e)h(more)f(than)f(established)f(that)j(the)50 3289 y(v)-5 b(ariance)38 b(of)g Fj(F)583 3303 y Fi(\014)s(;N)757 3289 y Fr(tends)f(to)i(zero)g(\(this)e(follo)m(ws)g(of)h(course)h(also) f(from)f(Gaussian)g(concen)m(tration)j(of)50 3434 y(measure)30 b(estimates)h([L)-8 b(T]\),)31 b(and)f(hence)h(the)f(result)g(of)g(the) h(Lemma)f(is)g(pro)m(v)m(en.)41 b Fk(})50 3652 y Fs(Remark:)46 b Fr(The)29 b(last)h(argumen)m(t)g(is)f(the)h(sole)g(reason)g(wh)m(y)g (w)m(e)g(need)g(to)h(exclude)e(the)h(situations)f(when)50 3798 y(equalit)m(y)g(holds)f(for)h(some)g(of)h(the)g(equations)f (\(1.9\).)42 b(In)28 b(fact,)j(in)d(this)g(case)j(the)e(mo)s(del)g(is)f (in)g(a)i(sense)f(at)50 3943 y(a)i(\\critical)e(p)s(oin)m(t")h(in)f (the)i(space)g(of)f(its)g(parameters.)159 4162 y(Finally)-8 b(,)39 b(w)m(e)f(substitute)f(the)i(result)e(of)h(Lemma)h(4.3)g(in)m (to)f(Lemma)h(4.2.)65 b(Using)37 b(in)g(the)i(resulting)50 4307 y(expression)29 b(the)i(fact)g(that)g(b)m(y)f(4.1)i(and)d (\(4.2\),)741 4570 y Fg(E)10 b Fj(\026)850 4584 y Fi(\014)t(;N)991 4570 y Fr(\()p Fj(X)1101 4584 y Fi(\033)1143 4593 y Fn(1)1179 4584 y Fi(:::)o(\033)1286 4593 y Fm(i)1320 4570 y Fr(\))p Fj(=)1400 4488 y Fk(p)p 1476 4488 83 4 v 82 x Fj(N)35 b Fr(=)25 b Fj(\014)1736 4507 y Fk(p)p 1812 4507 79 4 v 63 x Fj(a)1860 4584 y Fi(i)1891 4570 y Fr(\(1)c Fk(\000)f Fg(E)10 b Fj(\026)2192 4531 y Fh(\012)p Fu(2)2192 4597 y Fi(\014)t(;N)2333 4570 y Fr(\(1)-23 b(I)2424 4586 y Fi(\033)2466 4595 y Fn(1)2502 4586 y Fi(:::)n(\033)2608 4595 y Fm(i)2637 4586 y Fu(=)p Fi(\033)2737 4564 y Ff(0)2735 4608 y Fn(1)2771 4586 y Fi(:::)n(\033)2879 4564 y Ff(0)2877 4609 y Fm(i)2906 4586 y Fh(g)2948 4570 y Fr(\)\))459 b(\(4)p Fj(:)p Fr(10\))50 4833 y(w)m(e)31 b(obtain)e(the)i(assertion)f (of)h(Prop)s(osition)d(1.8.)p Fk(})50 5098 y Fv(References)-68 5316 y Fr([A)m(C])46 b(M.)e(Aizenmann)e(and)h(P)-8 b(.)44 b(Con)m(tucci,)f(On)g(the)h(stabilit)m(y)e(of)h(the)h(quenc)m(hed)f (state)i(in)d(mean)h(\014eld)159 5462 y(spin-glass)29 b(mo)s(dels,)g(J.)h(Statist.)h(Ph)m(ys.)f Fs(92)p Fr(,)h(765-783)i (\(1998\).)p eop %%Page: 38 38 38 37 bop 50 295 a Fo(38)3225 b Fl(R)l(efer)l(enc)l(es)-73 513 y Fr([BG])47 b(A.)30 b(Bo)m(vier,)g(V.)g(Ga)m(yrard,)g(The)f (Hop\014eld)f(mo)s(del)g(as)i(a)f(generalized)g(random)g(mean)h (\014eld)e(mo)s(del,)g(in)159 658 y(Mathematics)37 b(of)f(spin)d (glasses)j(and)f(neural)f(net)m(w)m(orks,)j(A.)f(Bo)m(vier,)h(P)-8 b(.)36 b(Picco.)g(Eds.,)f(Progress)h(in)159 804 y(Probablit)m(y)-8 b(,)30 b(Birkh\177)-45 b(auser,)29 b(Boston,)i(\(1997\).)-118 1022 y([BK1])47 b(A.)21 b(Bo)m(vier)g(and)f(I.)g(Kurk)m(o)m(v)-5 b(a,)21 b(Derrida's)f(Generalized)g(random)g(Energy)g(mo)s(dels)f(1.)i (P)m(oisson)f(cascades)159 1167 y(and)30 b(the)g(extremal)h(pro)s (cess.)f(preprin)m(t)e(\(2002\).)-118 1385 y([BK3])47 b(A.)32 b(Bo)m(vier)h(and)f(I.)g(Kurk)m(o)m(v)-5 b(a,)33 b(Derrida's)e(Generalized)h(Random)g(Energy)g(mo)s(dels)f(3.)i(Mo)s (dels)e(with)159 1531 y(con)m(tin)m(uous)f(hierarc)m(hies.)f(in)g (preparation)h(\(2002\).)-129 1749 y([BKL])46 b(A.)30 b(Bo)m(vier,)h(I.)g(Kurk)m(o)m(v)-5 b(a,)31 b(and)e(M.)i(L\177)-45 b(ow)m(e,)31 b(Fluctuations)f(of)g(the)g(free)h(energy)f(in)f(the)i (REM)f(and)g(the)159 1895 y Fj(p)p Fr(-spin)e(SK-mo)s(dels,)h(Ann.)h (Probab.)g Fs(30)h Fr(\(2002\).)-130 2113 y([CCP])45 b(D.)29 b(Cap)s(o)s(caccia,)h(M.)f(Cassandro,)f(and)g(P)-8 b(.)30 b(Picco,)f(On)f(the)h(existence)g(of)g(Thermo)s(dynamics)d(for)j (the)159 2258 y(Generalised)g(Random)h(Energy)g(Mo)s(del.)61 b(J.)30 b(Stat.)h(Ph)m(ysics)f Fs(46)p Fr(,)h(493{505)i(\(1987\).)-52 2476 y([D1])47 b(B.)39 b(Derrida,)f(Random)g(energy)g(mo)s(del:)g (limit)e(of)j(a)f(family)f(of)i(disordered)d(mo)s(dels,)76 b(Ph)m(ys.)38 b(Rev.)159 2622 y(Letts.)31 b Fs(45)p Fr(,)g(79-82)i (\(1980\).)-52 2840 y([D2])47 b(B.)23 b(Derrida,)e(Random)h(energy)g (mo)s(del:)f(An)h(exactly)h(solv)-5 b(able)21 b(mo)s(del)g(of)h (disordered)e(systems,)45 b(Ph)m(ys.)159 2985 y(Rev.)31 b(B)f Fs(24)p Fr(,)i(2613-2626)i(\(1981\))-52 3204 y([D3])47 b(B.)21 b(Derrida,)f(A)h(generalization)f(of)g(the)h(random)f(energy)h (mo)s(del)e(that)i(includes)d(correlations)i(b)s(et)m(w)m(een)159 3349 y(the)30 b(energies,)h(J.)f(Ph)m(ysique.)g(Lett.)h Fs(46)p Fr(,)g(401{407)j(\(1985\).)-123 3567 y([DG1])47 b(B.)30 b(Derrida)g(and)f(E.)h(Gardner,)f(Solution)f(of)j(the)f (generalized)f(random)g(energy)i(mo)s(del,)e(J.)g(Ph)m(ys.)h(C)159 3713 y Fs(19)p Fr(,)h(2253{2274)j(\(1986\).)-123 3931 y([DG2])47 b(B.)e(Derrida)f(and)g(E.)g(Gardner,)g(Magnetic)i(prop)s (erties)d(and)h(function)f Fj(q)s Fr(\()p Fj(x)p Fr(\))i(of)g(the)g (generalised)159 4076 y(random)29 b(energy)i(mo)s(del,)f(J.)g(Ph)m(ys.) g(C)g Fs(19)p Fr(,)h(5783-5798)j(\(1986\))-123 4295 y([DG3])47 b(E.)34 b(Gardner)f(and)h(B.)g(Derrida,)g(The)f(probabilit)m(y)f (distribution)e(of)k(the)g(partition)f(function)f(of)j(the)159 4440 y(random)29 b(energy)i(mo)s(del,)f(J.)g(Ph)m(ys.)g(A)p Fs(22)p Fr(,)h(1975-1981)k(\(1989\))-80 4658 y([GG])47 b(S.)38 b(Ghirlanda,)e(F.)j(Guerra.)76 b(General)39 b(prop)s(erties)d (of)j(the)f(o)m(v)m(erlap)h(probabilit)m(y)d(distributions)e(in)159 4804 y(disordered)28 b(spin)h(systems.T)-8 b(o)m(w)m(ards)31 b(P)m(arisi)e(ultrametricit)m(y,)h(J.)g(Ph)m(ys.)g(A)61 b(31\(1998\))34 b(,)c(9144-9155.)-53 5022 y([Ka])46 b(O.)30 b(Kallen)m(b)s(erg,)f(Random)h(measures,)h(4th)f(edition,)f(Ak)-5 b(ademie)31 b(V)-8 b(erlag,)31 b(Berlin,)e(1986.)-52 5240 y([L)-8 b(T])46 b(M.)22 b(Ledoux)g(and)f(L.)h(T)-8 b(alagrand,)21 b(Probabilit)m(y)f(in)g(Banac)m(h)j(space.)g(Spinger,)d (Berlin-Heidelb)s(erg-New)159 5385 y(Y)-8 b(ork,)31 b(1991.)p eop %%Page: 39 39 39 38 bop 50 295 a Fl(GREM,)28 b(p)l(art)f(2)3089 b Fo(39)-44 513 y Fr([Re])46 b(S.I.)28 b(Resnic)m(k,)g(Extreme)h(v)-5 b(alues,)28 b(regular)f(v)-5 b(ariation)28 b(,)h(and)e(p)s(oin)m(t)h (pro)s(cesses,)g(Springer,)e(New)j(Y)-8 b(ork,)159 658 y(1987.)-54 876 y([Ru])45 b(D.)29 b(Ruelle,)e(A)h(mathematical)g (reform)m(ulation)f(of)h(Derrida's)f(REM)h(and)g(GREM.)h(Comm)m(un.)e (Math.)159 1022 y(Ph)m(ys.)60 b Fs(108)p Fr(,)32 b(225-239)h(\(1987\).) -58 1240 y([SK])45 b(D.)28 b(Sherrington)e(and)h(S.)g(Kirkpatric)m(k,)f (\\Solv)-5 b(able)27 b(mo)s(del)g(of)g(a)h(spin)e(glass",)56 b(Ph)m(ys.)27 b(Rev.)h(Lett.)h Fs(35)p Fr(,)159 1385 y(1792-1796)34 b(\(1972\).)-48 1604 y([T1])46 b(M.)21 b(T)-8 b(alagrand,)21 b(Rigorous)g(results)f(for)h(the)g(Hop\014eld)f (mo)s(del)f(with)h(man)m(y)h(patterns,)h(Probab.)e(Theory)159 1749 y(Related)31 b(Fields)d Fs(110)p Fr(,)k(177-276)h(\(1998\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF