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(the)i(case)g(of)f(the)h(t)m(w)m(o-lev)m(el)g(tree,)i(ho)m(w)m(ev)m(er) 50 804 y(without)27 b(explaining)f(the)j(relation)e(to)j(an)m(y)e (thermo)s(dynamics)f(quan)m(tities)h(in)f(the)i(mo)s(del.)39 b(Most)29 b(of)g(the)50 949 y(in)m(tuitiv)m(e)h(insigh)m(t)g(in)m(to)i (the)g(mo)s(del)f(stems)g(from)h(a)g(pap)s(er)e(b)m(y)i(Ruelle)e([Ru])i (who)f(in)f(fact)j(form)m(ulates)e(a)50 1095 y(new)d(mo)s(del)f(based)h (on)g(P)m(oisson)g(p)s(oin)m(t)f(pro)s(cesses)h(that)h(are)g(suggested) g(to)g(represen)m(t)f(the)h(asymptotics)50 1240 y(of)k(the)h (\015uctuations)f(of)g(the)h(partition)e(function)g(and)h(the)h(Gibbs)d (measures)j(of)f(Derrida's)g(mo)s(dels)f(in)50 1385 y(the)24 b(thermo)s(dynamic)f(limit.)36 b(But)25 b(this)e(connection)h(w)m(as)h (established)d(rigorously)g(ev)m(en)j(in)e(the)i(simplest)50 1531 y(case)31 b(of)g(the)f(REM)h(only)e(v)m(ery)i(recen)m(tly)h ([BKL].)159 1749 y(An)23 b(imp)s(ortan)m(t)f(ingredien)m(t)f(in)h(the)h (analysis)f(of)h(the)g(REM)g(is)f(the)i(theory)f(of)g(con)m(v)m (ergence)i(to)f(P)m(oisson)50 1895 y(pro)s(cesses)33 b(of)h(the)g(extreme)g(v)-5 b(alue)33 b(statistics)h(of)f(\(i.i.d.\))50 b(random)33 b(v)-5 b(ariables)32 b(that)i(is,)g(of)g(course,)g(v)m(ery) 50 2040 y(w)m(ell)41 b(kno)m(wn)h(\(see)h(e.g.)78 b([LLR,Re]\).)f(In)42 b(the)g(GREM,)h(analogous)g(results)e(will)f(b)s(e)i(needed)g(in)f(the) 50 2185 y(correlated)g(case.)72 b(Here,)44 b(standard)c(results)f(on)i (extremal)f(pro)s(cesses)h(are)g(m)m(uc)m(h)f(rarer.)71 b(There)40 b(are)50 2331 y(criteria)45 b(under)e(whic)m(h)i(the)g (extremal)h(pro)s(cesses)f(ha)m(v)m(e)i(the)e(same)h(limits)d(as)j(in)e (the)i(indep)s(enden)m(t)50 2476 y(case.)71 b(Ho)m(w)m(ev)m(er,)45 b(here)40 b(w)m(e)g(will)e(b)s(e)h(mostly)h(in)m(terested)g(in)f (situations)g(when)g(this)g(is)g(not)h(the)g(case.)50 2622 y(The)c(results)f(of)h(this)g(\014rst)f(pap)s(er)h(will)d(b)s(e)j (general)g(P)m(oisson)g(con)m(v)m(ergence)j(results)c(for)h(the)h (extremes)50 2767 y(of)f(hierarc)m(hically)e(correlated)j(Gaussian)e (random)g(pro)s(cesses,)j(where)e(the)g(limits)e(will)f(in)i(general)h (b)s(e)50 2913 y(describ)s(ed)28 b(as)j(cascades)g(of)g(P)m(oisson)f (pro)s(cesses.)50 3131 y Fr(1.2.)59 b(De\014nition)40 b(of)f(the)g(mo)s(dels.)51 b Fq(W)-8 b(e)35 b(will)c(consider)i (Gaussian)g(pro)s(cesses)h Fo(X)3031 3145 y Fn(\033)3114 3131 y Fq(indexed)f(b)m(y)h(the)50 3276 y(h)m(yp)s(ercub)s(e)i Fp(S)552 3290 y Fn(N)658 3276 y Fp(\021)g(f\000)p Fq(1)p Fo(;)15 b Fq(1)p Fp(g)1056 3243 y Fn(N)1127 3276 y Fq(.)62 b(Let)38 b(us)f(equip)f(the)h(h)m(yp)s(ercub)s(e)f Fp(S)2425 3290 y Fn(N)2531 3276 y Fq(with)g(the)i(natural)f(ultra-metric)50 3422 y(v)-5 b(aluation)1177 3567 y Fo(d)1224 3581 y Fn(N)1293 3567 y Fq(\()p Fo(\033)n(;)15 b(\033)1473 3530 y Fg(0)1500 3567 y Fq(\))25 b Fp(\021)1686 3506 y Fq(1)p 1667 3546 83 4 v 1667 3630 a Fo(N)1776 3567 y Fq(\(min)14 b(\()p Fo(i)p Fp(j)p Fo(\033)2121 3581 y Fn(i)2177 3567 y Fp(6)p Fq(=)25 b Fo(\033)2328 3530 y Fg(0)2325 3590 y Fn(i)2356 3567 y Fq(\))20 b Fp(\000)g Fq(1\))941 b(\(1)p Fo(:)p Fq(1\))50 3773 y(\(Note)46 b(that)g(exp\()p Fp(\000)p Fo(d)823 3787 y Fn(N)892 3773 y Fq(\()p Fo(\033)n(;)15 b(\033)1072 3740 y Fg(0)1098 3773 y Fq(\)\))46 b(is)e(a)h(ultra-metric) f(on)g Fp(S)2125 3787 y Fn(N)2194 3773 y Fq(\).)85 b(W)-8 b(e)46 b(will)c(consider)h(pro)s(cesses)i(whose)50 3919 y(co)m(v)-5 b(ariance)31 b(is)f(giv)m(en)g(b)m(y)h(a)f (\(non-decreasing\))h(function)e(of)h Fo(d)2214 3933 y Fn(N)2284 3919 y Fq(,)g(i.e.)41 b(w)m(e)31 b(assume)f(that)h Fe(E)10 b Fo(X)3271 3933 y Fn(\033)3351 3919 y Fq(=)25 b(0,)31 b(and)1078 4166 y(co)m(v)16 b(\()p Fo(X)1334 4180 y Fn(\033)1384 4166 y Fo(;)f(X)1499 4180 y Fn(\033)1543 4162 y Fd(0)1573 4166 y Fq(\))26 b(=)f Fe(E)10 b Fo(X)1860 4180 y Fn(\033)1915 4166 y Fo(X)1990 4180 y Fn(\033)2034 4162 y Fd(0)2089 4166 y Fq(=)25 b Fo(A)p Fq(\()p Fo(d)2335 4180 y Fn(N)2404 4166 y Fq(\()p Fo(\033)n(;)15 b(\033)2584 4129 y Fg(0)2611 4166 y Fq(\)\))842 b(\(1)p Fo(:)p Fq(2\))50 4414 y(where)43 b Fo(A)p Fq(\()p Fo(x)p Fq(\))h(is)e(a)i(probabilit)m (y)d(distribution)f(function)i(on)i([0)p Fo(;)15 b Fq(1].)81 b(Note)45 b(that)f(the)g(mo)s(dels)e(of)i(the)50 4560 y(SK-family)30 b(can)i(b)s(e)f(constructed)h(in)f(an)h(analogous)g(w)m (a)m(y)-8 b(,)34 b(except)f(that)f Fo(d)2659 4574 y Fn(N)2729 4560 y Fq(\()p Fo(\033)n(;)15 b(\033)2909 4527 y Fg(0)2935 4560 y Fq(\))32 b(is)f(replaced)h(b)m(y)g(the)50 4705 y(o)m(v)m(erlap)f Fo(R)435 4719 y Fn(N)504 4705 y Fq(\()p Fo(\033)n(;)15 b(\033)684 4672 y Fg(0)710 4705 y Fq(\))26 b Fp(\021)f Fo(N)950 4672 y Fg(\000)p Ft(1)1063 4637 y Ff(P)1159 4659 y Fn(N)1159 4732 y(i)p Ft(=1)1296 4705 y Fo(\033)1348 4719 y Fn(i)1378 4705 y Fo(\033)1433 4672 y Fg(0)1430 4729 y Fn(i)1460 4705 y Fq(.)159 4923 y(The)k(prop)s(er)f (GREM)i(with)f Fm(\014nitely)h Fq(man)m(y)g(hierarc)m(hies)e(corresp)s (onds)h(to)h(the)g(sp)s(ecial)e(case)j(when)e Fo(A)50 5069 y Fq(is)k(the)h(distribution)d(function)i(of)h(a)h(measure)f(that) h(is)e(supp)s(orted)g(on)h(a)g(\014nite)f(n)m(um)m(b)s(er,)i Fo(n)p Fq(,)f(of)h(p)s(oin)m(ts)50 5214 y Fo(x)102 5228 y Ft(1)143 5214 y Fo(;)15 b(:)g(:)g(:)h(;)f(x)396 5228 y Fn(n)471 5214 y Fp(2)25 b Fq([0)p Fo(;)15 b Fq(1].)43 b(In)29 b(that)i(case)h(w)m(e)e(denote)h(the)g(mass)f(of)h(the)f(atoms) h Fo(x)2694 5228 y Fn(i)2755 5214 y Fq(b)m(y)f Fo(a)2929 5228 y Fn(i)2960 5214 y Fq(,)g(and)g(w)m(e)h(set)1141 5462 y(ln)14 b Fo(\013)1290 5476 y Fn(i)1346 5462 y Fq(=)25 b(\()p Fo(x)1529 5476 y Fn(i)1580 5462 y Fp(\000)19 b Fo(x)1722 5476 y Fn(i)p Fg(\000)p Ft(1)1846 5462 y Fq(\))c(ln)f(2)p Fo(;)107 b(i)25 b Fq(=)g(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(n)905 b Fq(\(1)p Fo(:)p Fq(3\))p eop %%Page: 4 4 4 3 bop 50 295 a Fk(4)3309 b Fh(Se)l(ction)27 b Fk(1)50 513 y Fq(where)36 b Fo(x)371 527 y Ft(0)447 513 y Fp(\021)f Fq(0.)60 b(Of)36 b(course)1103 445 y Ff(P)1199 467 y Fn(n)1199 540 y(i)p Ft(=1)1336 513 y Fo(a)1384 527 y Fn(i)1450 513 y Fq(=)f(1,)k(and)1848 445 y Ff(Q)1934 467 y Fn(n)1934 540 y(i)p Ft(=1)2071 513 y Fo(\013)2129 527 y Fn(i)2195 513 y Fq(=)c(2.)60 b(The)36 b(h)m(yp)s(ercub)s(e)f Fp(S)3125 527 y Fn(N)3230 513 y Fq(can)i(then)f(b)s(e)50 658 y(considered)30 b(as)i(a)g Fo(n)p Fq(-fold)e(pro)s(duct)g Fp(S)1348 672 y Fn(N)1444 658 y Fq(=)1542 590 y Ff(Q)1628 612 y Fn(n)1628 685 y(i)p Ft(=1)1765 658 y Fp(S)1820 675 y Fn(N)19 b Ft(ln)13 b Fn(\013)2017 684 y Fi(i)2046 675 y Fn(=)g Ft(ln)g(2)2210 658 y Fq(.)44 b(W)-8 b(e)32 b(will)d(write)i Fo(\033)f Fq(=)d Fo(\033)3075 672 y Ft(1)3115 658 y Fo(\033)3167 672 y Ft(2)3223 658 y Fo(:)15 b(:)g(:)h(\033)3396 672 y Fn(n)3477 658 y Fq(where)50 804 y Fo(\033)102 818 y Fn(i)164 804 y Fp(2)31 b(S)311 820 y Fn(N)19 b Ft(ln)13 b Fn(\013)508 829 y Fi(i)537 820 y Fn(=)g Ft(ln)g(2)701 804 y Fq(.)52 b(Usually)33 b(w)m(e)h(will)e(assume)i(that)h Fo(x)1995 818 y Ft(1)2067 804 y Fo(>)d Fq(0,)k Fo(x)2328 818 y Fn(n)2409 804 y Fq(=)31 b(1,)36 b(and)e(all)f Fo(a)2976 818 y Fn(i)3038 804 y Fo(>)e Fq(0,)36 b(although)e(at)50 949 y(some)d(later)f(stage)i (w)m(e)f(will)c(ha)m(v)m(e)32 b(to)f(lo)s(ok)f(also)g(at)h(what)g(happ) s(ens)d(when)i Fo(x)2713 963 y Ft(1)2779 949 y Fq(=)25 b(0)30 b(and)g Fo(a)3175 963 y Fn(n)3250 949 y Fq(=)25 b(0.)159 1167 y(Then)k(the)i(Gaussian)e(pro)s(cess)h Fo(X)1335 1181 y Fn(\033)1415 1167 y Fq(can)h(b)s(e)e(represen)m(ted)i (in)e(the)h(follo)m(wing)f(w)m(a)m(y)-8 b(.)159 1385 y(T)g(ak)m(e)23 b Fo(\013)428 1352 y Fn(N)428 1408 y Ft(1)501 1385 y Fq(+)s(\()p Fo(\013)668 1399 y Ft(1)710 1385 y Fo(\013)768 1399 y Ft(2)809 1385 y Fq(\))844 1352 y Fn(N)916 1385 y Fq(+)s Fp(\001)15 b(\001)g(\001)5 b Fq(+)s(\()p Fo(\013)1267 1399 y Ft(1)1309 1385 y Fo(\013)1367 1399 y Ft(2)1423 1385 y Fp(\001)15 b(\001)g(\001)h Fo(\013)1602 1399 y Fn(n)1651 1385 y Fq(\))1686 1352 y Fn(N)1778 1385 y Fq(indep)s(enden)m(t)k(standard)h(Gaussian)g(random)g(v)-5 b(ariables)50 1531 y(en)m(umerated)36 b(as)f(follo)m(ws:)51 b(The)35 b(\014rst)g Fo(\013)1452 1498 y Fn(N)1452 1553 y Ft(1)1556 1531 y Fq(of)h(them)g Fo(X)1978 1545 y Fn(\033)2020 1554 y Fj(1)2096 1531 y Fq(are)g(en)m(umerated)f(b)m(y)h Fo(\013)2940 1498 y Fn(N)2940 1553 y Ft(1)3045 1531 y Fq(con\014gurations)f(of)50 1676 y(spins)40 b Fo(\033)342 1690 y Ft(1)428 1676 y Fp(2)45 b(f\000)p Fq(1)p Fo(;)15 b Fq(+1)p Fp(g)896 1643 y Fn(N)20 b Ft(ln)14 b Fn(\013)1095 1652 y Fj(1)1130 1643 y Fn(=)f Ft(ln)g(2)1294 1676 y Fq(,)45 b(the)e(next)f(\()p Fo(\013)1842 1690 y Ft(1)1884 1676 y Fo(\013)1942 1690 y Ft(2)1983 1676 y Fq(\))2018 1643 y Fn(N)2129 1676 y Fq(of)h(them)f Fo(X)2564 1690 y Fn(\033)2606 1699 y Fj(1)2642 1690 y Fn(\033)2684 1699 y Fj(2)2767 1676 y Fq(|)g(b)m(y)g(con\014gurations)g(of)50 1822 y(spins)34 b Fo(\033)336 1836 y Ft(1)377 1822 y Fo(\033)429 1836 y Ft(2)505 1822 y Fp(2)h(f\000)p Fq(1)p Fo(;)15 b Fq(+1)p Fp(g)963 1789 y Fn(N)20 b Ft(ln)q(\()p Fn(\013)1177 1798 y Fj(1)1213 1789 y Fn(\013)1260 1798 y Fj(2)1296 1789 y Ft(\))p Fn(=)13 b Ft(ln)g(2)1488 1822 y Fq(,)38 b(etc)f Fo(:)15 b(:)g(:)32 b Fq(,)38 b(\014nally)c(the)j (last)f(\()p Fo(\013)2618 1836 y Ft(1)2659 1822 y Fo(\013)2717 1836 y Ft(2)2773 1822 y Fp(\001)15 b(\001)g(\001)h Fo(\013)2952 1836 y Fn(n)3002 1822 y Fq(\))3037 1789 y Fn(N)3142 1822 y Fq(=)35 b(2)3293 1789 y Fn(N)3398 1822 y Fq(of)i(them)50 1967 y Fo(X)125 1981 y Fn(\033)167 1990 y Fj(1)203 1981 y Fn(\033)245 1990 y Fj(2)280 1981 y Fn(;:::)11 b(;\033)443 1990 y Fi(n)522 1967 y Fq(are)31 b(en)m(umerated)g(b)m(y)f (con\014gurations)g Fo(\033)1932 1981 y Ft(1)1972 1967 y Fo(\033)2024 1981 y Ft(2)2080 1967 y Fo(:)15 b(:)g(:)h(\033)2253 1981 y Fn(n)2328 1967 y Fp(2)25 b(f\000)p Fq(1)p Fo(;)15 b Fq(+1)p Fp(g)2776 1934 y Fn(N)2847 1967 y Fq(.)159 2185 y(Then)29 b(the)i(pro)s(cess)f Fo(X)945 2199 y Fn(\033)1024 2185 y Fq(has)g(the)h(same)g(distribution)c(as)477 2456 y Fo(X)552 2470 y Fn(\033)626 2456 y Fp(\021)722 2393 y(p)p 798 2393 89 4 v 63 x Fo(a)846 2470 y Ft(1)887 2456 y Fo(X)962 2470 y Fn(\033)1004 2479 y Fj(1)1065 2456 y Fq(+)1155 2393 y Fp(p)p 1231 2393 V 63 x Fo(a)1279 2470 y Ft(2)1320 2456 y Fo(X)1395 2470 y Fn(\033)1437 2479 y Fj(1)1473 2470 y Fn(\033)1515 2479 y Fj(2)1575 2456 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)1883 2393 y Fp(p)p 1959 2393 98 4 v 63 x Fo(a)2007 2470 y Fn(n)2057 2456 y Fo(X)2132 2470 y Fn(\033)2174 2479 y Fj(1)2210 2470 y Fn(\033)2252 2479 y Fj(2)2287 2470 y Fn(:::)o(\033)2394 2479 y Fi(n)2443 2456 y Fo(;)107 b Fq(if)29 b Fo(\033)f Fq(=)d Fo(\033)2886 2470 y Ft(1)2927 2456 y Fo(\033)2979 2470 y Ft(2)3035 2456 y Fo(:)15 b(:)g(:)h(\033)3208 2470 y Fn(n)3258 2456 y Fo(;)240 b Fq(\(1)p Fo(:)p Fq(4\))50 2726 y(The)30 b(explicit)e(construction)i(in)f(terms)i(of)f(\(1.4\))i (will)c(b)s(e)i(quite)g(useful.)50 3010 y Fr(1.3.)47 b(Con)m(v)m(ergence)36 b(of)f(asso)s(ciated)g(p)s(oin)m(t)g(pro)s (cesses)159 3228 y Fq(W)-8 b(e)33 b(in)m(tro)s(duce)e(the)h(function)f Fo(u)1291 3242 y Ft(ln)13 b Fn(\013;N)1502 3228 y Fq(\()p Fo(x)p Fq(\),)33 b Fo(x)28 b Fp(2)f Fe(R)s Fq(,)39 b(suc)m(h)31 b(that)i Fo(P)13 b Fq(\()p Fo(X)36 b(>)27 b(u)2746 3242 y Ft(ln)13 b Fn(\013;N)2957 3228 y Fq(\()p Fo(x)p Fq(\)\))29 b Fp(\030)f Fo(\013)3300 3195 y Fg(\000)p Fn(N)3426 3228 y Fo(e)3468 3195 y Fg(\000)p Fn(x)3571 3228 y Fq(,)k(as)50 3373 y Fo(N)j Fp(")26 b(1)p Fq(,)h(where)f Fo(X)34 b Fq(is)26 b(a)h(standard)f(Gaussian)f(random)h(v)-5 b(ariable.)39 b(It)26 b(can)h(b)s(e)f(written)g(explicitly)e(as)j(\(see)50 3519 y(e.g.)42 b([LLR]\))698 3691 y Fo(u)750 3705 y Ft(ln)13 b Fn(\013;N)961 3691 y Fq(\()p Fo(x)p Fq(\))26 b(=)1205 3609 y Fp(p)p 1281 3609 293 4 v 82 x Fq(2)15 b(ln)f Fo(\013N)31 b Fq(+)1854 3630 y Fo(x)p 1696 3670 369 4 v 1696 3689 a Fp(p)p 1772 3689 293 4 v 77 x Fq(2)15 b(ln)f Fo(\013N)2096 3691 y Fp(\000)2197 3630 y Fq(ln)g Fo(N)31 b Fq(+)19 b(ln)14 b(ln)g Fo(\013)21 b Fq(+)f(ln)14 b(4)p Fo(\031)p 2197 3670 828 4 v 2404 3766 a Fq(2)2449 3689 y Fp(p)p 2525 3689 293 4 v 77 x Fq(2)h(ln)g Fo(\013N)3036 3691 y(:)462 b Fq(\(1)p Fo(:)p Fq(5\))50 4010 y Fc(Theorem)41 b(1.1:)105 b Fm(L)-5 b(et)35 b Fo(n)30 b Fp(2)g Fe(N)6 b Fm(,)42 b Fo(n)30 b Fp(\025)g Fq(1)p Fm(,)36 b Fq(0)31 b Fo(<)f(a)1807 4024 y Fn(i)1867 4010 y Fo(<)g Fq(1)p Fm(,)37 b Fo(\013)2136 4024 y Fn(i)2196 4010 y Fo(>)30 b Fq(1)p Fm(,)37 b Fo(i)30 b Fq(=)g(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(n)p Fm(.)50 b(Set)44 b Fq(\026)-54 b Fo(\013)31 b Fp(\021)3398 3942 y Ff(Q)3484 3965 y Fn(n)3484 4038 y(i)p Ft(=1)3621 4010 y Fo(\013)3679 4024 y Fn(i)50 4156 y Fm(and)i(assume)h(that)731 4088 y Ff(P)826 4110 y Fn(n)826 4183 y(i)p Ft(=1)964 4156 y Fo(a)1012 4170 y Fn(i)1068 4156 y Fq(=)24 b(1)p Fm(.)43 b(The)32 b(p)-5 b(oint)34 b(pr)-5 b(o)g(c)g(ess)970 4340 y Ff(X)581 4544 y Fn(\033)r Ft(=)p Fn(\033)723 4553 y Fj(1)759 4544 y Fn(:::)o(\033)866 4553 y Fi(n)911 4544 y Fg(2f\000)p Ft(1)p Fn(;)p Ft(+1)p Fg(g)1241 4526 y Fj(ln)17 b(\026)-38 b Fi(\013=)12 b Fj(ln)e(2)1505 4426 y Fo(\016)1545 4455 y Fn(u)1588 4425 y Fd(\000)p Fj(1)1588 4479 y(ln)18 b(\026)-38 b Fi(\013)o(;N)1772 4455 y Ft(\()1800 4413 y Fg(p)p 1861 4413 75 4 v 1861 4455 a Fn(a)1900 4464 y Fj(1)1935 4455 y Fn(X)1995 4464 y Fi(\033)2032 4478 y Fj(1)2072 4455 y Ft(+)2128 4413 y Fg(p)p 2188 4413 V 42 x Fn(a)2227 4464 y Fj(2)2263 4455 y Fn(X)2323 4464 y Fi(\033)2360 4478 y Fj(1)2395 4464 y Fi(\033)2432 4478 y Fj(2)2472 4455 y Ft(+)p Fg(\001\001\001)n Ft(+)2648 4413 y Fg(p)p 2708 4413 85 4 v 42 x Fn(a)2747 4464 y Fi(n)2792 4455 y Fn(X)2852 4464 y Fi(\033)2889 4478 y Fj(1)2924 4464 y Fi(\033)2961 4478 y Fj(2)2997 4464 y Fi(:::)o(\033)3096 4473 y(n)3145 4455 y Ft(\))50 4775 y Fm(c)-5 b(onver)g(ges)28 b(we)-5 b(akly)28 b(to)g(the)g(Poisson)g(p)-5 b(oint)29 b(pr)-5 b(o)g(c)g(ess)29 b Fp(P)35 b Fm(on)28 b Fe(R)36 b Fm(with)28 b(intensity)g(me)-5 b(asur)g(e)28 b Fo(K)7 b(e)3191 4742 y Fg(\000)p Fn(x)3294 4775 y Fo(dx)p Fm(,)28 b Fo(K)k Fp(2)25 b Fe(R)50 4921 y Fm(if)32 b(and)i(only)f(if)f(the)h (fol)5 b(lowing)34 b(ine)-5 b(qualities)33 b(hold)h(true:)517 5191 y Fo(a)565 5205 y Fn(i)616 5191 y Fq(+)20 b Fo(a)755 5205 y Fn(i)p Ft(+1)897 5191 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1253 5205 y Fn(n)1328 5191 y Fp(\025)25 b Fq(ln)o(\()p Fo(\013)1593 5205 y Fn(i)1624 5191 y Fo(\013)1682 5205 y Fn(i)p Ft(+1)1819 5191 y Fp(\001)15 b(\001)g(\001)h Fo(\013)1998 5205 y Fn(n)2048 5191 y Fq(\))p Fo(=)f Fq(ln)24 b(\026)-54 b Fo(\013)93 b Fm(for)34 b(al)5 b(l)33 b Fo(i)25 b Fq(=)g(2)p Fo(;)15 b Fq(3)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(n:)281 b Fq(\(1)p Fo(:)p Fq(6\))50 5462 y Fm(F)-7 b(urthermor)i(e,)34 b(if)e(al)5 b(l)33 b(ine)-5 b(qualities)32 b(in)g(\(1.6\))h(ar)-5 b(e)33 b(strict,)f(then)h(the)f(c)-5 b(onstant)34 b Fo(K)e Fq(=)25 b(1)p Fm(.)42 b(If)32 b(some)h(of)f(them)p eop %%Page: 5 5 5 4 bop 50 295 a Fh(GREM,)28 b(p)l(art)f(1)3127 b Fk(5)50 513 y Fm(ar)-5 b(e)33 b(e)-5 b(qualities,)33 b(then)g Fo(K)39 b Fm(is)33 b(given)f(as)546 716 y Fo(K)g Fq(=)25 b Fe(P)806 615 y Ff(\020)1100 629 y(\\)968 812 y Fi(k)q Fj(:)p Fi(a)1056 828 y(k)1097 812 y Fj(+)p Fd(\001\001\001)o Fj(+)p Fi(a)1288 821 y(n)887 873 y Fj(=ln)o(\()p Fi(\013)1053 889 y(k)1092 873 y Fd(\001\001\001)o Fi(\013)1196 882 y(n)1240 873 y Fj(\))p Fi(=)11 b Fj(ln)18 b(\026)-38 b Fi(\013)1425 615 y Ff(\020)1480 716 y Fo(Z)1542 730 y Fn(k)r Fg(\000)p Ft(1)1680 594 y Ff(r)p 1770 594 1294 4 v 2324 654 a Fo(a)2372 668 y Fn(k)r Fg(\000)p Ft(1)p 1781 695 1272 4 v 1781 778 a Fq(\()p Fo(a)1864 792 y Fn(k)r Fg(\000)p Ft(1)2023 778 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2379 792 y Fn(n)2428 778 y Fq(\)\()p Fo(a)2546 792 y Fn(k)2612 778 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2968 792 y Fn(n)3018 778 y Fq(\))1446 1064 y(+)g Fo(Z)1599 1078 y Fn(k)r Fg(\000)p Ft(2)1736 942 y Ff(r)p 1827 942 1387 4 v 2428 1002 a Fo(a)2476 1016 y Fn(k)r Fg(\000)p Ft(2)p 1838 1043 1365 4 v 1838 1126 a Fq(\()p Fo(a)1921 1140 y Fn(k)r Fg(\000)p Ft(2)2080 1126 y Fq(+)f Fp(\001)c(\001)g(\001)22 b Fq(+)e Fo(a)2436 1140 y Fn(n)2485 1126 y Fq(\)\()p Fo(a)2603 1140 y Fn(k)r Fg(\000)p Ft(1)2762 1126 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)3118 1140 y Fn(n)3168 1126 y Fq(\))1446 1270 y(+)g Fp(\001)15 b(\001)g(\001)1446 1467 y Fq(+)20 b Fo(Z)1599 1481 y Ft(1)1639 1346 y Ff(r)p 1730 1346 1193 4 v 2282 1406 a Fo(a)2330 1420 y Ft(1)p 1741 1446 1171 4 v 1741 1530 a Fq(\()p Fo(a)1824 1544 y Ft(1)1886 1530 y Fq(+)f Fp(\001)c(\001)g(\001)22 b Fq(+)e Fo(a)2242 1544 y Fn(n)2291 1530 y Fq(\)\()p Fo(a)2409 1544 y Ft(2)2471 1530 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2827 1544 y Fn(n)2877 1530 y Fq(\))2948 1467 y Fo(<)25 b Fq(0)3089 1366 y Ff(\021)q(\021)50 1679 y Fm(wher)-5 b(e)34 b Fo(Z)369 1693 y Ft(1)409 1679 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(Z)688 1693 y Fn(n)p Fg(\000)p Ft(1)864 1679 y Fm(ar)-5 b(e)33 b Fo(n)20 b Fp(\000)g Fq(1)33 b Fm(indep)-5 b(endent)34 b(standar)-5 b(d)36 b(Gaussian)d(r)-5 b(andom)35 b(variables.)50 1898 y Fr(Remark:)66 b Fq(The)38 b(sharp)g(conditions)g(on)h(the)g(parameters)g Fo(a)2231 1912 y Fn(i)2300 1898 y Fq(and)g Fo(\013)2544 1912 y Fn(i)2613 1898 y Fq(in)f(this)g(theorem)h(are)h Fm(strictly)50 2043 y(we)-5 b(aker)37 b Fq(than)g(the)g(su\016cien)m(t) f(conditions)f(one)i(obtains)f(e.g.)60 b(from)36 b(Slepian's)f (Gaussian)h(comparison)50 2189 y(lemma)30 b(\(see)h(e.g.)42 b([LLR],)30 b(Theorem)h(4.2.1\).)50 2407 y Fc(Theorem)42 b(1.2:)112 b Fm(L)-5 b(et)37 b Fo(\013)1050 2421 y Fn(i)1113 2407 y Fp(\025)32 b Fq(1)p Fm(,)38 b Fo(i)33 b Fq(=)f(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(k)s Fm(,)48 b Fq(\026)-54 b Fo(\013)33 b Fp(\021)2153 2339 y Ff(Q)2239 2361 y Fn(k)2239 2434 y(i)p Ft(=1)2376 2407 y Fo(\013)2434 2421 y Fn(i)2465 2407 y Fm(.)54 b(L)-5 b(et)37 b Fo(Y)2761 2421 y Fn(\033)2803 2430 y Fj(1)2843 2407 y Fo(;)15 b(Y)2936 2421 y Fn(\033)2978 2430 y Fj(1)3013 2421 y Fn(\033)3055 2430 y Fj(2)3095 2407 y Fo(;)g(:)g(:)g(:)32 b(;)15 b(Y)3365 2421 y Fn(\033)3407 2430 y Fj(1)3443 2421 y Fn(:::)o(\033)3550 2431 y Fi(k)3631 2407 y Fm(b)-5 b(e)50 2552 y Fo(\013)108 2519 y Fn(N)108 2575 y Ft(1)202 2552 y Fq(+)24 b Fp(\001)15 b(\001)g(\001)25 b Fq(+)f(\()p Fo(\013)615 2566 y Ft(1)672 2552 y Fp(\001)15 b(\001)g(\001)h Fo(\013)851 2566 y Fn(k)896 2552 y Fq(\))931 2519 y Fn(N)1038 2552 y Fm(identic)-5 b(al)5 b(ly)40 b(distribute)-5 b(d)40 b(r)-5 b(andom)40 b(variables)g(enumer)-5 b(ate)g(d)40 b(as)f(in)f(the)h(de\014-)50 2698 y(nition)26 b(of)h(the)f(GREM.)g(Assume)g(that)h Fq(1)6 b(+)g Fo(\013)1587 2665 y Fn(N)1587 2720 y Ft(1)1663 2698 y Fq(+)g Fp(\001)15 b(\001)g(\001)7 b Fq(+)f(\()p Fo(\013)2022 2712 y Ft(1)2078 2698 y Fp(\001)15 b(\001)g(\001)i Fo(\013)2258 2712 y Fn(k)r Fg(\000)p Ft(1)2396 2698 y Fq(\))2431 2665 y Fn(N)2526 2698 y Fm(ve)-5 b(ctors)27 b Fq(\()p Fo(Y)2913 2712 y Fn(\033)2955 2721 y Fj(1)2995 2698 y Fq(\))3030 2721 y Fn(\033)3072 2730 y Fj(1)3108 2721 y Fg(2f\000)p Ft(1)p Fn(;)p Ft(1)p Fg(g)3382 2700 y Fj(ln)11 b Fi(\013)3488 2714 y Fj(1)3523 2700 y Fi(=)h Fj(ln)17 b(\026)-38 b Fi(\013)3682 2698 y Fm(,)50 2843 y Fq(\()p Fo(Y)138 2857 y Fn(\033)180 2866 y Fj(1)216 2857 y Fn(\033)258 2866 y Fj(2)298 2843 y Fq(\))333 2866 y Fn(\033)375 2875 y Fj(2)411 2866 y Fg(2f\000)p Ft(1)p Fn(;)p Ft(1)p Fg(g)685 2846 y Fj(ln)11 b Fi(\013)791 2860 y Fj(2)826 2846 y Fi(=)g Fj(ln)18 b(\026)-38 b Fi(\013)1008 2843 y Fp(8)p Fo(\033)1111 2857 y Ft(1)1176 2843 y Fp(2)25 b(f\000)p Fq(1)p Fo(;)15 b Fq(1)p Fp(g)1553 2810 y Ft(ln)g Fn(\013)1675 2819 y Fj(1)1711 2810 y Fn(=)d Ft(ln)21 b(\026)-43 b Fn(\013)1885 2843 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b Fq(\()p Fo(Y)2190 2857 y Fn(\033)2232 2866 y Fj(1)2268 2857 y Fn(\033)2310 2866 y Fj(2)2346 2857 y Fn(:::)o(\033)2453 2867 y Fi(k)2497 2843 y Fq(\))2532 2869 y Fn(\033)2574 2879 y Fi(k)2614 2869 y Fg(2f\000)p Ft(1)p Fn(;)p Ft(1)p Fg(g)2888 2846 y Fj(ln)c Fi(\013)2994 2862 y(k)3033 2846 y(=)h Fj(ln)18 b(\026)-38 b Fi(\013)3215 2843 y Fp(8)23 b Fo(\033)3341 2857 y Ft(1)3397 2843 y Fo(:)15 b(:)g(:)h(\033)3570 2857 y Fn(k)r Fg(\000)p Ft(1)3733 2843 y Fp(2)50 2989 y(f\000)p Fq(1)p Fo(;)f Fq(1)p Fp(g)341 2956 y Ft(ln)r(\()p Fn(\013)478 2965 y Fj(1)515 2956 y Fg(\001\001\001)n Fn(\013)626 2966 y Fi(k)q Fd(\000)p Fj(1)747 2956 y Ft(\))p Fn(=)e Ft(ln)20 b(\026)-43 b Fn(\013)976 2989 y Fm(ar)-5 b(e)28 b(indep)-5 b(endent.)42 b(L)-5 b(et)27 b(also)h Fo(v)2040 3003 y Fn(N)s(;)p Ft(1)2163 2989 y Fq(\()p Fo(x)p Fq(\))p Fo(;)15 b(:)g(:)g(:)33 b(;)15 b(v)2547 3003 y Fn(N)s(;k)2674 2989 y Fq(\()p Fo(x)p Fq(\))27 b Fm(b)-5 b(e)27 b(functions)h(on)f Fe(R)36 b Fm(such)50 3134 y(that)e(the)f(fol)5 b(lowing)33 b(p)-5 b(oint)34 b(pr)-5 b(o)g(c)g(esses)1118 3217 y Ff(X)1145 3408 y Fn(\033)1187 3417 y Fj(1)1264 3303 y Fo(\016)1304 3320 y Fn(v)1340 3329 y Fi(N)r(;)p Fj(1)1448 3320 y Ft(\()p Fn(Y)1519 3329 y Fi(\033)1556 3343 y Fj(1)1596 3320 y Ft(\))1654 3303 y Fp(!)26 b(P)1834 3317 y Ft(1)1118 3471 y Ff(X)1145 3662 y Fn(\033)1187 3671 y Fj(2)1264 3558 y Fo(\016)1304 3574 y Fn(v)1340 3583 y Fi(N)r(;)p Fj(2)1448 3574 y Ft(\()p Fn(Y)1519 3583 y Fi(\033)1556 3597 y Fj(1)1592 3583 y Fi(\033)1629 3597 y Fj(2)1669 3574 y Ft(\))1727 3558 y Fp(!)f(P)1906 3572 y Ft(2)2040 3558 y Fp(8)p Fo(\033)2143 3572 y Ft(1)1118 3794 y Fp(\001)15 b(\001)g(\001)1118 3881 y Ff(X)1143 4072 y Fn(\033)1185 4082 y Fi(k)1264 3967 y Fo(\016)1304 3983 y Fn(v)1340 3993 y Fi(N)r(;k)1452 3983 y Ft(\()p Fn(Y)1523 3992 y Fi(\033)1560 4006 y Fj(1)1596 3992 y Fi(\033)1633 4006 y Fj(2)1669 3992 y Fi(:::)o(\033)1768 4008 y(k)1812 3983 y Ft(\))1871 3967 y Fp(!)25 b(P)2050 3981 y Fn(k)2188 3967 y Fp(8)p Fo(\033)2291 3981 y Ft(1)2346 3967 y Fo(:)15 b(:)g(:)h(\033)2519 3981 y Fn(k)r Fg(\000)p Ft(1)3523 3672 y Fq(\(1)p Fo(:)p Fq(7\))50 4210 y Fm(c)-5 b(onver)g(ge)47 b(we)-5 b(akly)48 b(to)g(the)f(Poisson)g(p)-5 b(oint)49 b(pr)-5 b(o)g(c)g(esses)49 b Fp(P)2084 4224 y Ft(1)2125 4210 y Fo(;)15 b(:)g(:)g(:)31 b(;)15 b Fp(P)2404 4224 y Fn(k)2497 4210 y Fm(on)47 b Fe(R)55 b Fm(with)48 b(intensity)f(me)-5 b(asur)g(es)50 4355 y Fo(K)127 4369 y Ft(1)168 4355 y Fo(e)210 4322 y Fg(\000)p Fn(x)313 4355 y Fo(dx;)15 b(:)g(:)g(:)32 b(;)15 b(K)706 4369 y Fn(k)751 4355 y Fo(e)793 4322 y Fg(\000)p Fn(x)896 4355 y Fo(dx)44 b Fm(with)g(some)g(c)-5 b(onstants)46 b Fo(K)1978 4369 y Ft(1)2019 4355 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(K)2313 4369 y Fn(k)2402 4355 y Fm(r)-5 b(esp)g(e)g(ctively.)76 b(Then)44 b(the)g(fol)5 b(lowing)50 4500 y(p)-5 b(oint)34 b(pr)-5 b(o)g(c)g(esses)35 b(on)e Fe(R)856 4467 y Fn(k)589 4693 y Fp(P)659 4646 y Ft(\()p Fn(k)r Ft(\))652 4720 y Fn(N)787 4693 y Fp(\021)883 4607 y Ff(X)909 4798 y Fn(\033)951 4807 y Fj(1)1029 4693 y Fo(\016)1069 4709 y Fn(v)1105 4718 y Fi(N)r(;)p Fj(1)1213 4709 y Ft(\()p Fn(Y)1284 4718 y Fi(\033)1321 4732 y Fj(1)1361 4709 y Ft(\))1409 4607 y Ff(X)1436 4798 y Fn(\033)1478 4807 y Fj(2)1556 4693 y Fo(\016)1596 4709 y Fn(v)1632 4718 y Fi(N)r(;)p Fj(2)1739 4709 y Ft(\()p Fn(Y)1810 4718 y Fi(\033)1847 4732 y Fj(1)1883 4718 y Fi(\033)1920 4732 y Fj(2)1960 4709 y Ft(\))2008 4693 y Fp(\001)15 b(\001)g(\001)2129 4607 y Ff(X)2154 4798 y Fn(\033)2196 4808 y Fi(k)2276 4693 y Fo(\016)2316 4709 y Fn(v)2352 4719 y Fi(N)r(;k)2464 4709 y Ft(\()p Fn(Y)2535 4718 y Fi(\033)2572 4732 y Fj(1)2608 4718 y Fi(\033)2645 4732 y Fj(2)2680 4718 y Fi(:::\033)2780 4734 y(k)2824 4709 y Ft(\))2882 4693 y Fp(!)25 b(P)3068 4655 y Ft(\()p Fn(k)r Ft(\))50 4953 y Fm(c)-5 b(onver)g(ge)25 b(we)-5 b(akly)26 b(to)f(p)-5 b(oint)26 b(pr)-5 b(o)g(c)g(ess)27 b Fp(P)1381 4920 y Ft(\()p Fn(k)r Ft(\))1507 4953 y Fm(on)e Fe(R)1689 4920 y Fn(k)1740 4953 y Fm(,)h(which)f(is)g(char)-5 b(acterise)g(d)27 b(by)e(the)f(fol)5 b(lowing)26 b(gener)-5 b(ating)50 5099 y(functions:)268 5294 y Fo(F)326 5308 y Ft(\001)386 5317 y Fj(1)422 5308 y Fg(\002\001\001\001)n(\002)p Ft(\001)660 5318 y Fi(k)704 5294 y Fq(\()p Fo(z)t Fq(\))26 b Fp(\021)f Fe(E)11 b Fo(z)1043 5172 y Ff(P)1145 5268 y Fi(x)1181 5282 y Fj(1)1234 5234 y Ft(1)-23 b(I)1273 5246 y Fd(f)p Fi(x)1342 5260 y Fj(1)1378 5246 y Fd(2)p Fj(\001)1471 5260 y(1)1506 5246 y Fd(g)1544 5234 y Fg(\001\001\001)1621 5172 y Ff(P)1717 5268 y Fi(x)1753 5284 y(k)1810 5234 y Ft(1)g(I)1849 5246 y Fd(f)p Fi(x)1918 5262 y(k)1958 5246 y Fd(2)p Fj(\001)2051 5262 y Fi(k)2091 5246 y Fd(g)846 5466 y Fq(=)25 b Fo(f)987 5480 y Ft(1)p Fn(;)p Ft(\001)1105 5489 y Fj(1)1144 5466 y Fq(\()p Fo(f)1224 5480 y Ft(2)p Fn(;)p Ft(\001)1342 5489 y Fj(2)1382 5466 y Fq(\()p Fo(f)1462 5480 y Ft(3)p Fn(;)p Ft(\001)1580 5489 y Fj(3)1634 5466 y Fp(\001)15 b(\001)g(\001)i Fq(\()p Fo(f)1836 5480 y Fn(k)r Fg(\000)p Ft(1)p Fn(;)p Ft(\001)2052 5490 y Fi(k)q Fd(\000)p Fj(1)2175 5466 y Fq(\()p Fo(f)2255 5480 y Fn(k)r(;)p Ft(\001)2378 5490 y Fi(k)2420 5466 y Fq(\()p Fo(z)t Fq(\)\)\))e Fp(\001)g(\001)g(\001)j Fq(\)\))p Fo(;)202 b Fp(j)p Fo(z)t Fp(j)26 b Fo(<)f Fq(1)3523 5358 y(\(1)p Fo(:)p Fq(8\))p eop %%Page: 6 6 6 5 bop 50 295 a Fk(6)3309 b Fh(Se)l(ction)27 b Fk(1)50 514 y Fm(wher)-5 b(e)34 b Fo(f)352 528 y Fn(i;)p Ft(\001)460 537 y Fi(i)491 514 y Fq(\()p Fo(z)t Fq(\))27 b(=)e Fo(e)772 481 y Fn(K)833 490 y Fi(i)862 481 y Ft(\()p Fn(z)s Fg(\000)p Ft(1\)\()p Fn(e)1112 453 y Fd(\000)p Fi(a)1196 468 y(i)1230 481 y Fg(\000)p Fn(e)1322 453 y Fd(\000)p Fi(b)1400 468 y(i)1432 481 y Ft(\))1465 514 y Fm(,)33 b Fq(\001)1602 528 y Fn(i)1657 514 y Fq(=)25 b(\()p Fo(a)1836 528 y Fn(i)1867 514 y Fo(;)15 b(b)1946 528 y Fn(i)1976 514 y Fq(])33 b Fm(with)h Fo(a)2280 528 y Fn(i)2310 514 y Fo(;)15 b(b)2389 528 y Fn(i)2445 514 y Fp(2)25 b Fe(R)41 b Fm(or)33 b Fo(b)2785 528 y Fn(i)2841 514 y Fq(=)25 b Fp(1)p Fm(,)32 b Fo(i)26 b Fq(=)f(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(k)s Fm(.)50 732 y(Mor)-5 b(e)g(over,)32 b(the)f(fol)5 b(lowing)31 b(indep)-5 b(endenc)g(e)32 b(pr)-5 b(op)g(erties)33 b(of)d(the)h(c)-5 b(ounting)31 b(r)-5 b(andom)33 b(variables)e(of)g(the)g(pr)-5 b(o-)50 877 y(c)g(ess)46 b Fp(P)319 844 y Ft(\()p Fn(k)r Ft(\))421 877 y Fm(,)499 809 y Ff(P)594 905 y Fn(x)635 914 y Fj(1)691 877 y Fq(1)-23 b(I)746 907 y Fg(f)p Fn(x)824 916 y Fj(1)860 907 y Fg(2)p Ft(\001)969 876 y Fi(j)969 929 y Fj(1)1004 907 y Fg(g)1061 877 y Fp(\001)15 b(\001)g(\001)1182 809 y Ff(P)1278 905 y Fn(x)1319 915 y Fi(k)1379 877 y Fq(1)-23 b(I)1434 907 y Fg(f)p Fn(x)1512 917 y Fi(k)1552 907 y Fg(2)p Ft(\001)1661 876 y Fi(j)1661 931 y(k)1701 907 y Fg(g)1742 877 y Fm(,)50 b(c)-5 b(orr)g(esp)g(onding)49 b(to)d(the)h(intervals)g Fq(\001)3154 834 y Fn(j)3154 902 y Ft(1)3224 877 y Fp(\002)30 b(\001)15 b(\001)g(\001)31 b(\002)e Fq(\001)3637 834 y Fn(j)3637 905 y(k)3682 877 y Fm(,)50 1026 y Fq(\001)126 983 y Fn(j)126 1052 y(i)189 1026 y Fq(=)c([)p Fo(a)358 983 y Fn(j)358 1052 y(i)397 1026 y Fo(;)15 b(b)476 983 y Fn(j)476 1052 y(i)514 1026 y Fq(\))p Fm(,)33 b Fo(j)e Fq(=)25 b(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(r)s Fm(,)32 b Fo(r)c(>)d Fq(1)p Fm(,)33 b Fo(i)26 b Fq(=)f(1)p Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(k)36 b Fm(hold)e(true:)50 1244 y(\(i\))29 b(If)g(the)h (\014rst)g(c)-5 b(omp)g(onents)31 b(of)f(these)f(intervals)h(ar)-5 b(e)31 b(disjoint,)f(i.e.)f Fo(a)2497 1211 y Ft(1)2497 1267 y(1)2563 1244 y Fp(\024)c Fo(b)2698 1211 y Ft(1)2698 1267 y(1)2764 1244 y Fp(\024)g Fo(a)2908 1211 y Ft(2)2908 1267 y(1)2974 1244 y Fp(\024)g Fo(b)3109 1211 y Ft(2)3109 1267 y(1)3175 1244 y Fp(\024)g(\001)15 b(\001)g(\001)h Fo(a)3440 1211 y Fn(r)3440 1267 y Ft(1)3506 1244 y Fp(\024)25 b Fo(b)3641 1211 y Fn(r)3641 1267 y Ft(1)3682 1244 y Fm(,)50 1390 y(then)33 b(these)g(r.v.)42 b(ar)-5 b(e)33 b(indep)-5 b(endent.)50 1608 y(\(ii\))25 b(If)f(the)i(\014rst)f Fo(l)5 b Fp(\000)s Fq(1)26 b Fm(c)-5 b(omp)g(onents)28 b(of)d(these)g(intervals)h(c)-5 b(oincide)25 b(and)h(the)g Fo(l)r(th)f Fm(c)-5 b(omp)g(onents)27 b(ar)-5 b(e)26 b(disjoint,)50 1754 y(i.e.)58 b Fq(\001)310 1721 y Ft(1)310 1777 y Fn(i)386 1754 y Fq(=)36 b Fp(\001)15 b(\001)g(\001)37 b Fq(=)e(\001)817 1721 y Fn(r)817 1777 y(i)895 1754 y Fm(for)k Fo(i)d Fq(=)f(1)p Fo(;)15 b(:)g(:)g(:)33 b(;)15 b(l)26 b Fp(\000)e Fq(1)39 b Fm(and)g Fo(a)1944 1721 y Ft(1)1944 1779 y Fn(l)2021 1754 y Fp(\024)c Fo(b)2166 1721 y Ft(1)2166 1779 y Fn(l)2243 1754 y Fp(\024)g Fo(a)2397 1721 y Ft(2)2397 1779 y Fn(l)2473 1754 y Fp(\024)h Fo(b)2619 1721 y Ft(2)2619 1779 y Fn(l)2695 1754 y Fp(\024)g(\001)15 b(\001)g(\001)h Fo(a)2971 1721 y Fn(k)2971 1779 y(l)3051 1754 y Fp(\024)36 b Fo(b)3197 1721 y Fn(r)3197 1779 y(l)3237 1754 y Fm(,)j(then)g(these)50 1899 y(r.v.)33 b(ar)-5 b(e)35 b(c)-5 b(onditional)5 b(ly)36 b(indep)-5 b(endent)35 b(under)f(c)-5 b(ondition)36 b(that)2266 1831 y Ff(P)2361 1926 y Fn(x)2402 1935 y Fj(1)2458 1899 y Fq(1)-23 b(I)2513 1915 y Fg(f)p Fn(x)2591 1924 y Fj(1)2627 1915 y Fg(2)p Ft(\001)2736 1924 y Fj(1)2771 1915 y Fg(g)2828 1899 y Fp(\001)15 b(\001)g(\001)2949 1831 y Ff(P)3045 1926 y Fn(x)3086 1936 y Fi(l)p Fd(\000)p Fj(1)3213 1899 y Fq(1)-23 b(I)3268 1915 y Fg(f)p Fn(x)3346 1925 y Fi(l)p Fd(\000)p Fj(1)3453 1915 y Fg(2)p Ft(\001)3562 1925 y Fi(l)p Fd(\000)p Fj(1)3668 1915 y Fg(g)50 2044 y Fm(is)32 b(\014xe)-5 b(d.)50 2263 y Fr(Remark:)57 b Fq(This)33 b(theorem)i(is)f(a)i (generalisation)e(of)h(Theorem)g(3)g(in)f([GMP]:)i(w)m(e)g(do)f(not)g (sp)s(ecify)f(the)50 2408 y(distribution)h(of)k(the)g(r.v.)g Fo(Y)1057 2422 y Fn(\033)1099 2431 y Fj(1)1135 2422 y Fn(:::)n(\033)1241 2431 y Fi(i)1314 2408 y Fq(in)e(the)i(assumptions,)h (and)e(do)h(not)g(imp)s(ose)f(their)g(indep)s(endence)50 2554 y(\(but)30 b(only)f(of)i(their)e(v)m(ectors\).)50 2772 y Fr(Remark:)43 b Fq(The)25 b(pro)s(cess)g Fp(P)1049 2739 y Ft(\()p Fn(k)r Ft(\))1176 2772 y Fq(is)g(called)f(a)i Fm(Poisson)j(c)-5 b(asc)g(ade)27 b Fq(with)d Fo(k)k Fq(lev)m(els.)39 b(It)25 b(is)g(b)s(est)g(understo)s(o)s(d)e(in)50 2917 y(terms)28 b(of)h(the)g(follo)m(wing)e(iterativ)m(e)i(construction.)39 b(If)28 b Fo(k)h Fq(=)c(1,)k(it)f(is)g(just)g(a)h(P)m(oisson)f(p)s(oin) m(t)g(pro)s(cess)g(on)g Fe(R)50 3063 y Fq(with)21 b(in)m(tensit)m(y)g (measure)h Fo(K)1035 3077 y Ft(1)1076 3063 y Fo(e)1118 3030 y Fg(\000)p Fn(x)1221 3063 y Fo(dx)p Fq(.)38 b(T)-8 b(o)23 b(construct)g Fp(P)1971 3030 y Ft(\(2\))2091 3063 y Fq(on)f Fe(R)2269 3030 y Ft(2)2316 3063 y Fq(,)i(w)m(e)e(place)g(the) h(pro)s(cess)f Fp(P)3242 3030 y Ft(\(1\))3362 3063 y Fq(for)g Fo(k)28 b Fq(=)d(1)50 3208 y(on)j(the)h(axis)f(of)g(the)h (\014rst)e(co)s(ordinate)i(and)e(through)h(eac)m(h)i(of)e(its)g(p)s (oin)m(ts)f(dra)m(w)h(a)h(straigh)m(t)g(line)e(parallel)50 3354 y(to)38 b(the)f(axis)g(of)h(the)f(second)h(co)s(ordinate.)61 b(Then)36 b(w)m(e)i(put)f(on)g(eac)m(h)h(of)g(these)g(lines)d(indep)s (enden)m(tly)f(a)50 3499 y(P)m(oisson)27 b(p)s(oin)m(t)f(pro)s(cess)g (with)g(in)m(tensit)m(y)g(measure)h Fo(K)1922 3513 y Ft(2)1963 3499 y Fo(e)2005 3466 y Fg(\000)p Fn(x)2108 3499 y Fo(dx)p Fq(.)40 b(These)27 b(p)s(oin)m(ts)f(on)h Fe(R)2985 3466 y Ft(2)3058 3499 y Fq(form)g(the)g(pro)s(cess)50 3644 y Fp(P)120 3611 y Ft(\(2\))248 3644 y Fq(This)i(pro)s(cedure)g(is) h(no)m(w)g(simply)e(iterated)j Fo(k)i Fq(times.)159 3863 y(Theorems)c(1.1)i(and)f(1.2)h(com)m(bined)e(giv)m(e)h(a)h(\014rst)e (imp)s(ortan)m(t)g(result)g(that)h(establish)e(what)i(di\013eren)m(t)50 4008 y(p)s(oin)m(t)f(pro)s(cesses)h(ma)m(y)h(b)s(e)f(constructed)h(in)e (the)h(GREM.)50 4226 y Fc(Theorem)44 b(1.3:)64 b Fm(L)-5 b(et)38 b Fo(\013)1005 4240 y Fn(i)1070 4226 y Fp(\025)c Fq(1)p Fm(,)39 b Fq(0)c Fo(<)f(a)1520 4240 y Fn(i)1584 4226 y Fo(<)g Fq(1)p Fm(,)39 b Fo(i)c Fq(=)f(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(n)p Fm(,)2441 4158 y Ff(Q)2527 4181 y Fn(n)2527 4254 y(i)p Ft(=1)2665 4226 y Fo(\013)2723 4240 y Fn(i)2787 4226 y Fq(=)34 b(2)p Fm(,)3004 4158 y Ff(P)3100 4181 y Fn(n)3100 4254 y(i)p Ft(=1)3238 4226 y Fo(a)3286 4240 y Fn(i)3350 4226 y Fq(=)g(1)p Fm(.)57 b(L)-5 b(et)50 4372 y Fo(J)100 4386 y Ft(1)141 4372 y Fo(;)15 b(J)231 4386 y Ft(2)273 4372 y Fo(;)g(:)g(:)g(:)31 b(;)15 b(J)539 4386 y Fn(m)640 4372 y Fp(2)31 b Fe(N)49 b Fm(b)-5 b(e)35 b(the)i(indic)-5 b(es)36 b(such)g(that)h Fq(0)32 b(=)e Fo(J)2035 4386 y Ft(0)2108 4372 y Fo(<)h(J)2260 4386 y Ft(1)2332 4372 y Fo(<)g(J)2484 4386 y Ft(2)2556 4372 y Fo(<)g Fp(\001)15 b(\001)g(\001)32 b Fo(<)f(J)2947 4386 y Fn(m)3048 4372 y Fq(=)g Fo(n)p Fm(.)51 b(We)35 b(denote)50 4517 y(by)h Fq(\026)-46 b Fo(a)219 4531 y Fn(l)276 4517 y Fp(\021)377 4449 y Ff(P)473 4471 y Fn(J)513 4481 y Fi(l)473 4544 y Fn(i)p Ft(=)p Fn(J)595 4554 y Fi(l)p Fd(\000)p Fj(1)701 4544 y Ft(+1)813 4517 y Fo(a)861 4531 y Fn(i)891 4517 y Fm(,)44 b Fq(\026)-54 b Fo(\013)1012 4531 y Fn(l)1070 4517 y Fp(\021)1170 4449 y Ff(Q)1256 4471 y Fn(J)1296 4481 y Fi(l)1256 4544 y Fn(i)p Ft(=)p Fn(J)1378 4554 y Fi(l)p Fd(\000)p Fj(1)1484 4544 y Ft(+1)1596 4517 y Fo(\013)1654 4531 y Fn(i)1685 4517 y Fm(,)35 b Fo(l)d Fq(=)d(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(m)p Fm(,)36 b(and)g(intr)-5 b(o)g(duc)g(e)36 b(standar)-5 b(d)38 b(Gaussian)50 4663 y(r)-5 b(andom)35 b(variables)154 4957 y Fq(\026)128 4980 y Fo(X)210 4916 y Fn(\033)252 4925 y Fj(1)288 4916 y Fn(:::)n(\033)394 4925 y Fi(J)429 4941 y(l)p Fd(\000)p Fj(1)203 4993 y Fn(\033)245 5002 y Fi(J)280 5018 y(l)p Fd(\000)p Fj(1)387 5002 y(+1)469 4993 y Fn(\033)511 5002 y Fi(J)546 5018 y(l)p Fd(\000)p Fj(1)653 5002 y(+2)735 4993 y Fg(\001\001\001)o Fn(\033)842 5002 y Fi(J)877 5018 y(l)938 4980 y Fp(\021)1034 4879 y Ff(\020)1088 4925 y Fp(p)p 1164 4925 292 4 v 55 x Fo(a)1212 4994 y Fn(J)1252 5004 y Fi(l)p Fd(\000)p Fj(1)1359 4994 y Ft(+1)1455 4980 y Fo(X)1530 4994 y Fn(\033)1572 5003 y Fi(J)1607 5017 y Fj(1)1648 4994 y Fn(:::)o(\033)1755 5003 y Fi(J)1790 5019 y(l)p Fd(\000)p Fj(1)1901 4994 y Fn(\033)1943 5003 y Fi(J)1978 5019 y(l)p Fd(\000)p Fj(1)2085 5003 y(+1)2192 4980 y Fq(+)2283 4925 y Fp(p)p 2358 4925 V 2358 4980 a Fo(a)2406 4994 y Fn(J)2446 5004 y Fi(l)p Fd(\000)p Fj(1)2553 4994 y Ft(+2)2650 4980 y Fo(X)2725 4994 y Fn(\033)2767 5003 y Fi(J)2802 5017 y Fj(1)2842 4994 y Fn(:::)o(\033)2949 5003 y Fi(J)2984 5019 y(l)p Fd(\000)p Fj(1)3095 4994 y Fn(\033)3137 5003 y Fi(J)3172 5019 y(l)p Fd(\000)p Fj(1)3279 5003 y(+1)3361 4994 y Fn(\033)3403 5003 y Fi(J)3438 5019 y(l)p Fd(\000)p Fj(1)3545 5003 y(+2)933 5180 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)1241 5122 y Fp(p)p 1316 5122 120 4 v 1316 5180 a Fo(a)1364 5194 y Fn(J)1404 5204 y Fi(l)1436 5180 y Fo(X)1511 5194 y Fn(\033)1553 5203 y Fi(J)1588 5217 y Fj(1)1628 5194 y Fn(:::)o(\033)1735 5203 y Fi(J)1770 5219 y(l)p Fd(\000)p Fj(1)1881 5194 y Fn(\033)1923 5203 y Fi(J)1958 5219 y(l)p Fd(\000)p Fj(1)2065 5203 y(+1)2147 5194 y Fn(:::)o(\033)2254 5203 y Fi(J)2289 5219 y(l)2325 5079 y Ff(\021)2379 5180 y Fo(=)2424 5111 y Fp(p)p 2500 5111 76 4 v 2501 5180 a Fq(\026)-46 b Fo(a)2548 5194 y Fn(l)2576 5180 y Fo(:)3523 5316 y Fq(\(1)p Fo(:)p Fq(9\))50 5462 y Fm(Assume)23 b(that)h(the)g(p)-5 b(artition)26 b Fo(J)1112 5476 y Ft(1)1153 5462 y Fo(;)15 b(J)1243 5476 y Ft(2)1285 5462 y Fo(;)g(:)g(:)g(:)31 b(;)15 b(J)1551 5476 y Fn(m)1644 5462 y Fm(satis\014es)25 b(the)e(fol)5 b(lowing)25 b(c)-5 b(ondition:)39 b(for)23 b(al)5 b(l)24 b Fo(l)j Fq(=)e(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(m)p eop %%Page: 7 7 7 6 bop 50 295 a Fh(GREM,)28 b(p)l(art)f(1)3127 b Fk(7)50 513 y Fm(and)33 b(al)5 b(l)33 b Fo(k)j Fm(such)d(that)h Fo(J)882 527 y Fn(l)p Fg(\000)p Ft(1)1023 513 y Fq(+)20 b(2)26 b Fp(\024)f Fo(k)j Fp(\024)d Fo(J)1502 527 y Fn(l)566 774 y Fq(\()p Fo(a)649 788 y Fn(k)714 774 y Fq(+)20 b Fo(a)853 788 y Fn(k)r Ft(+1)1005 774 y Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1270 788 y Fn(J)1310 798 y Fi(l)1337 788 y Fg(\000)p Ft(1)1455 774 y Fq(+)g Fo(a)1594 788 y Fn(J)1634 798 y Fi(l)1665 774 y Fq(\))p Fo(=)q Fq(\026)-46 b Fo(a)1793 788 y Fn(l)1847 774 y Fp(\025)25 b Fq(ln)o(\()p Fo(\013)2112 788 y Fn(k)2157 774 y Fo(\013)2215 788 y Fn(k)r Ft(+1)2367 774 y Fp(\001)15 b(\001)g(\001)h Fo(\013)2546 788 y Fn(J)2586 798 y Fi(l)2613 788 y Fg(\000)p Ft(1)2711 774 y Fo(\013)2769 788 y Fn(J)2809 798 y Fi(l)2840 774 y Fq(\))p Fo(=)f Fq(ln\()9 b(\026)-54 b Fo(\013)3105 788 y Fn(l)3133 774 y Fq(\))p Fo(:)284 b Fq(\(1)p Fo(:)p Fq(10\))50 1035 y Fm(Then)33 b(the)g(p)-5 b(oint)34 b(pr)-5 b(o)g(c)g(ess)65 1273 y Fp(P)135 1226 y Ft(\()p Fn(m)p Ft(\))128 1300 y Fn(N)90 1446 y Fp(\021)250 1360 y Ff(X)186 1550 y Fn(\033)228 1559 y Fj(1)264 1550 y Fn(:::)o(\033)371 1559 y Fi(J)406 1573 y Fj(1)461 1446 y Fo(\016)501 1474 y Fn(u)544 1445 y Fd(\000)p Fj(1)544 1499 y(ln)18 b(\026)-38 b Fi(\013)650 1513 y Fj(1)685 1499 y Fi(;N)764 1474 y Ft(\()813 1458 y(\026)792 1474 y Fn(X)852 1483 y Fi(\033)889 1497 y Fj(1)925 1483 y Fi(:::)o(\033)1024 1499 y(J)1059 1513 y Fj(1)1104 1474 y Ft(\))1275 1360 y Ff(X)1152 1550 y Fn(\033)1194 1559 y Fi(J)1229 1573 y Fj(1)1265 1559 y(+1)1347 1550 y Fn(:::)o(\033)1454 1559 y Fi(J)1489 1573 y Fj(2)1544 1446 y Fo(\016)1584 1485 y Fn(u)1627 1456 y Fd(\000)p Fj(1)1627 1509 y(ln)18 b(\026)-38 b Fi(\013)1733 1523 y Fj(2)1768 1509 y Fi(;N)1847 1485 y Ft(\()1896 1469 y(\026)1875 1485 y Fn(X)1940 1434 y Fi(\033)1977 1448 y Fj(1)2013 1434 y Fi(:::)o(\033)2112 1450 y(J)2147 1464 y Fj(1)1935 1497 y Fi(\033)1972 1513 y(J)2007 1527 y Fj(1)2043 1513 y(+1)2125 1497 y Fi(:::\033)2225 1513 y(J)2260 1527 y Fj(2)2305 1485 y Ft(\))2353 1446 y Fp(\001)15 b(\001)g(\001)2601 1360 y Ff(X)2413 1550 y Fn(\033)2455 1559 y Fi(J)2490 1574 y(m)p Fd(\000)p Fj(1)2631 1559 y(+1)2713 1550 y Fn(:::)o(\033)2820 1559 y Fi(J)2855 1568 y(m)2935 1446 y Fo(\016)2975 1494 y Fn(u)3018 1464 y Fd(\000)p Fj(1)3018 1518 y(ln)j(\026)-38 b Fi(\013)3124 1527 y(m)3184 1518 y(;N)3263 1494 y Ft(\()3312 1478 y(\026)3291 1494 y Fn(X)3356 1434 y Fi(\033)3393 1448 y Fj(1)3429 1434 y Fi(:::)o(\033)3528 1450 y(J)3563 1465 y(m)p Fd(\000)p Fj(1)3351 1506 y Fi(\033)3388 1521 y(J)3423 1536 y(m)p Fd(\000)p Fj(1)3564 1521 y(+1)3646 1506 y Fi(:::\033)3746 1521 y(J)3781 1530 y(m)3850 1494 y Ft(\))50 1797 y Fm(c)-5 b(onver)g(ges)24 b(we)-5 b(akly)24 b(to)g(the)g(pr)-5 b(o)g(c)g(ess)26 b Fp(P)1333 1764 y Ft(\()p Fn(m)p Ft(\))1482 1797 y Fm(on)e Fe(R)1663 1764 y Fn(m)1761 1797 y Fm(de\014ne)-5 b(d)24 b(in)g(The)-5 b(or)g(em)25 b(1.2)f(with)g(c)-5 b(onstants)25 b Fo(K)3328 1811 y Ft(1)3369 1797 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(K)3663 1811 y Fn(m)3733 1797 y Fm(.)50 1942 y(The)33 b(c)-5 b(onstant)1744 2094 y Fo(K)1821 2108 y Fn(l)1874 2094 y Fq(=)25 b(1)1462 b(\(1)p Fo(:)p Fq(11\))50 2310 y Fm(if)32 b(al)5 b(l)33 b Fo(J)318 2324 y Fn(l)367 2310 y Fp(\000)20 b Fo(J)508 2324 y Fn(l)p Fg(\000)p Ft(1)649 2310 y Fp(\000)g Fq(1)33 b Fm(ine)-5 b(qualities)33 b(in)g(\(1.10\))h(for)f Fo(k)28 b Fq(=)d Fo(J)2044 2324 y Fn(l)p Fg(\000)p Ft(1)2186 2310 y Fq(+)20 b(2)p Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(J)2589 2324 y Fn(l)2650 2310 y Fm(ar)-5 b(e)34 b(strict.)42 b(Otherwise)111 2587 y Fo(K)188 2601 y Fn(l)241 2587 y Fq(=)25 b Fe(P)392 2486 y Ff(\020)969 2500 y(\\)775 2685 y Fi(k)q Fj(:)p Fi(J)863 2701 y(l)p Fd(\000)p Fj(1)970 2685 y(+1)p Fd(\024)p Fi(k)q Fd(\024)p Fi(J)1216 2701 y(l)1243 2685 y(;)473 2753 y Fj(\()p Fi(a)533 2769 y(k)573 2753 y Fj(+)p Fd(\001\001\001)o Fj(+)p Fi(a)764 2769 y(J)799 2784 y(l)830 2753 y Fj(\))p Fi(=)r Fj(\026)-33 b Fi(a)923 2769 y(J)958 2784 y(l)989 2753 y Fj(=ln)o(\()p Fi(\013)1155 2769 y(k)1194 2753 y Fd(\001\001\001)p Fi(\013)1299 2769 y(J)1334 2784 y(l)1364 2753 y Fj(\))p Fi(=)12 b Fj(ln)17 b(\026)-38 b Fi(\013)1539 2769 y(l)1500 2486 y Ff(\020)1554 2587 y Fo(Z)1616 2601 y Fn(k)r Fg(\000)p Ft(1)1754 2465 y Ff(r)p 1845 2465 1337 4 v 2421 2525 a Fo(a)2469 2539 y Fn(k)r Fg(\000)p Ft(1)p 1856 2566 1315 4 v 1856 2649 a Fq(\()p Fo(a)1939 2663 y Fn(k)r Fg(\000)p Ft(1)2098 2649 y Fq(+)19 b Fp(\001)c(\001)g (\001)22 b Fq(+)e Fo(a)2454 2663 y Fn(J)2494 2673 y Fi(l)2525 2649 y Fq(\)\()p Fo(a)2643 2663 y Fn(k)2709 2649 y Fq(+)f Fp(\001)c(\001)g(\001)22 b Fq(+)e Fo(a)3065 2663 y Fn(J)3105 2673 y Fi(l)3136 2649 y Fq(\))1013 2959 y(+)g Fo(Z)1166 2973 y Fn(k)r Fg(\000)p Ft(2)1304 2837 y Ff(r)p 1394 2837 1430 4 v 2016 2897 a Fo(a)2064 2911 y Fn(k)r Fg(\000)p Ft(2)p 1405 2938 1409 4 v 1405 3021 a Fq(\()p Fo(a)1488 3035 y Fn(k)r Fg(\000)p Ft(2)1647 3021 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2003 3035 y Fn(J)2043 3045 y Fi(l)2074 3021 y Fq(\)\()p Fo(a)2192 3035 y Fn(k)r Fg(\000)p Ft(1)2351 3021 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2707 3035 y Fn(J)2747 3045 y Fi(l)2778 3021 y Fq(\))1013 3165 y(+)g Fp(\001)15 b(\001)g(\001)1013 3361 y Fq(+)20 b Fo(Z)1166 3375 y Fn(J)1206 3385 y Fi(l)p Fd(\000)p Fj(1)1313 3375 y Ft(+1)1409 3241 y Ff(r)p 1500 3241 1642 4 v 2175 3295 a Fo(a)2223 3309 y Fn(J)2263 3319 y Fi(l)p Fd(\000)p Fj(1)2370 3309 y Ft(+1)p 1511 3340 1620 4 v 1511 3423 a Fq(\()p Fo(a)1594 3437 y Fn(J)1634 3447 y Fi(l)p Fd(\000)p Fj(1)1741 3437 y Ft(+1)1858 3423 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2214 3437 y Fn(J)2254 3447 y Fi(l)2285 3423 y Fq(\)\()p Fo(a)2403 3437 y Fn(J)2443 3447 y Fi(l)p Fd(\000)p Fj(1)2551 3437 y Ft(+2)2668 3423 y Fq(+)f Fp(\001)c(\001)g(\001)22 b Fq(+)e Fo(a)3024 3437 y Fn(J)3064 3447 y Fi(l)3095 3423 y Fq(\))3166 3361 y Fo(<)25 b Fq(0)3307 3260 y Ff(\021)q(\021)3477 2979 y Fq(\(1)p Fo(:)p Fq(12\))50 3644 y Fm(wher)-5 b(e)34 b Fo(Z)369 3658 y Fn(J)409 3668 y Fi(l)p Fd(\000)p Fj(1)515 3658 y Ft(+1)612 3644 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(Z)891 3658 y Fn(J)931 3668 y Fi(l)995 3644 y Fm(ar)-5 b(e)33 b(indep)-5 b(endent)34 b(standar)-5 b(d)36 b(Gaussian)e(r.v.)50 3862 y Fr(Remark:)44 b Fq(Giv)m(en)28 b Fo(a)795 3876 y Fn(i)825 3862 y Fo(;)15 b(\013)923 3876 y Fn(i)954 3862 y Fq(,)28 b Fo(i)e Fq(=)f(1)p Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(n)p Fq(,)28 b(the)g(partition)e(of)i(indices)d Fo(J)2506 3876 y Ft(1)2547 3862 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(J)2814 3876 y Fn(m)2911 3862 y Fq(satisfying)27 b(the)g(condi-)50 4007 y(tion)22 b(\(1.10\))k(is)c(generally)g(not)i (unique.)36 b(Therefore)23 b(this)f(corollary)g(yields)g(a)h Fm(family)h Fq(of)f(con)m(v)m(ergen)m(t)j(p)s(oin)m(t)50 4153 y(pro)s(cesses)31 b(asso)s(ciated)g(with)f(the)i(GREM.)g(In)e(the) h(sequel)g(w)m(e)h(will)c(frequen)m(tly)i(drop)h(the)g(upp)s(er)e (index)50 4298 y(and)h(write)f Fp(P)k(\021)25 b(P)721 4265 y Ft(\()p Fn(m)p Ft(\))848 4298 y Fq(,)30 b(in)f(particular)g (when)g(the)i(v)-5 b(alue)30 b(of)g Fo(m)g Fq(can)h(b)s(e)f(read)g (o\013)h(from)f(the)g(argumen)m(ts.)50 4516 y Fr(1.4.)47 b(The)34 b(extremal)g(pro)s(cess.)159 4735 y Fq(W)-8 b(e)28 b(are)f(no)m(w)f(also)h(in)e(the)i(p)s(osition)d(to)k(form)m (ulate)e(a)h(result)f(on)g(the)h(extreme)g(order)f(statistics)h(of)g (the)50 4880 y(random)33 b(v)-5 b(ariables)32 b Fo(X)842 4894 y Fn(\033)891 4880 y Fq(.)50 b(It)34 b(is)f(clear)g(that)h(the)g (extremal)g(pro)s(cess)f(should)f(b)s(e)g(constructed)i(from)f(one)50 5025 y(of)c(the)f(cascade)j(pro)s(cesses)d(that)h(w)m(ere)g (constructed)g(ab)s(o)m(v)m(e.)42 b(It)29 b(remains)e(to)i(\014nd)e (out)i(whic)m(h)f(one.)40 b(The)50 5171 y(answ)m(er)28 b(is)g(of)g(course)h(simple:)38 b(it)28 b(is)f(the)i(one)g(that)g(pro)m (vides)e(the)i(largest)g(v)-5 b(alues,)28 b(but)g(still)e(exists.)40 b(It)29 b(is)50 5316 y(easy)23 b(to)h(see)f(ho)m(w)g(this)f(will)e(ha)m (v)m(e)k(to)g(b)s(e)e(done:)37 b(one)23 b(has)f(simply)f(to)i(try)g(to) h(group)e(as)h(man)m(y)g(in)m(termediate)50 5462 y(hierarc)m(hies)36 b(together)k(to)e(a)g(single)f(p)s(oin)m(t)g(pro)s(cess.)63 b(In)37 b(other)h(w)m(ords,)h(among)g(the)f(p)s(ossible)d(c)m(hoices)p eop %%Page: 8 8 8 7 bop 50 295 a Fk(8)3309 b Fh(Se)l(ction)27 b Fk(1)50 513 y Fq(for)i(the)h(in)m(tegers)g Fo(J)730 527 y Ft(1)772 513 y Fo(;)15 b(:)g(:)g(:)h(;)f(J)1023 527 y Fn(m)1123 513 y Fq(allo)m(w)m(ed)29 b(in)g(Theorem)h(1.3)g(w)m(e)h(m)m(ust)e(c)m (ho)s(ose)i(those)f(that)h(ha)m(v)m(e)g(maximal)50 658 y(spacing.)40 b(Algorithmically)-8 b(,)28 b(this)i(is)f(ac)m(hiev)m(ed) i(b)m(y)f(setting)h Fo(J)2183 672 y Ft(0)2249 658 y Fp(\021)25 b Fq(0,)31 b(and)812 896 y Fo(J)862 910 y Fn(l)915 896 y Fp(\021)25 b Fq(min)o Fp(f)p Fo(J)35 b(>)25 b(J)1439 910 y Fn(l)p Fg(\000)p Ft(1)1585 896 y Fq(:)h Fo(A)1704 910 y Fn(J)1744 920 y Fi(l)p Fd(\000)p Fj(1)1851 910 y Ft(+1)p Fn(;J)2040 896 y Fo(>)f(A)2204 910 y Fn(J)6 b Ft(+1)p Fn(;k)2439 896 y Fp(8)p Fo(k)28 b Fp(\025)d Fo(J)30 b Fq(+)20 b(1)p Fp(g)p Fo(:)530 b Fq(\(1)p Fo(:)p Fq(13\))50 1133 y(where)30 b(w)m(e)h(ha)m(v)m(e)g(put)f Fo(A)891 1147 y Fn(j;k)1012 1133 y Fp(\021)1108 1065 y Ff(P)1204 1087 y Fn(k)1204 1160 y(i)p Ft(=)p Fn(j)1339 1133 y Fo(a)1387 1147 y Fn(i)1418 1133 y Fo(=)p Fq(\(2)15 b(ln\()1670 1065 y Ff(Q)1756 1087 y Fn(k)1756 1160 y(i)p Ft(=)p Fn(j)1891 1133 y Fo(\013)1949 1147 y Fn(i)1979 1133 y Fq(\)\).)159 1351 y(Of)30 b(course)g(it)g(will)e(b)s(e)i(essen)m (tial)g(that)50 1569 y Fc(Prop)s(osition)k(1.4:)39 b Fm(The)30 b(se)-5 b(quenc)g(e)29 b Fo(J)1481 1583 y Ft(1)1522 1569 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(J)1789 1583 y Fn(m)1888 1569 y Fm(de\014ne)-5 b(d)31 b(by)f(\(1.13\))h(veri\014es)f (the)g(c)-5 b(onditions)31 b(\(1.10\))50 1715 y(for)i(al)5 b(l)33 b(k)f(such)h(that)h Fo(J)842 1729 y Fn(l)p Fg(\000)p Ft(1)984 1715 y Fq(+)20 b(2)25 b Fp(\024)g Fo(k)k Fp(\024)c Fo(J)1463 1729 y Fn(l)1523 1715 y Fm(and)34 b(al)5 b(l)33 b Fo(l)27 b Fq(=)e(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(m)p Fm(.)159 1933 y Fq(Set)24 b(no)m(w)h(\026)-46 b Fo(a)540 1947 y Fn(l)593 1933 y Fp(\021)689 1865 y Ff(P)785 1887 y Fn(J)825 1897 y Fi(l)785 1960 y Fn(i)p Ft(=)p Fn(J)907 1970 y Fi(l)p Fd(\000)p Fj(1)1013 1960 y Ft(+1)1125 1933 y Fo(a)1173 1947 y Fn(i)1203 1933 y Fq(,)35 b(\026)-55 b Fo(\013)1311 1947 y Fn(l)1365 1933 y Fp(\021)1461 1865 y Ff(Q)1547 1887 y Fn(J)1587 1897 y Fi(l)1547 1960 y Fn(i)p Ft(=)p Fn(J)1669 1970 y Fi(l)p Fd(\000)p Fj(1)1775 1960 y Ft(+1)1887 1933 y Fo(\013)1945 1947 y Fn(i)1975 1933 y Fq(,)25 b Fo(\015)2072 1947 y Fn(l)2126 1933 y Fp(\021)2222 1868 y(p)p 2297 1868 76 4 v 65 x Fq(\026)-46 b Fo(a)2345 1947 y Fn(l)2373 1933 y Fo(=)2418 1862 y Fp(p)p 2494 1862 238 4 v 71 x Fq(2)15 b(ln)24 b(\026)-54 b Fo(\013)2704 1947 y Fn(l)2732 1933 y Fq(,)25 b Fo(l)i Fq(=)e(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)34 b(;)15 b(m)p Fq(.)38 b(Clearly)50 2078 y(b)m(y)30 b(\(1.13\))j Fo(\015)486 2092 y Ft(1)552 2078 y Fo(>)25 b(\015)695 2092 y Ft(2)761 2078 y Fo(>)g Fp(\001)15 b(\001)g(\001)26 b Fo(>)f(\015)1131 2092 y Fn(m)1200 2078 y Fq(.)41 b(De\014ne)30 b(the)h(function)e Fo(U)2122 2092 y Fn(J)n(;N)2281 2078 y Fq(b)m(y)276 2356 y Fo(U)338 2370 y Fn(J)n(;N)467 2356 y Fq(\()p Fo(x)p Fq(\))d Fp(\021)744 2242 y Fn(m)711 2269 y Ff(X)719 2465 y Fn(l)p Ft(=1)857 2255 y Ff(\020)911 2271 y(p)p 1002 2271 397 4 v 85 x Fq(2)p Fo(N)11 b Fq(\026)-46 b Fo(a)1178 2370 y Fn(l)1222 2356 y Fq(ln)23 b(\026)-54 b Fo(\013)1371 2370 y Fn(l)1419 2356 y Fp(\000)20 b Fo(N)1593 2318 y Fg(\000)p Ft(1)p Fn(=)p Ft(2)1764 2356 y Fo(\015)1811 2370 y Fn(l)1839 2356 y Fq(\(ln)o(\()p Fo(N)10 b Fq(\(ln)24 b(\026)-54 b Fo(\013)2253 2370 y Fn(l)2281 2356 y Fq(\)\))21 b(+)e(ln)14 b(4)p Fo(\031)s Fq(\))p Fo(=)p Fq(2)2778 2255 y Ff(\021)2855 2356 y Fq(+)20 b Fo(N)3029 2318 y Fg(\000)p Ft(1)p Fn(=)p Ft(2)3200 2356 y Fo(x)225 b Fq(\(1)p Fo(:)p Fq(14\))50 2638 y(and)30 b(the)g(p)s(oin)m(t)g(pro)s(cess)1369 2784 y Fp(E)1417 2798 y Fn(N)1511 2784 y Fp(\021)1730 2697 y Ff(X)1607 2898 y Fn(\033)r Fg(2f\000)p Ft(1)p Fn(;)p Ft(1)p Fg(g)1925 2880 y Fi(N)1999 2784 y Fo(\016)2039 2812 y Fn(U)2096 2783 y Fd(\000)p Fj(1)2089 2836 y Fi(J)n(;N)2201 2812 y Ft(\()p Fn(X)2289 2821 y Fi(\033)2333 2812 y Ft(\))2366 2784 y Fo(:)1086 b Fq(\(1)p Fo(:)p Fq(15\))50 3051 y(Then)29 b(the)i(follo)m(wing)e(holds)f(true:)50 3269 y Fc(Theorem)37 b(1.5:)52 b Fm(\(i\))33 b(The)g(p)-5 b(oint)35 b(pr)-5 b(o)g(c)g(ess)35 b Fp(E)1674 3283 y Fn(N)1776 3269 y Fm(c)-5 b(onver)g(ges)33 b(we)-5 b(akly,)34 b(as)g Fo(N)h Fp(")27 b(1)p Fm(,)32 b(to)i(the)f(p)-5 b(oint)35 b(pr)-5 b(o)g(c)g(ess)50 3415 y(on)33 b Fe(R)1119 3560 y Fp(E)h(\021)1297 3437 y Ff(Z)1347 3643 y Fb(R)1397 3625 y Fi(m)1473 3560 y Fp(P)1543 3523 y Ft(\()p Fn(m)p Ft(\))1670 3560 y Fq(\()p Fo(dx)1804 3574 y Ft(1)1845 3560 y Fo(;)15 b(:)g(:)g(:)i(;)e(dx)2146 3574 y Fn(m)2215 3560 y Fq(\))p Fo(\016)2290 3529 y Ff(P)2387 3551 y Fi(m)2387 3625 y(l)p Fj(=1)2504 3590 y Fn(\015)2542 3600 y Fi(l)2568 3590 y Fn(x)2609 3600 y Fi(l)3477 3560 y Fq(\(1)p Fo(:)p Fq(16\))50 3789 y Fm(wher)-5 b(e)34 b Fp(P)377 3756 y Ft(\()p Fn(m)p Ft(\))537 3789 y Fm(is)f(the)h (Poisson)g(c)-5 b(asc)g(ade)35 b(intr)-5 b(o)g(duc)g(e)g(d)37 b(in)c(The)-5 b(or)g(em)35 b(1.3)f(c)-5 b(orr)g(esp)g(onding)37 b(to)d(the)f(p)-5 b(artition)50 3935 y Fo(J)100 3949 y Ft(1)141 3935 y Fo(;)15 b(:)g(:)g(:)i(;)e(J)393 3949 y Fn(m)495 3935 y Fm(given)32 b(by)g(\(1.13\).)159 4153 y(\(ii\))39 b Fp(E)47 b Fm(exists)40 b(pr)-5 b(ovide)g(d)41 b(by)e(the)h(fact)g(that)g Fo(\015)1752 4167 y Ft(1)1830 4153 y Fo(>)d Fp(\001)15 b(\001)g(\001)38 b Fo(>)f(\015)2236 4167 y Fn(m)2305 4153 y Fm(.)62 b(It)39 b(is)g(the)h(cluster)f(p)-5 b(oint)41 b(pr)-5 b(o)g(c)g(ess)41 b(on)50 4298 y Fe(R)j Fm(c)-5 b(ontaining)37 b(an)f(a.s.)52 b(\014nite)36 b(numb)-5 b(er)36 b(of)g(p)-5 b(oints)37 b(in)f(any)g(interval)h Fq([)p Fo(a;)15 b Fp(1)p Fq(\))p Fm(,)37 b Fo(a)31 b Fp(2)g Fe(R)r Fm(.)57 b(The)36 b(pr)-5 b(ob)g(ability)50 4444 y(that)37 b(ther)-5 b(e)37 b(exists)f(at)h(le)-5 b(ast)37 b(one)f(p)-5 b(oint)38 b(of)e Fp(E)44 b Fm(in)36 b(the)g(interval)h Fq([)p Fo(a;)15 b Fp(1)p Fq(\))37 b Fm(is)f(de)-5 b(cr)g(e)g(asing)37 b(exp)-5 b(onential)5 b(ly)38 b(as)50 4589 y Fo(a)25 b Fp(!)g Fq(+)p Fp(1)p Fm(.)159 4807 y(\(iii\))k(F)-7 b(urthermor)i(e,)32 b(we)d(have)h Fq(max)1402 4821 y Fn(\033)1451 4807 y Fq(\()p Fo(X)1561 4821 y Fn(\033)1610 4807 y Fo(=)1655 4730 y Fp(p)p 1731 4730 83 4 v 77 x Fo(N)10 b Fq(\))26 b Fp(!)1991 4737 y(p)p 2067 4737 340 4 v 2068 4807 a Fq(\026)-46 b Fo(a)2115 4821 y Ft(1)2156 4807 y Fq(2)15 b(ln)23 b(\026)-54 b Fo(\013)2365 4821 y Ft(1)2419 4807 y Fq(+)12 b Fp(\001)j(\001)g(\001)e Fq(+)2703 4737 y Fp(p)p 2779 4737 396 4 v 2780 4807 a Fq(\026)-46 b Fo(a)2827 4821 y Fn(m)2896 4807 y Fq(2)15 b(ln)23 b(\026)-54 b Fo(\013)3105 4821 y Fn(m)3204 4807 y Fm(a.s.)41 b(and)29 b(also)50 4953 y Fe(E)10 b Fq(\()q(max)315 4967 y Fn(\033)379 4953 y Fo(X)454 4967 y Fn(\033)503 4953 y Fo(=)548 4876 y Fp(p)p 625 4876 83 4 v 625 4953 a Fo(N)g Fq(\))25 b Fp(!)884 4882 y(p)p 960 4882 340 4 v 71 x Fq(2)q(\026)-46 b Fo(a)1053 4967 y Ft(1)1109 4953 y Fq(ln)23 b(\026)-54 b Fo(\013)1258 4967 y Ft(1)1320 4953 y Fq(+)19 b Fp(\001)c(\001)g(\001)22 b Fq(+)1628 4882 y Fp(p)p 1703 4882 396 4 v 1703 4953 a Fq(2)q(\026)-46 b Fo(a)1796 4967 y Fn(m)1881 4953 y Fq(ln)23 b(\026)-54 b Fo(\013)2030 4967 y Fn(m)2099 4953 y Fm(.)50 5171 y Fr(Remark:)40 b Fq(Note)22 b(that)f(if)e(some)i(of)g(the)f (inequalities)e(in)h(the)i(condition)e(\(1.10\))k(hold)c(with)g (equalit)m(y)-8 b(,)23 b(there)50 5316 y(are)k(t)m(w)m(o)g(\(or)g(sev)m (eral\))g(of)g(the)f(p)s(oin)m(t)g(pro)s(cesses)g(that)h(to)g(leading)e (order)h(giv)m(e)h(the)f(same)h(con)m(tribution)e(to)50 5462 y(the)31 b(extremes.)41 b(The)30 b(degeneracy)i(is)e(only)f(w)m (eakly)i(lifted)e(in)g(fa)m(v)m(our)i(of)g(the)g(pro)s(cess)f(with)f (the)i(longest)p eop %%Page: 9 9 9 8 bop 50 295 a Fh(GREM,)28 b(p)l(art)f(1)3127 b Fk(9)50 513 y Fq(incremen)m(ts)33 b(in)f(the)i Fo(J)830 527 y Fn(i)895 513 y Fq(through)f(the)g(ln)14 b Fo(N)44 b Fq(term)34 b(in)e(the)i(functions)e Fo(u)2546 527 y Ft(ln)13 b Fn(\013;N)2791 513 y Fq(de\014ned)32 b(in)h(\(1.5\).)52 b(If)33 b(one)50 658 y(considers)28 b(what)i(happ)s(ens)e(here)i(as)g(a)g(function)f(of) h(the)g(parameters,)h(one)f(sees)g(that)h(the)f(pro)s(cess)f(with)50 804 y(few)m(er)k(lev)m(els)f('dies)f(out',)j(and)e(the)h(pro)s(cess)f (with)f(extra)i(lev)m(els)f(tak)m(es)i(o)m(v)m(er.)49 b(While)31 b(the)i(v)-5 b(alues)32 b(of)h(the)50 949 y(extremes)26 b(are)f(in)f(leading)f(order)i(the)g(same)h(at)f(the)h (co)s(existence)g(p)s(oin)m(t,)f(as)g(the)g(inequalit)m(y)f(gets)i (strictly)50 1095 y(violated,)k(these)h(v)-5 b(alues)30 b(no)m(w)g(b)s(egin)f(to)i(drop)e(substan)m(tially)-8 b(.)159 1313 y(T)g(o)42 b(our)f(kno)m(wledge)g(this)f(result)g(is)h (new,)j(and)c(one)i(of)g(the)f(rather)g(few)g(explicit)f(results)g(on)i (the)50 1458 y(con)m(v)m(ergence)32 b(of)d(the)h(extremal)f(pro)s(cess) g(in)f(the)i(case)g(when)f(correlations)g(are)h(strong)f(enough)g(to)i (ha)m(v)m(e)50 1604 y(an)26 b(e\013ect)h(on)f(the)g(extremal)g(pro)s (cess.)38 b(Of)26 b(course)g(our)f(result)g(sho)m(ws)g(the)i(exp)s (ected)f(e\013ect)h(of)f(clustering)50 1749 y(of)41 b(extremes)i(in)d (this)g(case.)75 b(Note)43 b(that)f(the)g(case)h Fo(m)g Fq(=)h(1)d(arises)g(when)g(w)m(e)h(are)g(in)e(the)i(standard)50 1895 y(situation)29 b(of)i(con)m(v)m(ergence)h(to)g(the)e(simple)e(P)m (oisson)i(pro)s(cess.)159 2113 y(In)g(the)g(remainder)f(of)h(this)g (pap)s(er)f(w)m(e)i(will)d(presen)m(t)i(the)h(pro)s(ofs)e(of)i(the)f (theorems)h(stated)g(ab)s(o)m(v)m(e.)50 2331 y Fr(Ac)m(kno)m(wledgemen) m(ts:)77 b Fq(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 2476 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 2708 y Fu(2.)55 b(Pro)s(ofs)42 b(of)f(p)s(oin)m(t)g(pro)s(cess)h(con)m(v)m(ergence)e (results.)159 2926 y Fq(In)29 b(this)f(section)i(w)m(e)g(will)d(pro)s (of)i(all)g(the)h(results)e(related)i(to)g(the)g(con)m(v)m(ergence)i (of)e(the)g(v)-5 b(arious)29 b(p)s(oin)m(t)50 3071 y(pro)s(cesses,)d (including)c(the)k(extremal)f(pro)s(cess)g(the)h(partition)e(function)g (and)g(Ruelle's)h(pro)s(cesses)g Fp(W)3550 3085 y Fn(\014)s(;N)3685 3071 y Fq(.)50 3217 y(Throughout)37 b(the)h(rest)g(of)g(the)g(pap)s(er) f(w)m(e)h(will)d(use)j(the)g(relation)f Fo(A)p Fq(\()p Fo(N)10 b Fq(\))39 b Fp(\030)e Fo(B)5 b Fq(\()p Fo(N)10 b Fq(\))39 b(to)f(indicate)f(that)50 3362 y(lim)176 3376 y Fn(N)7 b Fg("1)382 3318 y Fn(A)p Ft(\()p Fn(N)g Ft(\))p 381 3341 180 4 v 381 3394 a Fn(B)s Ft(\()p Fn(N)g Ft(\))596 3362 y Fq(=)25 b(1.)159 3581 y(The)30 b(tec)m(hnical)g(k)m(ey)h(to)g (Theorem)f(1.1)i(is)d(the)i(follo)m(wing)e(Lemma:)50 3799 y Fc(Lemma)45 b(2.1:)65 b Fm(L)-5 b(et)38 b Fo(n)c Fp(2)h Fe(N)6 b Fm(,)45 b Fo(n)34 b Fp(\025)h Fq(1)p Fm(,)k Fq(0)c Fo(<)f(a)1725 3813 y Fn(i)1790 3799 y Fo(<)h Fq(1)p Fm(,)k Fo(\013)2066 3813 y Fn(i)2131 3799 y Fo(>)34 b Fq(1)p Fm(,)40 b Fo(i)35 b Fq(=)f(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(n)p Fm(,)39 b(and)48 b Fq(\026)-54 b Fo(\013)35 b Fp(\021)3370 3731 y Ff(Q)3456 3753 y Fn(n)3456 3826 y(i)p Ft(=1)3594 3799 y Fo(\013)3652 3813 y Fn(i)3682 3799 y Fm(.)50 3944 y(Assume)d(that)575 3876 y Ff(P)671 3898 y Fn(n)671 3971 y(i)p Ft(=1)808 3944 y Fo(a)856 3958 y Fn(i)912 3944 y Fq(=)25 b(1)p Fm(.)42 b(If)33 b(al)5 b(l)33 b(ine)-5 b(qualities)33 b(\(1.6\))g(hold)h(true,)f(then)g (for)g(al)5 b(l)33 b Fo(x)26 b Fp(2)f Fe(R)50 4147 y(P)105 4046 y Ff(\020)160 4147 y Fp(8)p Fo(\033)j Fp(2)c(f\000)p Fq(1)p Fo(;)15 b Fq(+1)p Fp(g)738 4109 y Ft(ln)23 b(\026)-44 b Fn(\013)q(=)12 b Ft(ln)h(2)1049 4147 y Fq(:)1100 4084 y Fp(p)p 1176 4084 89 4 v 63 x Fo(a)1224 4161 y Ft(1)1264 4147 y Fo(X)1339 4161 y Fn(\033)1381 4170 y Fj(1)1423 4147 y Fq(+)1496 4084 y Fp(p)p 1571 4084 V 1571 4147 a Fo(a)1619 4161 y Ft(2)1660 4147 y Fo(X)1735 4161 y Fn(\033)1777 4170 y Fj(1)1813 4161 y Fn(\033)1855 4170 y Fj(2)1897 4147 y Fq(+)r Fp(\001)i(\001)g(\001)r Fq(+)2150 4084 y Fp(p)p 2225 4084 98 4 v 2225 4147 a Fo(a)2273 4161 y Fn(n)2322 4147 y Fo(X)2397 4161 y Fn(\033)2439 4170 y Fj(1)2475 4161 y Fn(\033)2517 4170 y Fj(2)2553 4161 y Fn(:::)n(\033)2659 4170 y Fi(n)2734 4147 y Fo(<)25 b(u)2882 4161 y Ft(ln)c(\026)-44 b Fn(\013;N)3093 4147 y Fq(\()p Fo(x)p Fq(\))3215 4046 y Ff(\021)3295 4147 y Fp(\030)25 b Fo(e)3433 4109 y Fg(\000)p Fn(K)t(e)3590 4082 y Fd(\000)p Fi(x)3685 4147 y Fo(;)3523 4292 y Fq(\(2)p Fo(:)p Fq(1\))50 4438 y Fm(as)31 b Fo(N)k Fp(")26 b(1)p Fm(,)31 b(wher)-5 b(e)32 b Fo(K)38 b Fm(is)30 b(de\014ne)-5 b(d)32 b(in)f(The)-5 b(or)g(em)33 b(1.1.)42 b(If)31 b(at)g(le)-5 b(ast)32 b(one)f(of)g(ine)-5 b(qualities)32 b(\(1.6\))g(fails,)f(then) 118 4640 y Fe(P)173 4539 y Ff(\020)228 4640 y Fp(8)p Fo(\033)d Fp(2)c(f\000)p Fq(1)p Fo(;)15 b Fq(+1)p Fp(g)806 4603 y Ft(ln)23 b(\026)-44 b Fn(\013=)13 b Ft(ln)g(2)1117 4640 y Fq(:)1168 4577 y Fp(p)p 1244 4577 89 4 v 63 x Fo(a)1292 4654 y Ft(1)1332 4640 y Fo(X)1407 4654 y Fn(\033)1449 4663 y Fj(1)1510 4640 y Fq(+)1601 4577 y Fp(p)p 1677 4577 V 63 x Fo(a)1725 4654 y Ft(2)1766 4640 y Fo(X)1841 4654 y Fn(\033)1883 4663 y Fj(1)1919 4654 y Fn(\033)1961 4663 y Fj(2)2021 4640 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)2329 4577 y Fp(p)p 2405 4577 98 4 v 63 x Fo(a)2453 4654 y Fn(n)2502 4640 y Fo(X)2577 4654 y Fn(\033)2619 4663 y Fj(1)2655 4654 y Fn(\033)2697 4663 y Fj(2)2733 4654 y Fn(:::)o(\033)2840 4663 y Fi(n)2914 4640 y Fo(<)k(u)3062 4654 y Ft(ln)c(\026)-44 b Fn(\013;N)3273 4640 y Fq(\()p Fo(x)p Fq(\))3395 4539 y Ff(\021)3475 4640 y Fp(\030)25 b Fq(1)p Fo(:)3523 4786 y Fq(\(2)p Fo(:)p Fq(2\))50 5171 y Fm(Pr)-5 b(o)g(of.)68 b Fq(If)39 b Fo(n)h Fq(=)g(1,)i(then)d(all)47 b(\026)-54 b Fo(\013)1183 5138 y Fn(N)1292 5171 y Fq(random)39 b(v)-5 b(ariables)37 b(in)h(the)i(ev)m(en)m(t)h(of)e(\(2.1\))i(are)f (indep)s(enden)m(t.)65 b(In)50 5316 y(this)34 b(case)j(the)e(statemen)m (t)i(of)f(the)f(lemma)g(is)g(a)h(classical)e(result)g(of)i(extreme)g(v) -5 b(alues)35 b(theory)-8 b(,)37 b(see)f(e.g.)50 5462 y(Theorem)30 b(5.7.2)i(in)d([LLR].)p eop %%Page: 10 10 10 9 bop 50 295 a Fk(10)3270 b Fh(Se)l(ction)27 b Fk(2)159 513 y Fq(Let)k(no)m(w)f Fo(n)25 b Fp(\025)g Fq(2.)41 b(W)-8 b(e)32 b(de\014ne)199 729 y Fo(P)13 b Fq(\()p Fo(i;)i(y)s Fq(\))26 b Fp(\021)f Fe(P)636 628 y Ff(\020)691 729 y Fp(9)p Fo(\033)794 743 y Fn(i)p Ft(+1)941 729 y Fp(2)g(f\000)p Fq(1)p Fo(;)15 b Fq(+1)p Fp(g)1389 692 y Ft(ln)g Fn(\013)1511 701 y Fi(i)1540 692 y Fn(=)d Ft(ln)i(2)1703 729 y Fo(;)h(:)g(:)g(:)32 b(;)15 b(\033)1972 743 y Fn(n)2047 729 y Fp(2)25 b(f\000)p Fq(1)p Fo(;)15 b Fq(+1)p Fp(g)2495 692 y Ft(ln)g Fn(\013)2617 701 y Fi(n)2662 692 y Fn(=)d Ft(ln)h(2)2850 729 y Fq(:)702 832 y Fp(p)p 778 832 79 4 v 58 x Fo(a)826 904 y Fn(i)856 890 y Fo(X)28 b Fq(+)1050 836 y Fp(p)p 1125 836 171 4 v 1125 890 a Fo(a)1173 904 y Fn(i)p Ft(+1)1296 890 y Fo(X)1371 904 y Fn(\033)1413 913 y Fi(i)p Fj(+1)1544 890 y Fq(+)1635 836 y Fp(p)p 1711 836 V 54 x Fo(a)1759 904 y Fn(i)p Ft(+2)1881 890 y Fo(X)1956 904 y Fn(\033)1998 913 y Fi(i)p Fj(+1)2105 904 y Fn(\033)2147 913 y Fi(i)p Fj(+2)2279 890 y Fq(+)19 b Fp(\001)c(\001)g(\001)22 b Fq(+)2587 832 y Fp(p)p 2662 832 98 4 v 2662 890 a Fo(a)2710 904 y Fn(n)2760 890 y Fo(X)2835 904 y Fn(\033)2877 913 y Fi(i)p Fj(+1)2984 904 y Fn(:::)o(\033)3091 913 y Fi(n)p 702 937 2438 4 v 1631 956 a Fp(p)p 1707 956 505 4 v 65 x Fo(a)1755 1035 y Fn(i)1805 1021 y Fq(+)e Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2161 1035 y Fn(n)3176 958 y Fo(>)25 b(y)3320 857 y Ff(\021)3523 856 y Fq(\(2)p Fo(:)p Fq(3\))50 1205 y(where)36 b Fo(X)r(;)15 b(X)511 1219 y Fn(\033)553 1228 y Fi(i)p Fj(+1)666 1205 y Fo(;)g(:)g(:)g(:)h(;)f(X)942 1219 y Fn(\033)984 1228 y Fi(i)p Fj(+1)1091 1219 y Fn(:::)o(\033)1198 1228 y Fi(n)1284 1205 y Fq(are)38 b(all)e(i.i.d.)59 b(standard)36 b(normal)g(r.v.'s.)61 b(W)-8 b(e)38 b(will)c(estimate)k(these)50 1351 y(probabilities)27 b(b)m(y)j(induction)e(do)m(wn)m(w)m(ards)i (from)g Fo(i)25 b Fq(=)g Fo(n)20 b Fp(\000)g Fq(1,)31 b(using)e(the)i(iden)m(tit)m(y)196 1609 y Fo(P)13 b Fq(\()p Fo(i;)i(y)s Fq(\))27 b(=)655 1547 y(1)p 590 1588 177 4 v 590 1606 a Fp(p)p 665 1606 101 4 v 665 1681 a Fq(2)p Fo(\031)792 1485 y Ff(Z)883 1507 y Fg(1)842 1691 y(\0001)991 1508 y Ff(\020)1046 1609 y Fq(1)20 b Fp(\000)1202 1508 y Ff(h)1245 1609 y Fq(1)h Fp(\000)f Fo(P)1473 1508 y Ff(\020)1527 1609 y Fo(i)h Fq(+)e(1)p Fo(;)1766 1479 y Fp(p)p 1842 1479 505 4 v 66 x Fo(a)1890 1559 y Fn(i)1941 1545 y Fq(+)g Fp(\001)c(\001)g(\001)22 b Fq(+)e Fo(a)2297 1559 y Fn(n)2346 1545 y Fo(y)j Fp(\000)2505 1486 y(p)p 2581 1486 79 4 v 59 x Fo(a)2629 1559 y Fn(i)2659 1545 y Fo(t)p 1766 1588 926 4 v 1893 1609 a Fp(p)p 1969 1609 597 4 v 62 x Fo(a)2017 1685 y Fn(i)p Ft(+1)2159 1671 y Fq(+)d Fp(\001)15 b(\001)g(\001)22 b Fq(+)e Fo(a)2516 1685 y Fn(n)2703 1508 y Ff(\021i)2800 1527 y Fn(\013)2847 1499 y Fi(N)2847 1545 y(i)p Fj(+1)2958 1508 y Ff(\021)3013 1609 y Fo(e)3055 1571 y Fg(\000)p Fn(t)3139 1544 y Fj(2)3175 1571 y Fn(=)p Ft(2)3268 1609 y Fq(d)o Fo(t:)147 b Fq(\(2)p Fo(:)p Fq(4\))159 1939 y(First)32 b(w)m(e)g(pro)m(v)m(e)i(the)e (assertion)g(of)h(the)f(lemma)g(under)f(assumption)g(that)i(all)e (inequalities)f(in)h(\(1.6\))50 2085 y(are)g(strict.)40 b(In)30 b(this)f(case)i(w)m(e)g(will)d(pro)m(v)m(e)j(inductiv)m(ely)d (for)i Fo(i)c Fq(=)f Fo(n)20 b Fp(\000)g Fq(1)p Fo(;)15 b(n)20 b Fp(\000)g Fq(2)p Fo(;)15 b(:)g(:)g(:)32 b(;)15 b Fq(1)32 b(that)1257 2353 y Fo(P)13 b Fq(\()p Fo(i;)i(y)s Fq(\))27 b Fp(\030)1650 2292 y Fq(\()p Fo(\013)1743 2306 y Fn(i)p Ft(+1)1881 2292 y Fp(\001)15 b(\001)g(\001)i Fo(\013)2061 2306 y Fn(n)2110 2292 y Fq(\))2145 2259 y Fn(N)2215 2292 y Fo(e)2257 2259 y Fg(\000)p Fn(y)2353 2231 y Fj(2)2388 2259 y Fn(=)p Ft(2)p 1650 2332 816 4 v 1946 2350 a Fp(p)p 2022 2350 101 4 v 76 x Fq(2)p Fo(\031)t(y)2477 2353 y(;)1021 b Fq(\(2)p Fo(:)p Fq(5\))50 2628 y(as)30 b Fo(N)36 b Fp(")25 b(1)p Fq(,)31 b(uniformly)c(for)j Fo(y)e Fp(2)d Fq([)1224 2550 y Ff(p)p 1315 2550 1106 4 v 78 x Fq(2\(ln)f(\026)-54 b Fo(\013)21 b Fp(\000)f Fo(\016)1697 2642 y Fn(i)1727 2628 y Fq(\))p Fo(N)10 b Fq(\()p Fo(a)1928 2642 y Fn(i)1980 2628 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2336 2642 y Fn(n)2385 2628 y Fq(\))q Fo(;)15 b Fp(1)p Fq(\))31 b(with)e(some)h Fo(\016)3092 2642 y Fn(i)3148 2628 y Fo(>)25 b Fq(0.)159 2846 y(Note)33 b(that)g Fo(u)631 2860 y Ft(ln)21 b(\026)-44 b Fn(\013;N)842 2846 y Fq(\()p Fo(x)p Fq(\))29 b Fp(2)e Fq([)1106 2768 y Ff(p)p 1197 2768 541 4 v 78 x Fq(2\(ln)d(\026)-54 b Fo(\013)21 b Fp(\000)f Fo(\016)1579 2860 y Ft(1)1620 2846 y Fq(\))p Fo(N)10 b(;)15 b Fp(1)p Fq(\))33 b(for)f(an)m(y)g Fo(x)c Fp(2)g Fe(R)s Fq(,)38 b(and)32 b(Then)f(inserting)f(in)m(to)i (\(2.5\))50 2991 y Fo(i)25 b Fq(=)g(1)31 b(and)f Fo(y)e Fq(=)d Fo(u)676 3005 y Ft(ln)c(\026)-44 b Fn(\013;N)887 2991 y Fq(\()p Fo(x)p Fq(\))31 b(implies)c(that)479 3227 y Fo(P)13 b Fq(\(1)p Fo(;)i(u)722 3241 y Ft(ln)21 b(\026)-43 b Fn(\013;N)934 3227 y Fq(\()p Fo(x)p Fq(\)\))26 b Fp(\030)f Fq(\()p Fo(\013)1306 3241 y Ft(2)1362 3227 y Fp(\001)15 b(\001)g(\001)i Fo(\013)1542 3241 y Fn(n)1591 3227 y Fq(\))1626 3190 y Fn(N)1696 3227 y Fo(e)1738 3190 y Fg(\000)p Fn(u)1838 3162 y Fj(2)1838 3208 y(ln)g(\026)-38 b Fi(\013;N)2022 3190 y Ft(\()p Fn(x)p Ft(\))p Fn(=)p Ft(2)2198 3227 y Fo(=)2243 3148 y Fp(p)p 2319 3148 101 4 v 79 x Fq(2)p Fo(\031)t(u)2472 3241 y Ft(ln)20 b(\026)-43 b Fn(\013;N)2683 3227 y Fq(\()p Fo(x)p Fq(\))26 b Fp(\030)f Fo(\013)2985 3188 y Fg(\000)p Fn(N)2985 3251 y Ft(1)3111 3227 y Fo(e)3153 3190 y Fg(\000)p Fn(x)3256 3227 y Fo(:)50 3463 y Fq(Since)d(the)h (probabilit)m(y)d(in)h(the)j(l.h.s.)e(of)h(\(2.1\))h(equals)e(\(1)5 b Fp(\000)g Fo(P)13 b Fq(\(1)p Fo(;)i(u)2321 3477 y Ft(ln)23 b(\026)-43 b Fn(\013;N)2534 3463 y Fq(\()p Fo(x)p Fq(\)\)\))2726 3430 y Fn(\013)2773 3403 y Fi(N)2773 3448 y Fj(1)2863 3463 y Fp(\030)25 b Fo(e)3001 3430 y Fn(\013)3048 3403 y Fi(N)3048 3448 y Fj(1)3120 3430 y Ft(ln\(1)p Fg(\000)p Fn(\013)3348 3400 y Fd(\000)p Fi(N)3348 3452 y Fj(1)3457 3430 y Fn(e)3492 3403 y Fd(\000)p Fi(x)3581 3430 y Ft(\))3639 3463 y Fp(\030)50 3608 y Fo(e)92 3575 y Fg(\000)p Fn(e)184 3548 y Fd(\000)p Fi(x)278 3608 y Fo(;)30 b Fq(this)g(pro)m(v)m(es)h (the)f(lemma)g(in)f(this)g(case.)159 3827 y(Let)e(us)e(no)m(w)i(turn)e (to)i(the)f(pro)s(of)g(of)g(\(2.5\).)41 b(Note)28 b(that)f Fo(P)13 b Fq(\()p Fo(i;)i(y)s Fq(\))26 b Fp(\024)f Fq(\()p Fo(\013)2541 3841 y Fn(i)p Ft(+1)2679 3827 y Fo(:)15 b(:)g(:)h(\013)2858 3841 y Fn(N)2927 3827 y Fq(\))2962 3794 y Fn(N)3032 3827 y Fo(P)d Fq(\()p Fo(X)33 b(>)25 b(y)s Fq(\),)i(where)50 3972 y Fo(X)42 b Fq(is)33 b(a)i(standard)f (Gaussian)f(random)h(v)-5 b(ariable,)35 b(and)e(so)i(the)g(upp)s(er)d (b)s(ound)h(for)h(\(2.5\))i(is)e(immediate)50 4117 y(b)m(y)c(Prop)s (osition)e(3.1.)159 4336 y(W)-8 b(e)30 b(will)d(establish)g(the)i(lo)m (w)m(er)h(b)s(ound)d(for)i(\(2.5\))i(b)m(y)e(induction.)38 b(Let)29 b(\014rst)g Fo(i)c Fq(=)g Fo(n)18 b Fp(\000)f Fq(1.)41 b(The)28 b(iden)m(tit)m(y)50 4481 y(\(2.4\))k(reads)147 4744 y Fo(P)13 b Fq(\()p Fo(n)20 b Fp(\000)g Fq(1)p Fo(;)15 b(y)s Fq(\))26 b(=)785 4682 y(1)p 720 4723 177 4 v 720 4741 a Fp(p)p 796 4741 101 4 v 75 x Fq(2)p Fo(\031)922 4620 y Ff(Z)1013 4643 y Fg(1)973 4827 y(\0001)1122 4643 y Ff(\020)1176 4744 y Fq(1)21 b Fp(\000)1333 4643 y Ff(h)1375 4744 y Fq(1)g Fp(\000)f Fe(P)1587 4643 y Ff(\020)1642 4744 y Fo(X)33 b(>)1856 4614 y Fp(p)p 1932 4614 400 4 v 62 x Fo(a)1980 4690 y Fn(n)p Fg(\000)p Ft(1)2143 4676 y Fq(+)20 b Fo(a)2282 4690 y Fn(n)2331 4676 y Fo(y)k Fp(\000)2490 4621 y(p)p 2566 4621 191 4 v 55 x Fo(a)2614 4690 y Fn(n)p Fg(\000)p Ft(1)2757 4676 y Fo(t)p 1856 4723 934 4 v 2236 4748 a Fp(p)p 2312 4748 98 4 v 58 x Fo(a)2360 4820 y Fn(n)2801 4643 y Ff(\021i)2898 4662 y Fn(\013)2945 4634 y Fi(N)2945 4680 y(n)3008 4643 y Ff(\021)3062 4744 y Fo(e)3104 4706 y Fg(\000)p Fn(t)3188 4679 y Fj(2)3224 4706 y Fn(=)p Ft(2)3317 4744 y Fq(d)p Fo(t:)97 b Fq(\(2)p Fo(:)p Fq(6\))50 5003 y(Let)32 b(us)e(\014x)h Fo(\016)504 4970 y Fg(0)501 5025 y Fn(n)p Fg(\000)p Ft(1)671 5003 y Fo(>)c Fq(0)32 b(suc)m(h)f(that)h(ln)14 b Fo(\013)1399 5017 y Fn(n)1470 5003 y Fp(\000)20 b Fq(\(ln)j(\026)-54 b Fo(\013)22 b Fp(\000)e Fo(\016)1901 4970 y Fg(0)1898 5025 y Fn(n)p Fg(\000)p Ft(1)2041 5003 y Fq(\))p Fo(a)2124 5017 y Fn(n)2201 5003 y Fo(<)27 b Fq(0,)32 b(whic)m(h)f(is)f(p)s (ossible)f(only)i(due)f(to)j(the)50 5148 y(strict)d(inequalit)m(y)f(in) g(\(1.6\))j(for)e Fo(i)c Fq(=)f Fo(n)p Fq(.)40 b(F)-8 b(or)31 b(all)e Fo(y)s(;)15 b(t)31 b Fq(satisfying)937 5278 y Fp(p)p 1013 5278 400 4 v 61 x Fo(a)1061 5353 y Fn(n)p Fg(\000)p Ft(1)1224 5339 y Fq(+)20 b Fo(a)1363 5353 y Fn(n)1412 5339 y Fo(y)j Fp(\000)1571 5284 y(p)p 1647 5284 191 4 v 55 x Fo(a)1695 5353 y Fn(n)p Fg(\000)p Ft(1)1837 5339 y Fo(t)p 937 5386 934 4 v 1317 5411 a Fp(p)p 1393 5411 98 4 v 59 x Fo(a)1441 5484 y Fn(n)1906 5407 y Fp(\025)2002 5302 y Ff(q)p 2093 5302 741 4 v 105 x Fq(2\(ln)h(\026)-54 b Fo(\013)21 b Fp(\000)f Fo(\016)2478 5376 y Fg(0)2475 5431 y Fn(n)p Fg(\000)p Ft(1)2618 5407 y Fq(\))p Fo(N)10 b(a)2784 5421 y Fn(n)3523 5407 y Fq(\(2)p Fo(:)p Fq(7\))p eop %%Page: 11 11 11 10 bop 50 295 a Fh(GREM,)28 b(p)l(art)f(1)3089 b Fk(11)50 513 y Fq(b)m(y)30 b(Prop)s(osition)e(3.1)k(w)m(e)f(ha)m(v)m(e)430 751 y Fo(\013)488 713 y Fn(N)488 773 y(n)558 751 y Fe(P)613 650 y Ff(\020)667 751 y Fo(X)i(>)882 621 y Fp(p)p 958 621 400 4 v 62 x Fo(a)1006 697 y Fn(n)p Fg(\000)p Ft(1)1169 683 y Fq(+)20 b Fo(a)1308 697 y Fn(n)1357 683 y Fo(y)j Fp(\000)1516 628 y(p)p 1592 628 191 4 v 55 x Fo(a)1640 697 y Fn(n)p Fg(\000)p Ft(1)1782 683 y Fo(t)p 882 730 934 4 v 1262 755 a Fp(p)p 1338 755 98 4 v 58 x Fo(a)1386 827 y Fn(n)1826 650 y Ff(\021)1906 751 y Fp(\024)i Fo(\013)2060 713 y Fn(N)2060 773 y(n)2129 751 y Fo(e)2171 713 y Ft(\(ln)c(\026)-43 b Fn(\013)p Fg(\000)p Fn(\016)2412 686 y Fd(0)2410 732 y Fi(n)p Fd(\000)p Fj(1)2535 713 y Ft(\))p Fn(N)7 b(a)2667 722 y Fi(n)2741 751 y Fp(!)26 b Fq(0)p Fo(;)106 b(N)36 b Fp(")25 b(1)p Fo(:)50 989 y Fq(Then,)k(using)g(the)i(elemen)m(tary)g (inequalities)520 1209 y Fp(\000)p Fo(x)20 b Fp(\000)g Fo(x)806 1171 y Ft(2)872 1209 y Fp(\024)25 b Fq(ln)o(\(1)c Fp(\000)f Fo(x)p Fq(\))25 b Fp(\024)g(\000)p Fo(x;)106 b Fq(1)21 b(+)f Fo(x)25 b Fp(\024)g Fo(e)2070 1171 y Fn(x)2141 1209 y Fp(\024)g Fq(1)c(+)f Fo(x)g Fq(+)g Fo(x)2609 1171 y Ft(2)2740 1209 y Fq(for)30 b Fp(j)p Fo(x)p Fp(j)c Fo(<)f Fq(1)p Fo(=)p Fq(2)285 b(\(2)p Fo(:)p Fq(8\))50 1429 y(and)30 b(again)g(Prop)s(osition)e(3.1,)k(it)e(is)f(easy)i(to)h (deduce)e(the)g(asymptotics)h(of)f(the)h(in)m(tegrand)f(in)f(\(2.6\):) 964 1676 y(1)20 b Fp(\000)1120 1575 y Ff(h)1163 1676 y Fq(1)h Fp(\000)f Fe(P)1375 1575 y Ff(\020)1430 1676 y Fo(X)32 b(>)1644 1546 y Fp(p)p 1720 1546 400 4 v 62 x Fo(a)1768 1622 y Fn(n)p Fg(\000)p Ft(1)1931 1608 y Fq(+)20 b Fo(a)2070 1622 y Fn(n)2119 1608 y Fo(y)j Fp(\000)2278 1553 y(p)p 2354 1553 191 4 v 55 x Fo(a)2402 1622 y Fn(n)p Fg(\000)p Ft(1)2545 1608 y Fo(t)p 1644 1655 934 4 v 2024 1679 a Fp(p)p 2100 1679 98 4 v 59 x Fo(a)2148 1752 y Fn(n)2588 1575 y Ff(\021)q(i)2686 1593 y Fn(\013)2733 1566 y Fi(N)2733 1612 y(n)393 1991 y Fp(\030)940 1927 y Fo(\013)998 1894 y Fn(N)998 1949 y(n)1067 1868 y Fp(p)p 1143 1868 V 59 x Fo(a)1191 1941 y Fn(n)p 500 1970 1181 4 v 500 1988 a Fp(p)p 576 1988 101 4 v 75 x Fq(2)p Fo(\031)s Fq(\()711 2001 y Fp(p)p 787 2001 400 4 v 62 x Fo(a)835 2077 y Fn(n)p Fg(\000)p Ft(1)998 2063 y Fq(+)d Fo(a)1137 2077 y Fn(n)1187 2063 y Fo(y)j Fp(\000)1346 2008 y(p)p 1421 2008 191 4 v 1421 2063 a Fo(a)1469 2077 y Fn(n)p Fg(\000)p Ft(1)1612 2063 y Fo(t)p Fq(\))1691 1991 y Fo(e)1733 1951 y Fg(\000)p Ft(\()1818 1894 y Fp(p)p 1894 1894 305 4 v 57 x Fn(a)1933 1960 y Fi(n)p Fd(\000)p Fj(1)2059 1951 y Ft(+)p Fn(a)2154 1960 y Fi(n)2198 1951 y Fn(y)r Fg(\000)2294 1913 y(p)p 2354 1913 165 4 v 38 x Fn(a)2393 1960 y Fi(n)p Fd(\000)p Fj(1)2518 1951 y Fn(t)p Ft(\))2573 1923 y Fj(2)2610 1951 y Fn(=)p Ft(\(2)p Fn(a)2750 1960 y Fi(n)2796 1951 y Ft(\))2828 1991 y Fo(;)107 b Fq(as)30 b Fo(N)36 b Fp(")25 b(1)p Fo(;)50 2221 y Fq(uniformly)19 b(for)j(all)f Fo(y)s(;)15 b(t)23 b Fq(satisfying)e(\(2.7\).)39 b(Observ)m(e)23 b(that)g(condition)d(\(2.7\))25 b(is)c(equiv)-5 b(alen)m(t)22 b(to)h Fo(t)i Fp(\024)g Fo(t)3423 2235 y Fn(cr)3495 2221 y Fq(\()p Fo(y)s Fq(\))h Fp(\021)50 2367 y Fq(\()85 2305 y Fp(p)p 161 2305 400 4 v 62 x Fo(a)209 2381 y Fn(n)p Fg(\000)p Ft(1)372 2367 y Fq(+)20 b Fo(a)511 2381 y Fn(n)560 2367 y Fo(y)k Fp(\000)721 2293 y Ff(p)p 812 2293 643 4 v 74 x Fq(2\(ln)g(\026)-55 b Fo(\013)21 b Fp(\000)f Fo(\016)1196 2335 y Fg(0)1193 2391 y Fn(n)p Fg(\000)p Ft(1)1336 2367 y Fq(\))p Fo(N)10 b(a)1502 2381 y Fn(n)1552 2367 y Fq(\))p Fo(=)1632 2312 y Fp(p)p 1708 2312 191 4 v 55 x Fo(a)1756 2381 y Fn(n)p Fg(\000)p Ft(1)1899 2367 y Fo(:)32 b Fq(Th)m(us)f(in)f(view)h(of)h(\(2.6\),)i Fo(P)13 b Fq(\()p Fo(n)21 b Fp(\000)g Fq(1)p Fo(;)15 b(y)s Fq(\))32 b(ob)s(eys)g(the)50 2512 y(follo)m(wing)d(asymptotic)h (lo)m(w)m(er)h(b)s(ound)50 2762 y Fo(P)13 b Fq(\()p Fo(n)c Fp(\000)g Fq(1)p Fo(;)15 b(y)s Fq(\))25 b Fp(\025)600 2701 y Fo(\013)658 2668 y Fn(N)658 2723 y(n)p 600 2741 128 4 v 613 2825 a Fq(2)p Fo(\031)753 2639 y Ff(Z)844 2661 y Fn(t)871 2670 y Fi(cr)938 2661 y Ft(\()p Fn(y)r Ft(\))804 2845 y Fg(\0001)1568 2640 y Fp(p)p 1644 2640 98 4 v 58 x Fo(a)1692 2712 y Fn(n)p 1064 2741 1181 4 v 1064 2760 a Fp(p)p 1140 2760 101 4 v 75 x Fq(2)p Fo(\031)t Fq(\()1276 2773 y Fp(p)p 1352 2773 400 4 v 62 x Fo(a)1400 2849 y Fn(n)p Fg(\000)p Ft(1)1563 2835 y Fq(+)19 b Fo(a)1701 2849 y Fn(n)1751 2835 y Fo(y)k Fp(\000)1910 2780 y(p)p 1986 2780 191 4 v 55 x Fo(a)2034 2849 y Fn(n)p Fg(\000)p Ft(1)2176 2835 y Fo(t)p Fq(\))2255 2762 y Fo(e)2297 2722 y Fg(\000)p Ft(\()2382 2665 y Fp(p)p 2459 2665 305 4 v 2459 2722 a Fn(a)2498 2731 y Fi(n)p Fd(\000)p Fj(1)2623 2722 y Ft(+)p Fn(a)2718 2731 y Fi(n)2763 2722 y Fn(y)r Fg(\000)2859 2685 y(p)p 2918 2685 165 4 v 2918 2722 a Fn(a)2957 2731 y Fi(n)p Fd(\000)p Fj(1)3083 2722 y Fn(t)p Ft(\))3138 2695 y Fj(2)3174 2722 y Fn(=)p Ft(\(2)p Fn(a)3314 2731 y Fi(n)3360 2722 y Ft(\))p Fg(\000)p Fn(t)3472 2695 y Fj(2)3508 2722 y Fn(=)p Ft(2)3601 2762 y Fq(d)p Fo(t;)50 3014 y Fq(as)30 b Fo(N)36 b Fp(")25 b(1)p Fq(,)31 b(uniformly)c(in)i Fo(y)f Fp(2)d Fe(R)s Fq(.)47 b(After)30 b(the)h(c)m(hange)g(of)g(v)-5 b(ariables)50 3159 y Fo(s)25 b Fq(=)g(\()249 3097 y Fp(p)p 325 3097 400 4 v 62 x Fo(a)373 3173 y Fn(n)p Fg(\000)p Ft(1)536 3159 y Fq(+)19 b Fo(a)674 3173 y Fn(n)724 3159 y Fo(t)h Fp(\000)868 3104 y(p)p 944 3104 191 4 v 55 x Fo(a)992 3173 y Fn(n)p Fg(\000)p Ft(1)1134 3159 y Fo(y)s Fq(\))p Fo(=)1262 3101 y Fp(p)p 1339 3101 98 4 v 1339 3159 a Fo(a)1387 3173 y Fn(n)1467 3159 y Fq(the)30 b(last)g(in)m (tegral)g(b)s(ecomes)1147 3365 y Fo(\013)1205 3332 y Fn(N)1205 3388 y(n)1274 3365 y Fo(e)1316 3332 y Fg(\000)p Fn(y)1412 3305 y Fj(2)1448 3332 y Fn(=)p Ft(2)p 1147 3406 380 4 v 1224 3424 a Fp(p)p 1300 3424 101 4 v 75 x Fq(2)p Fo(\031)t(y)1613 3365 y Fq(1)p 1548 3406 177 4 v 1548 3424 a Fp(p)p 1624 3424 101 4 v 75 x Fq(2)p Fo(\031)1750 3303 y Ff(Z)1841 3325 y Fn(s)1875 3334 y Fi(cr)1941 3325 y Ft(\()p Fn(y)r Ft(\))1801 3509 y Fg(\0001)2163 3365 y Fo(e)2205 3332 y Fg(\000)p Fn(s)2296 3305 y Fj(2)2332 3332 y Fn(=)p Ft(2)p 2068 3406 437 4 v 2068 3511 a Fq(1)21 b Fp(\000)2236 3423 y Fg(p)p 2295 3423 165 4 v 2295 3461 a Fn(a)2334 3470 y Fi(n)p Fd(\000)p Fj(1)2459 3461 y Fn(s)p 2236 3491 259 4 v 2273 3501 a Fg(p)p 2333 3501 85 4 v 42 x Fn(a)2372 3552 y Fi(n)2417 3543 y Fn(y)2531 3427 y Fq(d)o Fo(s)50 3726 y Fq(where)31 b Fo(s)357 3740 y Fn(cr)430 3726 y Fq(\()p Fo(y)s Fq(\))d(=)675 3652 y Ff(\000)716 3726 y Fo(y)d Fp(\000)878 3653 y Ff(p)p 969 3653 1113 4 v 73 x Fq(2\(ln)e(\026)-54 b Fo(\013)21 b Fp(\000)f Fo(\016)1353 3694 y Fg(0)1350 3750 y Fn(n)p Fg(\000)p Ft(1)1493 3726 y Fq(\))p Fo(N)10 b Fq(\()p Fo(a)1694 3740 y Fn(n)p Fg(\000)p Ft(1)1857 3726 y Fq(+)20 b Fo(a)1996 3740 y Fn(n)2046 3726 y Fq(\))2081 3652 y Ff(\001)2123 3667 y Fp(p)p 2198 3667 98 4 v 2198 3726 a Fo(a)2246 3740 y Fn(n)2296 3726 y Fo(=)2341 3671 y Fp(p)p 2417 3671 191 4 v 55 x Fo(a)2465 3740 y Fn(n)p Fg(\000)p Ft(1)2608 3726 y Fq(.)45 b(It)32 b(yields)f(the)h(necessary)g (as-)50 3871 y(ymptotic)25 b(lo)m(w)m(er)g(b)s(ound)e(for)i(\(2.5\))i (as)f Fo(N)35 b Fp(")26 b(1)f Fq(uniform)e(for)i Fo(y)j Fp(2)c Fq([)2339 3793 y Ff(p)p 2431 3793 1113 4 v 2431 3871 a Fq(2\(ln)f(\026)-54 b Fo(\013)21 b Fp(\000)f Fo(\016)2812 3885 y Fn(n)p Fg(\000)p Ft(1)2955 3871 y Fq(\))p Fo(N)10 b Fq(\()p Fo(a)3156 3885 y Fn(n)p Fg(\000)p Ft(1)3319 3871 y Fq(+)20 b Fo(a)3458 3885 y Fn(n)3508 3871 y Fq(\))p Fo(;)15 b Fp(1)p Fq(\))50 4017 y(if)29 b(w)m(e)i(set)g(0)26 b Fo(<)f(\016)616 4031 y Fn(n)p Fg(\000)p Ft(1)784 4017 y Fo(<)g(\016)923 3984 y Fg(0)920 4039 y Fn(n)p Fg(\000)p Ft(1)1063 4017 y Fq(.)159 4235 y(Assuming)i(that)i(\(2.1\))i(is)c(pro)m (v)m(ed)i(for)g Fo(i)c Fq(=)g Fo(n)17 b Fp(\000)f Fq(1)p Fo(;)f(n)i Fp(\000)g Fq(2)p Fo(;)e(:)g(:)g(:)32 b(;)15 b(l)k Fq(+)e(1,)29 b(w)m(e)h(will)c(pro)m(v)m(e)j(it)f(for)h Fo(i)c Fq(=)g Fo(l)r Fq(.)40 b(Let)50 4380 y(us)29 b(\014x)h Fo(\016)338 4347 y Fg(0)335 4406 y Fn(l)389 4380 y Fo(>)25 b Fq(0)31 b(suc)m(h)f(that)g Fo(\016)1005 4347 y Fg(0)1002 4406 y Fn(l)1057 4380 y Fp(\024)25 b Fo(\016)1193 4394 y Fn(l)p Ft(+1)1343 4380 y Fq(and)30 b(ln)o(\()p Fo(\013)1689 4394 y Fn(l)p Ft(+1)1824 4380 y Fp(\001)15 b(\001)g(\001)h Fo(\013)2003 4394 y Fn(n)2053 4380 y Fq(\))k Fp(\000)g Fq(\(ln)j(\026)-54 b Fo(\013)20 b Fp(\000)g Fo(\016)2537 4347 y Fg(0)2534 4406 y Fn(l)2563 4380 y Fq(\)\()p Fo(a)2681 4394 y Fn(l)p Ft(+1)2821 4380 y Fq(+)g Fp(\001)15 b(\001)g(\001)20 b Fq(+)g Fo(a)3176 4394 y Fn(n)3225 4380 y Fq(\))26 b Fo(<)f Fq(0)31 b(due)e(to)50 4526 y(the)h(strict)h(inequalit)m(y)d(in)h (\(1.6\))j(for)f Fo(i)25 b Fq(=)g Fo(l)d Fq(+)e(1.)41 b(Then)30 b(for)g(all)f Fo(y)k Fq(and)d Fo(t)g Fq(v)m(erifying)f(the)i (condition)716 4639 y Fp(p)p 792 4639 502 4 v 66 x Fo(a)840 4719 y Fn(l)888 4705 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)22 b Fq(+)d Fo(a)1244 4719 y Fn(n)1294 4705 y Fo(y)k Fp(\000)1453 4646 y(p)p 1529 4646 76 4 v 59 x Fo(a)1577 4719 y Fn(l)1605 4705 y Fo(t)p 716 4748 922 4 v 842 4770 a Fp(p)p 918 4770 594 4 v 61 x Fo(a)966 4845 y Fn(l)p Ft(+1)1106 4831 y Fq(+)d Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1462 4845 y Fn(n)1674 4769 y Fp(\025)1770 4662 y Ff(q)p 1861 4662 1194 4 v 107 x Fq(2\(ln)j(\026)-54 b Fo(\013)21 b Fp(\000)f Fo(\016)2245 4738 y Fg(0)2242 4796 y Fn(l)2271 4769 y Fq(\))p Fo(N)10 b Fq(\()p Fo(a)2472 4783 y Fn(l)p Ft(+1)2613 4769 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2969 4783 y Fn(n)3019 4769 y Fq(\))469 b(\(2)p Fo(:)p Fq(9\))50 5009 y(w)m(e)31 b(ha)m(v)m(e)g(b)m(y)g(the)f (induction)e(h)m(yp)s(othesis)h(\(2.1\))j(for)e Fo(i)c Fq(=)f Fo(l)d Fq(+)e(1)93 5252 y Fo(\013)151 5214 y Fn(N)151 5274 y(l)p Ft(+1)271 5252 y Fo(P)342 5151 y Ff(\020)396 5252 y Fo(l)i Fq(+)e(1)p Fo(;)633 5122 y Fp(p)p 709 5122 502 4 v 66 x Fo(a)757 5202 y Fn(l)805 5188 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1161 5202 y Fn(n)1210 5188 y Fo(y)j Fp(\000)1369 5129 y(p)p 1445 5129 76 4 v 59 x Fo(a)1493 5202 y Fn(l)1521 5188 y Fo(t)p 633 5231 922 4 v 758 5252 a Fp(p)p 834 5252 594 4 v 62 x Fo(a)882 5328 y Fn(l)p Ft(+1)1022 5314 y Fq(+)d Fp(\001)15 b(\001)g(\001)22 b Fq(+)e Fo(a)1379 5328 y Fn(n)1565 5151 y Ff(\021)1644 5252 y Fp(\024)25 b Fq(\()p Fo(\013)1833 5266 y Fn(l)p Ft(+1)1969 5252 y Fp(\001)15 b(\001)g(\001)h Fo(\013)2148 5266 y Fn(n)2198 5252 y Fq(\))2233 5214 y Fn(N)2302 5252 y Fo(e)2344 5214 y Fg(\000)p Ft(\()2429 5167 y Fg(p)p 2489 5167 327 4 v 47 x Fn(a)2528 5224 y Fi(l)2555 5214 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)2771 5223 y Fi(n)2815 5214 y Fn(y)r Fg(\000)2911 5173 y(p)p 2971 5173 66 4 v 41 x Fn(a)3010 5224 y Fi(l)3037 5214 y Fn(t)p Ft(\))3092 5187 y Fj(2)3128 5214 y Fn(=)p Ft(2\()p Fn(a)3268 5224 y Fi(l)p Fj(+1)3374 5214 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)3589 5223 y Fi(n)3634 5214 y Ft(\))1644 5466 y Fp(\024)25 b Fq(\()p Fo(\013)1833 5480 y Fn(l)p Ft(+1)1969 5466 y Fp(\001)15 b(\001)g(\001)h Fo(\013)2148 5480 y Fn(n)2198 5466 y Fq(\))2233 5429 y Fn(N)2302 5466 y Fo(e)2344 5429 y Fg(\000)p Ft(\(ln)21 b(\026)-43 b Fn(\013)p Fg(\000)p Fn(\016)2642 5401 y Fd(0)2640 5447 y Fi(l)2667 5429 y Ft(\))p Fn(N)7 b Ft(\()p Fn(a)2827 5439 y Fi(l)p Fj(+1)2931 5429 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)3147 5438 y Fi(n)3192 5429 y Ft(\))3225 5466 y Fo(;)p eop %%Page: 12 12 12 11 bop 50 295 a Fk(12)3270 b Fh(Se)l(ction)27 b Fk(2)50 513 y Fq(whic)m(h)g(tends)i(to)g(zero,)h(as)f Fo(N)35 b Fp(")26 b(1)p Fq(.)40 b(F)-8 b(urthermore,)29 b(again)g(b)m(y)g(the)g (induction)d(h)m(yp)s(othesis)i(for)g Fo(i)e Fq(=)f Fo(l)19 b Fq(+)d(1)50 658 y(and)30 b(elemen)m(tary)g(exp)s(onen)m(tial)g (inequalities)e(\(2.8\))510 958 y(1)21 b Fp(\000)667 857 y Ff(h)710 958 y Fq(1)f Fp(\000)g Fo(P)937 857 y Ff(\020)991 958 y Fo(l)j Fq(+)c(1)p Fo(;)1228 828 y Fp(p)p 1304 828 502 4 v 66 x Fo(a)1352 908 y Fn(l)1400 894 y Fq(+)h Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1756 908 y Fn(n)1806 894 y Fo(y)j Fp(\000)1965 835 y(p)p 2040 835 76 4 v 2040 894 a Fo(a)2088 908 y Fn(l)2116 894 y Fo(t)p 1228 937 922 4 v 1354 959 a Fp(p)p 1430 959 594 4 v 61 x Fo(a)1478 1034 y Fn(l)p Ft(+1)1618 1020 y Fq(+)d Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1974 1034 y Fn(n)2160 857 y Ff(\021i)2257 876 y Fn(\013)2304 849 y Fi(N)2304 895 y(l)p Fj(+1)535 1220 y Fp(\030)642 1159 y Fq(\()p Fo(\013)735 1173 y Fn(l)p Ft(+1)871 1159 y Fp(\001)15 b(\001)g(\001)h Fo(\013)1050 1173 y Fn(n)1100 1159 y Fq(\))1135 1126 y Fn(N)1204 1097 y Fp(p)p 1280 1097 V 62 x Fo(a)1328 1173 y Fn(l)p Ft(+1)1468 1159 y Fq(+)k Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1824 1173 y Fn(n)p 642 1199 1232 4 v 674 1218 a Fp(p)p 750 1218 101 4 v 75 x Fq(2)p Fo(\031)s Fq(\()885 1227 y Fp(p)p 961 1227 502 4 v 66 x Fo(a)1009 1307 y Fn(l)1057 1293 y Fq(+)g Fp(\001)15 b(\001)g(\001)22 b Fq(+)e Fo(a)1414 1307 y Fn(n)1463 1293 y Fo(y)j Fp(\000)1622 1234 y(p)p 1698 1234 76 4 v 59 x Fo(a)1746 1307 y Fn(l)1774 1293 y Fo(t)p Fq(\))1884 1220 y Fo(e)1926 1183 y Fg(\000)p Ft(\()2011 1135 y Fg(p)p 2072 1135 327 4 v 2072 1183 a Fn(a)2111 1193 y Fi(l)2138 1183 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)2353 1192 y Fi(n)2398 1183 y Fn(y)r Fg(\000)2494 1141 y(p)p 2554 1141 66 4 v 42 x Fn(a)2593 1193 y Fi(l)2619 1183 y Fn(t)p Ft(\))2674 1155 y Fj(2)2711 1183 y Fn(=)p Ft(2\()p Fn(a)2851 1193 y Fi(l)p Fj(+1)2956 1183 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)3172 1192 y Fi(n)3217 1183 y Ft(\))50 1526 y Fq(uniformly)k(for)j(all)g Fo(y)s(;)15 b(t)30 b Fq(satisfying)f(\(2.9\),)j(that)f(is)f(for)484 1800 y Fo(t)25 b Fp(\024)g Fo(t)671 1816 y Fn(cr)r Ft(\()p Fn(y)r Ft(\))865 1800 y Fp(\021)g Fq(\()996 1730 y Fp(p)p 1072 1730 502 4 v 70 x Fo(a)1120 1814 y Fn(l)1168 1800 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)22 b Fq(+)e Fo(a)1525 1814 y Fn(n)1574 1800 y Fo(y)j Fp(\000)1733 1693 y Ff(q)p 1824 1693 529 4 v 107 x Fq(2\(ln)h(\026)-54 b Fo(\013)20 b Fp(\000)g Fo(\016)2208 1769 y Fg(0)2205 1827 y Fn(l)2235 1800 y Fq(\))p Fo(N)10 b Fq(\()p Fo(a)2436 1814 y Fn(l)p Ft(+1)2576 1800 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)22 b Fq(+)e Fo(a)2933 1814 y Fn(n)2982 1800 y Fq(\)\))p Fo(=)3097 1737 y Fp(p)p 3174 1737 76 4 v 3174 1800 a Fo(a)3222 1814 y Fn(l)3250 1800 y Fo(:)50 2073 y Fq(In)k(view)f(of)i (this)f(estimate)h(the)g(iden)m(tit)m(y)f(\(2.4\))i(for)e Fo(i)i Fq(=)f Fo(l)h Fq(giv)m(es)f(the)g(follo)m(wing)e(asymptotic)i (lo)m(w)m(er)g(b)s(ound)50 2219 y(for)30 b Fo(P)13 b Fq(\()p Fo(l)r(;)i(y)s Fq(\))31 b(as)g Fo(N)k Fp(")26 b(1)k Fq(uniform)e(for)i Fo(y)e Fp(2)d Fe(R)s Fq(:)66 2529 y Fo(P)13 b Fq(\()p Fo(l)r(;)i(y)s Fq(\))26 b Fp(\025)457 2468 y Fq(\()p Fo(\013)550 2482 y Fn(l)p Ft(+1)686 2468 y Fp(\001)15 b(\001)g(\001)h Fo(\013)865 2482 y Fn(n)915 2468 y Fq(\))950 2435 y Fn(N)1019 2406 y Fp(p)p 1095 2406 594 4 v 62 x Fo(a)1143 2482 y Fn(l)p Ft(+1)1283 2468 y Fq(+)k Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1639 2482 y Fn(n)p 457 2508 1232 4 v 1023 2592 a Fq(2)p Fo(\031)1715 2405 y Ff(Z)1806 2428 y Fn(t)1833 2437 y Fi(cr)1899 2428 y Ft(\()p Fn(y)r Ft(\))1765 2612 y Fg(\0001)2026 2468 y Fo(e)2068 2435 y Fg(\000)p Ft(\()2153 2387 y Fg(p)p 2213 2387 327 4 v 48 x Fn(a)2252 2445 y Fi(l)2279 2435 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)2494 2444 y Fi(n)2539 2435 y Fn(y)r Fg(\000)2635 2393 y(p)p 2695 2393 66 4 v 42 x Fn(a)2734 2445 y Fi(l)2761 2435 y Fn(t)p Ft(\))2816 2407 y Fj(2)2852 2435 y Fn(=)p Ft(2\()p Fn(a)2992 2445 y Fi(l)p Fj(+1)3097 2435 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)3313 2444 y Fi(n)3358 2435 y Ft(\))p Fg(\000)p Fn(t)3470 2407 y Fj(2)3506 2435 y Fn(=)p Ft(2)p 2026 2508 1559 4 v 2344 2526 a Fp(p)p 2420 2526 502 4 v 66 x Fo(a)2468 2606 y Fn(l)2516 2592 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2872 2606 y Fn(n)2922 2592 y Fo(y)j Fp(\000)3081 2533 y(p)p 3157 2533 76 4 v 59 x Fo(a)3205 2606 y Fn(l)3233 2592 y Fo(t)3610 2529 y Fq(d)p Fo(t)350 2808 y Fq(=)457 2747 y(\()p Fo(\013)550 2761 y Fn(l)p Ft(+1)686 2747 y Fp(\001)15 b(\001)g(\001)h Fo(\013)865 2761 y Fn(n)915 2747 y Fq(\))950 2714 y Fn(N)1019 2747 y Fo(e)1061 2714 y Fg(\000)p Fn(y)1157 2686 y Fj(2)1193 2714 y Fn(=)p Ft(2)p 457 2787 814 4 v 752 2806 a Fp(p)p 828 2806 101 4 v 75 x Fq(2)p Fo(\031)s(y)1358 2747 y Fq(1)p 1293 2787 177 4 v 1293 2806 a Fp(p)p 1368 2806 101 4 v 1368 2881 a Fq(2)p Fo(\031)1495 2685 y Ff(Z)1586 2707 y Fn(s)1620 2716 y Fi(cr)1686 2707 y Ft(\()p Fn(y)r Ft(\))1545 2891 y Fg(\0001)2038 2747 y Fo(e)2080 2714 y Fg(\000)p Fn(s)2171 2686 y Fj(2)2207 2714 y Fn(=)p Ft(2)p 1813 2787 698 4 v 1813 2893 a Fq(1)21 b Fp(\000)2160 2804 y Fg(p)p 2220 2804 66 4 v 42 x Fn(a)2259 2856 y Fi(l)2285 2846 y Fn(s)p 1980 2872 519 4 v 1980 2883 a Fp(p)p 2056 2883 404 4 v 57 x Fn(a)2095 2950 y Fi(l)p Fj(+1)2200 2940 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)2415 2949 y Fi(n)2460 2940 y Fn(y)2536 2808 y Fq(d)p Fo(s;)50 3170 y Fq(where)554 3334 y Fo(s)597 3348 y Fn(cr)669 3334 y Fq(\()p Fo(y)s Fq(\))26 b Fp(\021)909 3261 y Ff(\000)951 3334 y Fo(y)d Fp(\000)1110 3227 y Ff(q)p 1201 3227 1057 4 v 107 x Fq(\(ln)g(\026)-54 b Fo(\013)21 b Fp(\000)f Fo(\016)1540 3303 y Fg(0)1537 3362 y Fn(l)1566 3334 y Fq(\))p Fo(N)10 b Fq(\()p Fo(a)1767 3348 y Fn(l)1816 3334 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2172 3348 y Fn(n)2221 3334 y Fq(\))2257 3261 y Ff(\001)2298 3259 y(p)p 2389 3259 594 4 v 75 x Fo(a)2437 3348 y Fn(l)p Ft(+1)2577 3334 y Fq(+)g Fp(\001)15 b(\001)g(\001)22 b Fq(+)e Fo(a)2934 3348 y Fn(n)2983 3334 y Fo(=)3028 3271 y Fp(p)p 3104 3271 76 4 v 63 x Fo(a)3152 3348 y Fn(l)3180 3334 y Fo(:)272 b Fq(\(2)p Fo(:)p Fq(10\))50 3562 y(This)28 b(b)s(ound)h(has)h(the)g(asymptotics)h (stated)g(in)e(\(2.5\),)k(uniformly)27 b(in)50 3707 y Fo(y)h Fp(2)d Fq([)234 3630 y Ff(p)p 325 3630 1055 4 v 77 x Fq(\(ln)e(\026)-54 b Fo(\013)21 b Fp(\000)f Fo(\016)661 3721 y Fn(l)689 3707 y Fq(\))p Fo(N)10 b Fq(\()p Fo(a)890 3721 y Fn(l)939 3707 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1295 3721 y Fn(n)1345 3707 y Fq(\))p Fo(;)15 b Fp(1)p Fq(\))31 b(if)e(only)h Fo(\016)1900 3721 y Fn(l)1953 3707 y Fo(>)25 b Fq(0)31 b(is)e(c)m(hosen)i(to)g(b)s(e)f Fo(\016)2782 3721 y Fn(l)2836 3707 y Fo(<)25 b(\016)2975 3674 y Fg(0)2972 3733 y Fn(l)3001 3707 y Fq(.)159 3926 y(The)45 b(analysis)f(is)g(more)h(delicate)h(whenev)m(er)f(some)g(of)h (the)f(inequalities)e(\(1.6\))k(degenerate)g(in)m(to)50 4071 y(equalities.)80 b(Assume)43 b(that)i Fo(k)51 b Fq(=)c(max)p Fp(f)p Fo(i)i(>)e Fq(1)p Fo(;)15 b(a)1850 4085 y Fn(i)1911 4071 y Fq(+)29 b Fp(\001)15 b(\001)g(\001)30 b Fq(+)f Fo(a)2294 4085 y Fn(n)2391 4071 y Fq(=)48 b(ln)n(\()p Fo(\013)2678 4085 y Fn(i)2724 4071 y Fp(\001)15 b(\001)g(\001)h Fo(\013)2903 4085 y Fn(n)2953 4071 y Fq(\))p Fo(=)f Fq(ln)24 b(\026)-54 b Fo(\013)p Fp(g)p Fq(.)82 b(Then)43 b(for)50 4216 y Fo(i)29 b Fq(=)g Fo(n)21 b Fp(\000)g Fq(1)p Fo(;)15 b(n)22 b Fp(\000)g Fq(2)p Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(k)36 b Fq(\(2.5\))e(remains)e(v)-5 b(alid)30 b(in)i(the)g(resp)s (ectiv)m(e)h(domains)e(of)i Fo(y)s Fq(.)47 b(F)-8 b(or)33 b Fo(i)d Fq(=)e Fo(k)d Fp(\000)c Fq(1)33 b(w)m(e)50 4362 y(will)28 b(pro)m(v)m(e)j(the)f(follo)m(wing)f(t)m(w)m(o)j(assertions)e (\()p Fr(a)1681 4376 y Fn(k)r Fg(\000)p Ft(1)1819 4362 y Fq(\))g(and)g(\()p Fr(b)2155 4376 y Fn(k)r Fg(\000)p Ft(1)2292 4362 y Fq(\).)50 4580 y(\()p Fr(a)136 4594 y Fn(k)r Fg(\000)p Ft(1)274 4580 y Fq(\))g(F)-8 b(or)32 b(an)m(y)e Fo(C)i Fp(2)25 b Fe(R)39 b Fq(w)m(e)31 b(ha)m(v)m(e)88 4891 y Fo(P)13 b Fq(\()p Fo(k)24 b Fp(\000)19 b Fq(1)p Fo(;)c(y)s Fq(\))27 b Fp(\030)657 4829 y Fq(\()p Fo(\013)750 4843 y Fn(k)810 4829 y Fp(\001)15 b(\001)g(\001)h Fo(\013)989 4843 y Fn(n)1039 4829 y Fq(\))1074 4796 y Fn(N)1143 4829 y Fo(e)1185 4796 y Fg(\000)p Fn(y)1281 4769 y Fj(2)1317 4796 y Fn(=)p Ft(2)p 657 4870 739 4 v 914 4888 a Fp(p)p 989 4888 101 4 v 989 4963 a Fq(2)p Fo(\031)t(y)545 5154 y Fp(\002)k Fe(P)691 5053 y Ff(\020)745 5154 y Fo(Z)807 5168 y Fn(k)r Fg(\000)p Ft(1)971 5154 y Fo(>)1066 5080 y Ff(\002)1104 5154 y Fo(y)j Fp(\000)1263 5072 y Ff(p)p 1354 5072 1031 4 v 82 x Fq(2\(ln)h(\026)-54 b Fo(\013)p Fq(\)\()p Fo(a)1702 5168 y Fn(k)r Fg(\000)p Ft(1)1861 5154 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2217 5168 y Fn(n)2267 5154 y Fq(\))p Fo(N)2385 5080 y Ff(\003)2423 5025 y(r)p 2514 5025 541 4 v 2525 5092 a Fo(a)2573 5106 y Fn(k)2638 5092 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2994 5106 y Fn(n)p 2525 5133 519 4 v 2691 5216 a Fo(a)2739 5230 y Fn(k)r Fg(\000)p Ft(1)3054 5053 y Ff(\021)3230 5154 y Fq(as)91 b Fo(N)35 b Fp(")26 b(1)3477 5316 y Fq(\(2)p Fo(:)p Fq(11\))50 5462 y(uniformly)h(in)i Fo(y)f Fp(2)d Fq([)754 5384 y Ff(p)p 845 5384 976 4 v 78 x Fq(2)15 b(ln)24 b(\026)-54 b Fo(\013)p Fq(\()p Fo(a)1138 5476 y Fn(k)r Fg(\000)p Ft(1)1297 5462 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1653 5476 y Fn(n)1702 5462 y Fq(\))p Fo(N)31 b Fq(+)20 b Fo(C)q(;)15 b Fp(1)p Fq(\))31 b(.)p eop %%Page: 13 13 13 12 bop 50 295 a Fh(GREM,)28 b(p)l(art)f(1)3089 b Fk(13)50 513 y Fq(\()p Fr(b)143 527 y Fn(k)r Fg(\000)p Ft(1)281 513 y Fq(\))32 b(F)-8 b(or)33 b(an)m(y)g Fo(\017)28 b(>)g Fq(0,)33 b(there)f(exist)g(a)h(negativ)m(e)g(constan)m(t)2232 490 y(~)2212 513 y Fo(C)2277 527 y Fn(k)r Fg(\000)p Ft(1)2414 513 y Fq(\()p Fo(\017)p Fq(\))c Fp(2)f Fe(R)41 b Fq(of)32 b(large)g(enough)g(absolute)50 658 y(v)-5 b(alue,)30 b(and)g Fo(N)10 b Fq(\()p Fo(\017)p Fq(\))25 b Fp(2)g Fe(N)7 b Fq(,)37 b(suc)m(h)30 b(that,)h(for)f(all)f Fo(N)35 b Fp(\025)25 b Fo(N)10 b Fq(\()p Fo(\017)p Fq(\))31 b(and)f(all)50 804 y Fo(y)e Fp(2)d Fq(\(0)15 b Fo(;)360 726 y Ff(p)p 451 726 976 4 v 78 x Fq(2)g(ln)24 b(\026)-54 b Fo(\013)p Fq(\()p Fo(a)744 818 y Fn(k)r Fg(\000)p Ft(1)903 804 y Fq(+)19 b Fp(\001)c(\001)g(\001)22 b Fq(+)e Fo(a)1259 818 y Fn(n)1308 804 y Fq(\))p Fo(N)31 b Fq(+)1558 781 y(~)1538 804 y Fo(C)1603 818 y Fn(k)r Fg(\000)p Ft(1)1740 804 y Fq(\()p Fo(\017)p Fq(\))1847 703 y Ff(i)1190 1136 y Fo(P)13 b Fq(\()p Fo(k)23 b Fp(\000)d Fq(1)p Fo(;)15 b(y)s Fq(\))27 b Fp(\024)1759 1074 y Fo(\017)p Fq(\()p Fo(\013)1889 1088 y Fn(k)1949 1074 y Fp(\001)15 b(\001)g(\001)h Fo(\013)2128 1088 y Fn(n)2178 1074 y Fq(\))2213 1041 y Fn(N)2282 1074 y Fo(e)2324 1041 y Fg(\000)p Fn(y)2420 1014 y Fj(2)2456 1041 y Fn(=)p Ft(2)p 1759 1115 776 4 v 2034 1133 a Fp(p)p 2110 1133 101 4 v 75 x Fq(2)p Fo(\031)s(y)2545 1136 y(:)907 b Fq(\(2)p Fo(:)p Fq(12\))50 1416 y(The)24 b(pro)s(of)f(is)h(based)g(again)g(on)g(\(2.4\))i(for)e Fo(i)i Fq(=)f Fo(k)11 b Fp(\000)d Fq(1.)39 b(First)24 b(of)g(all,)h(let)f(us)g(remark)g(that)h(the)g(asymptotics)50 1562 y(of)h Fo(P)13 b Fq(\()p Fo(k)s(;)i(y)s Fq(\))26 b(is)f(determined)f(b)m(y)i(\(2.5\))h(uniformly)c(in)i(the)g(in)m(terv) -5 b(al)25 b(of)h Fo(y)i Fp(2)d Fq([)p Fo(u)2667 1578 y Ft(ln)q(\()p Fn(\013)2803 1588 y Fi(k)2843 1578 y Fg(\001\001\001)o Fn(\013)2955 1587 y Fi(n)3000 1578 y Ft(\))p Fn(;N)3119 1562 y Fq(\()p Fo(x)p Fq(\))p Fo(;)15 b Fp(1)p Fq(\))27 b(for)e(an)m(y)50 1707 y Fo(x)g Fp(2)g Fe(R)s Fq(.)46 b(This)29 b(is)g(ensured)g(b)m(y)i(the)f(equalit)m(y)g(\(1.1\))i(for)e Fo(i)c Fq(=)f Fo(k)s Fq(,)31 b(since)e(consequen)m(tly)663 1965 y Fo(u)715 1982 y Ft(ln)q(\()p Fn(\013)851 1992 y Fi(k)891 1982 y Fn(:::)o(\013)1003 1991 y Fi(n)1048 1982 y Ft(\))p Fn(;N)1167 1965 y Fq(\()p Fo(x)p Fq(\))d(=)1411 1883 y Ff(p)p 1502 1883 938 4 v 82 x Fq(2\(ln)d(\026)-54 b Fo(\013)q Fq(\))p Fo(N)10 b Fq(\()p Fo(a)1933 1979 y Fn(k)1998 1965 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)22 b Fq(+)d Fo(a)2354 1979 y Fn(n)2404 1965 y Fq(\))i(+)e Fo(o)p Fq(\(1\))93 b Fo(N)35 b Fp(")25 b(1)p Fo(:)381 b Fq(\(2)p Fo(:)p Fq(13\))50 2224 y(Therefore)30 b(for)g(an)m(y)h Fo(x)25 b Fp(2)g Fe(R)340 2500 y Fo(\013)398 2463 y Fn(N)398 2523 y(k)467 2500 y Fo(P)538 2399 y Ff(\020)592 2500 y Fo(k)s(;)694 2371 y Fp(p)p 770 2371 612 4 v 62 x Fo(a)818 2447 y Fn(k)r Fg(\000)p Ft(1)976 2433 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1332 2447 y Fn(n)1381 2433 y Fo(y)j Fp(\000)1540 2378 y(p)p 1616 2378 186 4 v 55 x Fo(a)1664 2447 y Fn(k)r Fg(\000)p Ft(1)1802 2433 y Fo(t)p 694 2479 1141 4 v 967 2498 a Fp(p)p 1043 2498 519 4 v 65 x Fo(a)1091 2577 y Fn(k)1156 2563 y Fq(+)d Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1512 2577 y Fn(n)1846 2399 y Ff(\021)365 2759 y Fp(\030)625 2698 y Fq(\()p Fo(\013)718 2712 y Fn(k)779 2698 y Fp(\001)15 b(\001)g(\001)h Fo(\013)958 2712 y Fn(n)1008 2698 y Fq(\))1043 2665 y Fn(N)1112 2632 y Fp(p)p 1188 2632 V 66 x Fo(a)1236 2712 y Fn(k)1301 2698 y Fq(+)j Fp(\001)c(\001)g(\001)22 b Fq(+)e Fo(a)1657 2712 y Fn(n)p 472 2738 1388 4 v 472 2757 a Fp(p)p 548 2757 101 4 v 75 x Fq(2)p Fo(\031)s Fq(\()683 2770 y Fp(p)p 759 2770 612 4 v 62 x Fo(a)807 2846 y Fn(k)r Fg(\000)p Ft(1)965 2832 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1321 2846 y Fn(n)1371 2832 y Fo(y)j Fp(\000)1530 2777 y(p)p 1606 2777 186 4 v 55 x Fo(a)1654 2846 y Fn(k)r Fg(\000)p Ft(1)1791 2832 y Fo(t)p Fq(\))1871 2759 y(7)p Fo(e)1958 2719 y Fg(\000)p Ft(\()2043 2662 y Fp(p)p 2119 2662 420 4 v 57 x Fn(a)2158 2729 y Fi(k)q Fd(\000)p Fj(1)2279 2719 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)2494 2728 y Fi(n)2539 2719 y Fn(y)r Fg(\000)2635 2682 y(p)p 2695 2682 160 4 v 37 x Fn(a)2734 2729 y Fi(k)q Fd(\000)p Fj(1)2854 2719 y Fn(t)p Ft(\))2909 2692 y Fj(2)2945 2719 y Fn(=)p Ft(2\()p Fn(a)3085 2729 y Fi(k)3127 2719 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)3342 2728 y Fi(n)3387 2719 y Ft(\))365 3066 y Fp(\024)472 3004 y Fq(\()p Fo(\013)565 3018 y Fn(k)625 3004 y Fp(\001)15 b(\001)g(\001)h Fo(\013)804 3018 y Fn(n)854 3004 y Fq(\))889 2971 y Fn(N)958 3004 y Fo(e)1000 2957 y Fg(\000)p Fn(u)1100 2930 y Fj(2)1100 2979 y(ln\()p Fi(\013)1220 2994 y(k)1259 2979 y Fd(\001\001\001)o Fi(\013)1363 2988 y(n)1407 2979 y Fj(\))p Fi(;N)1511 2957 y Ft(\()p Fn(x)p Ft(\))p Fn(=)p Ft(2)p 472 3045 1216 4 v 678 3063 a Fp(p)p 754 3063 101 4 v 75 x Fq(2)p Fo(\031)t(u)907 3155 y Ft(ln\()p Fn(\013)1042 3165 y Fi(k)1083 3155 y Fg(\001\001\001)n Fn(\013)1194 3164 y Fi(n)1239 3155 y Ft(\))p Fn(;N)1358 3138 y Fq(\()p Fo(x)p Fq(\))1723 3066 y Fp(\030)25 b Fo(e)1861 3028 y Fg(\000)p Fn(x)1964 3066 y Fo(;)107 b(N)35 b Fp(")26 b(1)50 3356 y Fq(uniformly)h(for)j Fo(y)s(;)15 b(t)31 b Fq(suc)m(h)f(that)923 3503 y Fp(p)p 999 3503 612 4 v 61 x Fo(a)1047 3578 y Fn(k)r Fg(\000)p Ft(1)1205 3564 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1561 3578 y Fn(n)1610 3564 y Fo(y)j Fp(\000)1769 3510 y(p)p 1845 3510 186 4 v 54 x Fo(a)1893 3578 y Fn(k)r Fg(\000)p Ft(1)2031 3564 y Fo(t)p 923 3611 1141 4 v 1196 3630 a Fp(p)p 1272 3630 519 4 v 65 x Fo(a)1320 3709 y Fn(k)1385 3695 y Fq(+)d Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1741 3709 y Fn(n)2100 3632 y Fp(\025)25 b Fo(u)2248 3649 y Ft(ln)q(\()p Fn(\013)2384 3659 y Fi(k)2424 3649 y Fg(\001\001\001)o Fn(\013)2536 3658 y Fi(n)2581 3649 y Ft(\))p Fn(;N)2700 3632 y Fq(\()p Fo(x)p Fq(\))p Fo(:)630 b Fq(\(2)p Fo(:)p Fq(14\))50 3907 y(Then,)25 b(giv)m(en)g(an)g(arbitrary)f Fo(\017)h(>)g Fq(0,)i(using)c(the)i(inequalities)e(\(2.8\),)28 b(one)d(can)h(c)m(ho)s(ose)g(a)f(su\016cien)m(tly)f(large)50 4052 y Fo(x)102 4066 y Ft(0)168 4052 y Fq(=)h Fo(x)316 4066 y Ft(0)357 4052 y Fq(\()p Fo(\017)p Fq(\))g Fp(2)g Fe(R)39 b Fq(and)30 b Fo(N)10 b Fq(\()p Fo(\017;)15 b(x)1095 4066 y Ft(0)1136 4052 y Fq(\))31 b(to)g(b)s(ound)d(the)j(in)m(tegrand)f (in)f(\(2.4\))j(for)e Fo(i)c Fq(=)f Fo(k)e Fp(\000)d Fq(1)31 b(as)f(follo)m(ws:)114 4335 y(\(1)21 b Fp(\000)f Fo(\017)p Fq(\))543 4274 y(\()p Fo(\013)636 4288 y Fn(k)696 4274 y Fp(\001)15 b(\001)g(\001)h Fo(\013)875 4288 y Fn(n)925 4274 y Fq(\))960 4241 y Fn(N)1029 4208 y Fp(p)p 1105 4208 V 66 x Fo(a)1153 4288 y Fn(k)1218 4274 y Fq(+)k Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1574 4288 y Fn(n)p 389 4314 1388 4 v 389 4332 a Fp(p)p 465 4332 101 4 v 76 x Fq(2)p Fo(\031)t Fq(\()601 4346 y Fp(p)p 677 4346 612 4 v 62 x Fo(a)725 4422 y Fn(k)r Fg(\000)p Ft(1)883 4408 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1239 4422 y Fn(n)1288 4408 y Fo(y)j Fp(\000)1447 4353 y(p)p 1523 4353 186 4 v 55 x Fo(a)1571 4422 y Fn(k)r Fg(\000)p Ft(1)1709 4408 y Fo(t)p Fq(\))1788 4335 y Fo(e)1830 4295 y Fg(\000)p Ft(\()1915 4238 y Fp(p)p 1991 4238 420 4 v 57 x Fn(a)2030 4305 y Fi(k)q Fd(\000)p Fj(1)2151 4295 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)2366 4304 y Fi(n)2411 4295 y Fn(y)r Fg(\000)2507 4257 y(p)p 2567 4257 160 4 v 38 x Fn(a)2606 4305 y Fi(k)q Fd(\000)p Fj(1)2726 4295 y Fn(t)p Ft(\))2781 4268 y Fj(2)2817 4295 y Fn(=)p Ft(2\()p Fn(a)2957 4305 y Fi(k)2999 4295 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)3214 4304 y Fi(n)3259 4295 y Ft(\))139 4617 y Fp(\024)i Fq(1)c Fp(\000)392 4516 y Ff(h)435 4617 y Fq(1)g Fp(\000)e Fo(P)662 4516 y Ff(\020)717 4617 y Fo(k)s(;)818 4487 y Fp(p)p 894 4487 612 4 v 62 x Fo(a)942 4563 y Fn(k)r Fg(\000)p Ft(1)1100 4549 y Fq(+)h Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1456 4563 y Fn(n)1506 4549 y Fo(y)j Fp(\000)1665 4494 y(p)p 1740 4494 186 4 v 1740 4549 a Fo(a)1788 4563 y Fn(k)r Fg(\000)p Ft(1)1926 4549 y Fo(t)p 818 4596 1141 4 v 1091 4614 a Fp(p)p 1167 4614 519 4 v 66 x Fo(a)1215 4694 y Fn(k)1280 4680 y Fq(+)d Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1636 4694 y Fn(n)1970 4516 y Ff(\021i)2067 4535 y Fn(\013)2114 4507 y Fi(N)2114 4554 y(k)139 4876 y Fp(\024)25 b Fq(\(1)c(+)f Fo(\017)p Fq(\))664 4814 y(\()p Fo(\013)757 4828 y Fn(k)817 4814 y Fp(\001)15 b(\001)g(\001)i Fo(\013)997 4828 y Fn(n)1046 4814 y Fq(\))1081 4781 y Fn(N)1150 4749 y Fp(p)p 1226 4749 V 65 x Fo(a)1274 4828 y Fn(k)1339 4814 y Fq(+)j Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1695 4828 y Fn(n)p 511 4855 1388 4 v 511 4873 a Fp(p)p 586 4873 101 4 v 586 4948 a Fq(2)p Fo(\031)t Fq(\()722 4886 y Fp(p)p 798 4886 612 4 v 62 x Fo(a)846 4962 y Fn(k)r Fg(\000)p Ft(1)1004 4948 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1360 4962 y Fn(n)1410 4948 y Fo(y)j Fp(\000)1569 4893 y(p)p 1644 4893 186 4 v 1644 4948 a Fo(a)1692 4962 y Fn(k)r Fg(\000)p Ft(1)1830 4948 y Fo(t)p Fq(\))1909 4876 y Fo(e)1951 4836 y Fg(\000)p Ft(\()2036 4779 y Fp(p)p 2113 4779 420 4 v 2113 4836 a Fn(a)2152 4846 y Fi(k)q Fd(\000)p Fj(1)2272 4836 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)2487 4845 y Fi(n)2532 4836 y Fn(y)r Fg(\000)2628 4798 y(p)p 2688 4798 160 4 v 38 x Fn(a)2727 4846 y Fi(k)q Fd(\000)p Fj(1)2847 4836 y Fn(t)p Ft(\))2902 4808 y Fj(2)2939 4836 y Fn(=)p Ft(2\()p Fn(a)3079 4846 y Fi(k)3120 4836 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)3335 4845 y Fi(n)3380 4836 y Ft(\))3477 4612 y Fq(\(2)p Fo(:)p Fq(15\))50 5171 y(for)49 b(all)f Fo(N)67 b Fp(\025)57 b Fo(N)10 b Fq(\()p Fo(\017;)15 b(x)868 5185 y Ft(0)909 5171 y Fq(\))50 b(and)e(all)h Fo(y)s(;)15 b(t)49 b Fq(under)f(condition)g(\(2.14\),)56 b(that)50 b(is)f(for)g Fo(t)g Fq(up)f(to)i Fo(t)3391 5185 y Fn(cr)3464 5171 y Fq(\()p Fo(y)s Fq(\))57 b Fp(\021)50 5316 y Fq(\()p Fo(y)133 5255 y Fp(p)p 209 5255 612 4 v 61 x Fo(a)257 5330 y Fn(k)r Fg(\000)p Ft(1)415 5316 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)771 5330 y Fn(n)835 5316 y Fp(\000)14 b Fo(u)972 5333 y Ft(ln)q(\()p Fn(\013)1108 5343 y Fi(k)1148 5333 y Fg(\001\001\001)o Fn(\013)1260 5342 y Fi(n)1305 5333 y Ft(\))p Fn(;N)1424 5316 y Fq(\()p Fo(x)1511 5330 y Ft(0)1552 5316 y Fq(\))1587 5251 y Fp(p)p 1663 5251 519 4 v 65 x Fo(a)1711 5330 y Fn(k)1776 5316 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2132 5330 y Fn(n)2182 5316 y Fq(\))p Fo(=)2262 5261 y Fp(p)p 2338 5261 186 4 v 55 x Fo(a)2386 5330 y Fn(k)r Fg(\000)p Ft(1)2524 5316 y Fq(.)40 b(After)27 b(making)g(this)g(c)m(hoice,)i(let) 50 5462 y(us)j(decomp)s(ose)i(the)f(in)m(tegral)g(\(2.4\))i(for)e Fo(i)e Fq(=)e Fo(k)d Fp(\000)c Fq(1)33 b(in)m(to)h(t)m(w)m(o)g(terms)f Fo(I)2549 5476 y Ft(1)2623 5462 y Fq(and)g Fo(I)2843 5476 y Ft(2)2884 5462 y Fq(:)46 b(in)32 b(the)h(\014rst)g(one)h(the)p eop %%Page: 14 14 14 13 bop 50 295 a Fk(14)3270 b Fh(Se)l(ction)27 b Fk(2)50 513 y Fq(in)m(tegration)h(is)g(from)g Fp(\0001)g Fq(to)i Fo(t)1146 527 y Fn(cr)1218 513 y Fq(\()p Fo(y)s Fq(\))f(and)f(in)g(the) h(second)f(from)g Fo(t)2337 527 y Fn(cr)2410 513 y Fq(\()p Fo(y)s Fq(\))h(to)g(+)p Fp(1)p Fq(.)40 b(In)28 b(view)g(of)g(\(2.15\),) k(for)50 658 y(all)d Fo(N)35 b Fp(\025)25 b Fo(N)10 b Fq(\()p Fo(\017;)15 b(x)627 672 y Ft(0)669 658 y Fq(\),)31 b Fo(I)800 672 y Ft(1)871 658 y Fq(is)e(b)s(ounded)f(from)i(b)s(elo)m (w)g(and)f(ab)s(o)m(v)m(e)j(b)m(y)120 854 y(1)p 93 895 101 4 v 93 978 a(2)p Fo(\031)219 792 y Ff(Z)310 814 y Fn(t)337 823 y Fi(cr)404 814 y Ft(\()p Fn(y)r Ft(\))270 998 y Fg(\0001)535 854 y Fq(\()p Fo(\013)628 868 y Fn(k)689 854 y Fp(\001)15 b(\001)g(\001)h Fo(\013)868 868 y Fn(n)917 854 y Fq(\))952 821 y Fn(N)1022 788 y Fp(p)p 1097 788 519 4 v 1097 854 a Fo(a)1145 868 y Fn(k)1210 854 y Fq(+)k Fp(\001)15 b(\001)g(\001)22 b Fq(+)d Fo(a)1566 868 y Fn(n)p 470 895 1212 4 v 470 978 a Fq(\()505 916 y Fp(p)p 581 916 612 4 v 62 x Fo(a)629 992 y Fn(k)r Fg(\000)p Ft(1)787 978 y Fq(+)h Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1143 992 y Fn(n)1193 978 y Fo(y)j Fp(\000)1352 923 y(p)p 1427 923 186 4 v 1427 978 a Fo(a)1475 992 y Fn(k)r Fg(\000)p Ft(1)1613 978 y Fo(t)p Fq(\))1692 915 y Fo(e)1734 875 y Fg(\000)p Ft(\()1819 818 y Fp(p)p 1896 818 420 4 v 1896 875 a Fn(a)1935 885 y Fi(k)q Fd(\000)p Fj(1)2055 875 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)2270 884 y Fi(n)2315 875 y Fn(y)r Fg(\000)2411 838 y(p)p 2471 838 160 4 v 37 x Fn(a)2510 885 y Fi(k)q Fd(\000)p Fj(1)2630 875 y Fn(t)p Ft(\))2685 848 y Fj(2)2722 875 y Fn(=)p Ft(2\()p Fn(a)2862 885 y Fi(k)2903 875 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)3118 884 y Fi(n)3163 875 y Ft(\))p Fg(\000)p Fn(t)3275 848 y Fj(2)3311 875 y Fn(=)p Ft(2)3405 915 y Fq(d)o Fo(t)107 1225 y Fq(=)214 1164 y(\()p Fo(\013)307 1178 y Fn(k)367 1164 y Fp(\001)15 b(\001)g(\001)i Fo(\013)547 1178 y Fn(n)596 1164 y Fq(\))631 1131 y Fn(N)700 1164 y Fo(e)742 1131 y Fg(\000)p Fn(y)838 1103 y Fj(2)874 1131 y Fn(=)p Ft(2)p 214 1204 739 4 v 509 1288 a Fq(2)p Fo(\031)s(y)978 1101 y Ff(Z)1069 1050 y(\000)1111 1124 y Fn(y)1150 1076 y Fg(p)p 1210 1076 340 4 v 48 x Fn(a)1249 1134 y Fi(k)1289 1124 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)1505 1133 y Fi(n)1549 1124 y Fg(\000)p Fn(u)1649 1136 y Fj(ln\()p Fi(\013)1769 1152 y(k)1807 1136 y Fd(\001\001\001)p Fi(\013)1912 1145 y(n)1956 1136 y Fj(\))p Fi(;N)2060 1124 y Ft(\()p Fn(x)2129 1133 y Fj(0)2165 1124 y Ft(\))2193 1067 y Fp(p)p 2269 1067 420 4 v 57 x Fn(a)2308 1134 y Fi(k)q Fd(\000)p Fj(1)2429 1124 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)2644 1133 y Fi(n)2689 1050 y Ff(\001)2731 1124 y Fn(=)2768 1086 y Fg(p)p 2828 1086 160 4 v 38 x Fn(a)2867 1134 y Fi(k)q Fd(\000)p Fj(1)1029 1308 y Fg(\0001)3127 1164 y Fo(e)3169 1131 y Fg(\000)p Fn(s)3260 1103 y Fj(2)3296 1131 y Fn(=)p Ft(2)p 2942 1204 617 4 v 2942 1310 a Fq(1)j Fp(\000)3202 1221 y Fg(p)p 3262 1221 160 4 v 38 x Fn(a)3301 1269 y Fi(k)q Fd(\000)p Fj(1)3421 1259 y Fn(s)p 3109 1289 439 4 v 3109 1299 a Fg(p)p 3169 1299 340 4 v 47 x Fn(a)3208 1356 y Fi(k)3248 1346 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)3464 1355 y Fi(n)3509 1346 y Fn(y)3585 1225 y Fq(d)o Fo(s)50 1513 y Fq(times)28 b(the)g(factors)i(\(1)17 b Fp(\000)e Fo(\017)p Fq(\))29 b(and)f(\(1)17 b(+)f Fo(\017)p Fq(\))28 b(resp)s(ectiv)m(ely)-8 b(.)40 b(Com)m(bined)27 b(with)g(the)h(observ)-5 b(ation)29 b(\(2.13\),)i(this)50 1659 y(in)m(tegral)25 b(yields)f(the)i(asymptotics)g(stated)h(in)d(\()p Fr(a)1735 1673 y Fn(k)r Fg(\000)p Ft(1)1872 1659 y Fq(\).)40 b(Moreo)m(v)m(er)28 b(for)d(an)m(y)i Fo(\017)e(>)g Fq(0)h(one)g(can)g (\014nd)3420 1636 y(~)3399 1659 y Fo(C)3464 1673 y Fn(k)r Fg(\000)p Ft(1)3602 1659 y Fq(\()p Fo(\017)p Fq(\))50 1804 y(and)32 b Fo(N)10 b Fq(\()p Fo(\017)p Fq(\))32 b(suc)m(h)g(that)h(for)f(all)g Fo(y)f Fp(2)d Fq(\(0)p Fo(;)1412 1726 y Ff(p)p 1504 1726 1031 4 v 1504 1804 a Fq(2\(ln)c(\026)-54 b Fo(\013)p Fq(\))p Fo(N)10 b Fq(\()p Fo(a)1935 1818 y Fn(k)r Fg(\000)p Ft(1)2094 1804 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2450 1818 y Fn(n)2500 1804 y Fq(\))i(+)2669 1781 y(~)2649 1804 y Fo(C)2714 1818 y Fn(k)r Fg(\000)p Ft(1)2852 1804 y Fq(\()p Fo(\017)p Fq(\)])33 b(and)f Fo(N)38 b Fp(\025)28 b Fo(N)10 b Fq(\()p Fo(\017)p Fq(\))33 b Fo(I)3669 1818 y Ft(1)50 1950 y Fq(ob)s(eys)d(the)g(b)s(ound)f(of)h(\()p Fr(b)937 1964 y Fn(k)r Fg(\000)p Ft(1)1074 1950 y Fq(\).)159 2168 y(The)40 b(term)g Fo(I)623 2182 y Ft(2)705 2168 y Fq(brings)e(no)j(signi\014can)m(t)e(con)m(tribution)g(to)i(the)g (asymptotics)g(of)f(\()p Fr(a)3096 2182 y Fn(k)r Fg(\000)p Ft(1)3234 2168 y Fq(\))h(as)g(it)f(v)m(eri-)50 2313 y(\014es)f(the)h(b) s(ound)e(\(2.12\))k(uniformly)37 b(on)j(the)g(whole)f(in)m(terv)-5 b(al)39 b Fo(y)44 b Fp(2)c Fq(\(0)p Fo(;)15 b Fp(1)p Fq(\).)71 b(T)-8 b(o)40 b(sho)m(w)g(this,)h(w)m(e)g(sim-)50 2459 y(ply)e(b)s(ound)g(its)h(in)m(tegrand)h(from)f(ab)s(o)m(v)m(e)i(b) m(y)f Fo(e)1732 2426 y Fg(\000)p Fn(t)1816 2398 y Fj(2)1852 2426 y Fn(=)p Ft(2)1971 2459 y Fq(and)f(consider)g(separately)h(t)m(w)m (o)h(in)m(terv)-5 b(als)40 b(of)h Fo(y)s Fq(:)50 2604 y(\001)126 2618 y Ft(1)215 2604 y Fq(=)334 2503 y Ff(\020)389 2604 y Fq(0)p Fo(;)474 2494 y Ff(q)p 565 2494 1120 4 v 576 2559 a Ft(2\()p Fn(a)679 2569 y Fi(k)720 2559 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)936 2568 y Fi(n)981 2559 y Ft(\))12 b(ln)q(\()p Fn(\013)1157 2569 y Fi(k)1198 2559 y Fg(\001\001\001)n Fn(\013)1309 2568 y Fi(n)1354 2559 y Ft(\)+)1441 2542 y(~)1438 2559 y Fn(\016)1471 2569 y Fi(k)q Fd(\000)p Fj(1)p 576 2583 1015 4 v 874 2636 a Fn(a)913 2646 y Fi(k)q Fd(\000)p Fj(1)1033 2636 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)1249 2645 y Fi(n)1602 2604 y Fo(N)1685 2503 y Ff(i)1772 2604 y Fq(and)44 b(\001)2039 2618 y Ft(2)2129 2604 y Fq(=)2248 2503 y Ff(\020)2302 2494 y(q)p 2393 2494 1120 4 v 2404 2559 a Ft(2\()p Fn(a)2507 2569 y Fi(k)2548 2559 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)2763 2568 y Fi(n)2808 2559 y Ft(\))13 b(ln)q(\()p Fn(\013)2985 2569 y Fi(k)3025 2559 y Fg(\001\001\001)o Fn(\013)3137 2568 y Fi(n)3182 2559 y Ft(\)+)3269 2542 y(~)3266 2559 y Fn(\016)3299 2569 y Fi(k)q Fd(\000)p Fj(1)p 2404 2583 1015 4 v 2702 2636 a Fn(a)2741 2646 y Fi(k)q Fd(\000)p Fj(1)2861 2636 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)3077 2645 y Fi(n)3430 2604 y Fo(N)25 b(;)31 b Fp(1)p Fq(\))50 2757 y(where)314 2733 y(~)310 2757 y Fo(\016)350 2771 y Fn(k)r Fg(\000)p Ft(1)516 2757 y Fq(is)c(an)m(y)h(\014xed)f(n)m(um)m(b)s(er)f(0)g Fo(<)1490 2733 y Fq(~)1486 2757 y Fo(\016)1526 2771 y Fn(k)r Fg(\000)p Ft(1)1689 2757 y Fo(<)f(a)1833 2771 y Fn(k)r Fg(\000)p Ft(1)1971 2757 y Fq(.)40 b(The)27 b(estimate)i(of)f Fo(I)2723 2771 y Ft(2)2791 2757 y Fq(in)e(\001)2970 2771 y Ft(1)3039 2757 y Fq(is)g(trivial:)38 b(for)27 b(an)m(y)50 2903 y Fo(\017)e(>)g Fq(0)31 b(one)f(can)h(\014nd)e Fo(N)10 b Fq(\()p Fo(\017)p Fq(\))31 b(suc)m(h)f(that)h(for)f(all)f Fo(N)35 b Fp(\025)25 b Fo(N)10 b Fq(\()p Fo(\017)p Fq(\))31 b(and)f(all)f Fo(y)f Fp(2)d Fq(\001)2650 2917 y Ft(1)602 3165 y Fo(I)642 3179 y Ft(2)708 3165 y Fp(\024)815 3103 y Fo(\017)15 b Fq(ln)o(\()p Fo(\013)1036 3117 y Fn(k)1096 3103 y Fp(\001)g(\001)g(\001)h Fo(\013)1275 3117 y Fn(n)1325 3103 y Fq(\))1360 3070 y Fn(N)1429 3103 y Fo(e)1471 3070 y Fg(\000)p Fn(N)7 b Ft(\(\()p Fn(a)1688 3080 y Fi(k)1729 3070 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)1944 3079 y Fi(n)1989 3070 y Ft(\))13 b(ln)q(\()p Fn(\013)2166 3080 y Fi(k)2206 3070 y Fg(\001\001\001)o Fn(\013)2318 3079 y Fi(n)2363 3070 y Ft(\)+)2450 3054 y(~)2447 3070 y Fn(\016)2480 3080 y Fi(k)q Fd(\000)p Fj(1)2600 3070 y Ft(\))p Fn(=)p Ft(\()p Fn(a)2732 3080 y Fi(k)q Fd(\000)p Fj(1)2853 3070 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)3069 3079 y Fi(n)3114 3070 y Ft(\))p 815 3144 2332 4 v 1325 3162 a Ff(q)p 1416 3162 1221 4 v 111 x Fq(2)p Fo(\031)1527 3227 y Ft(2\()p Fn(a)1630 3237 y Fi(k)1671 3227 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)1887 3236 y Fi(n)1932 3227 y Ft(\))g(ln\()p Fn(\013)2108 3237 y Fi(k)2149 3227 y Fg(\001\001\001)o Fn(\013)2261 3236 y Fi(n)2305 3227 y Ft(\)+)2392 3210 y(~)2389 3227 y Fn(\016)2422 3237 y Fi(k)q Fd(\000)p Fj(1)p 1528 3252 1015 4 v 1825 3304 a Fn(a)1864 3314 y Fi(k)q Fd(\000)p Fj(1)1985 3304 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(a)2200 3313 y Fi(n)2553 3273 y Fo(N)708 3510 y Fp(\024)815 3448 y Fo(\017)i Fq(ln)o(\()p Fo(\013)1036 3462 y Fn(k)1096 3448 y Fp(\001)g(\001)g(\001)h Fo(\013)1275 3462 y Fn(n)1325 3448 y Fq(\))1360 3415 y Fn(N)1429 3448 y Fo(e)1471 3415 y Fg(\000)p Fn(y)1567 3388 y Fj(2)1603 3415 y Fn(=)p Ft(2)p 815 3489 867 4 v 1136 3507 a Fp(p)p 1212 3507 101 4 v 75 x Fq(2)p Fo(\031)s(y)1692 3510 y(:)3477 3328 y Fq(\(2)p Fo(:)p Fq(16\))50 3770 y(Since)291 3746 y(~)287 3770 y Fo(\016)327 3784 y Fn(k)r Fg(\000)p Ft(1)491 3770 y Fo(>)25 b Fq(0,)31 b(then)f(for)g(all)f Fo(y)f Fp(2)d Fq(\001)1395 3784 y Ft(2)1466 3770 y Fo(\034)1506 3784 y Fn(cr)1578 3770 y Fq(\()p Fo(y)s Fq(\))h Fo(>)f Fq(0.)41 b(Here)31 b(Prop)s(osition)e(3.1)i(applies:)636 4025 y Fo(I)676 4039 y Ft(2)742 4025 y Fp(\024)1027 3963 y Fq(1)p 849 4004 401 4 v 849 4022 a Fp(p)p 925 4022 101 4 v 75 x Fq(2)p Fo(\031)s(t)1058 4111 y Fn(cr)1131 4097 y Fq(\()p Fo(y)s Fq(\))1260 4025 y Fo(e)1302 3987 y Fg(\000)p Fn(t)1386 3996 y Fi(cr)1453 3987 y Ft(\()p Fn(y)r Ft(\))1548 3960 y Fj(2)1585 3987 y Fn(=)p Ft(2)1688 4025 y Fp(\024)1795 3963 y Fq(\()p Fo(\013)1888 3977 y Fn(k)1948 3963 y Fp(\001)15 b(\001)g(\001)h Fo(\013)2127 3977 y Fn(n)2177 3963 y Fq(\))2212 3930 y Fn(N)2281 3963 y Fo(C)7 b Fq(\()2392 3939 y(~)2388 3963 y Fo(\016)2428 3977 y Fn(k)r Fg(\000)p Ft(1)2566 3963 y Fq(\))p 1795 4004 807 4 v 2086 4022 a Fp(p)p 2162 4022 101 4 v 75 x Fq(2)p Fo(\031)t(y)2613 4025 y(e)2655 3987 y Fg(\000)p Fn(y)2751 3960 y Fj(2)2786 3987 y Fn(=)p Ft(2)2864 4025 y Fo(e)2906 3987 y Fn(f)g Ft(\()p Fn(y)r Ft(\))p Fn(=)p Ft(2)3477 4025 y Fq(\(2)p Fo(:)p Fq(17\))50 4269 y(where)93 4433 y Fo(f)j Fq(\()p Fo(y)s Fq(\))25 b(=)387 4359 y Ff(\000)449 4433 y Fp(\000)20 b Fo(y)588 4395 y Ft(2)629 4433 y Fq(\()p Fo(a)712 4447 y Fn(k)777 4433 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1133 4447 y Fn(n)1182 4433 y Fq(\))h Fp(\000)f Fo(u)1381 4395 y Ft(2)1381 4458 y(ln)q(\()p Fn(\013)1517 4468 y Fi(k)1557 4458 y Fg(\001\001\001)o Fn(\013)1669 4467 y Fi(n)1714 4458 y Ft(\))p Fn(;N)1833 4433 y Fq(\()p Fo(x)1920 4447 y Ft(0)1961 4433 y Fq(\)\()p Fo(a)2079 4447 y Fn(k)2144 4433 y Fq(+)g Fp(\001)15 b(\001)g(\001)22 b Fq(+)e Fo(a)2501 4447 y Fn(n)2550 4433 y Fq(\))286 4606 y(+)g(2)p Fo(u)474 4622 y Ft(ln)q(\()p Fn(\013)610 4632 y Fi(k)651 4622 y Fg(\001\001\001)n Fn(\013)762 4631 y Fi(n)807 4622 y Ft(\))p Fn(;N)926 4606 y Fq(\()p Fo(x)1013 4620 y Ft(0)1054 4606 y Fq(\))p Fo(y)1137 4530 y Ff(p)p 1228 4530 612 4 v 76 x Fo(a)1276 4620 y Fn(k)r Fg(\000)p 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Fo(a)2280 5028 y Fn(n)2329 5014 y Fo(=)2374 4944 y Fp(p)p 2451 4944 519 4 v 2451 5014 a Fo(a)2499 5028 y Fn(k)2564 5014 y Fq(+)f Fp(\001)c(\001)g(\001)22 b Fq(+)e Fo(a)2920 5028 y Fn(n)50 5236 y Fq(where)30 b(it)g(equals)210 5458 y Fo(u)262 5420 y Ft(2)262 5482 y(ln)q(\()p Fn(\013)398 5492 y Fi(k)438 5482 y Fg(\001\001\001)o Fn(\013)550 5491 y Fi(n)595 5482 y Ft(\))p Fn(;N)714 5458 y Fq(\()p Fo(x)801 5472 y Ft(0)842 5458 y Fq(\))20 b Fp(\000)g Fq(2)p Fo(N)26 b Fq(ln)n(\()p Fo(\013)1300 5472 y Fn(k)1361 5458 y Fp(\001)15 b(\001)g(\001)h Fo(\013)1540 5472 y Fn(n)1590 5458 y Fq(\))25 b(=)g Fp(\000)15 b Fq(ln)f Fo(N)30 b Fp(\000)20 b Fq(ln)14 b(ln)o(\()p Fo(\013)2377 5472 y Fn(k)2437 5458 y Fp(\001)h(\001)g(\001)h Fo(\013)2616 5472 y Fn(n)2666 5458 y Fq(\))21 b Fp(\000)e Fq(ln)14 b(4)p Fo(\031)24 b Fq(+)c(2)p Fo(x)3212 5472 y Ft(0)3273 5458 y Fq(+)g Fo(o)p Fq(\(1\))p Fo(:)p eop %%Page: 15 15 15 14 bop 50 295 a Fh(GREM,)28 b(p)l(art)f(1)3089 b Fk(15)50 513 y Fq(Hence,)28 b Fo(e)384 480 y Fn(f)7 b Ft(\()p Fn(y)r Ft(\))p Fn(=)p Ft(2)626 513 y Fq(b)s(eing)25 b(of)h(the)g(order) f(at)i(most)f Fo(N)1759 480 y Fg(\000)p Ft(1)p Fn(=)p Ft(2)1930 513 y Fq(,)h Fo(I)2022 527 y Ft(2)2089 513 y Fq(do)s(es)e(not)h(con)m(tribute)g(to)g(the)h(asymptotics)f(of)50 658 y(\()p Fr(a)136 672 y Fn(k)r Fg(\000)p Ft(1)274 658 y Fq(\))k(and)g(satis\014es)f(the)i(b)s(ound)d(of)i(\()p Fr(b)1483 672 y Fn(k)r Fg(\000)p Ft(1)1621 658 y Fq(\))h(for)e(all)g Fo(y)g Fp(2)24 b Fq(\001)2186 672 y Ft(2)2257 658 y Fq(as)30 b(w)m(ell)f(when)h Fo(N)40 b Fq(is)29 b(bigger)g(some)i Fo(N)10 b Fq(\()p Fo(\017)p Fq(\).)50 804 y(This)28 b(concludes)i(the)h (pro)s(of)e(of)i(\()p Fr(a)1251 818 y Fn(k)r Fg(\000)p Ft(1)1389 804 y Fq(\))f(and)g(\()p Fr(b)1725 818 y Fn(k)r Fg(\000)p Ft(1)1862 804 y Fq(\).)159 1022 y(T)-8 b(o)29 b(con)m(tin)m(ue)g(the)g(pro)s(of)e(of)i(the)g(lemma,)g(let)f(us)g (assume)g(that)i(the)e(inequalities)e(\(1.6\))31 b(are)e(strict)f(for) 50 1167 y Fo(i)d Fq(=)g Fo(k)17 b Fp(\000)c Fq(1)p Fo(;)i(k)i Fp(\000)c Fq(2)p Fo(;)i(:)g(:)g(:)32 b(;)15 b(l)g Fp(\000)e Fq(1)27 b(and)g(\()p Fr(a)1303 1134 y Ft(1)1303 1191 y Fn(i)1344 1167 y Fq(\),)h(\()p Fr(b)1525 1181 y Fn(i)1555 1167 y Fq(\))g(\(see)f(b)s(elo)m(w\))g(hold)e(true)i(for)g Fo(i)e Fq(=)g Fo(k)17 b Fp(\000)c Fq(2)p Fo(;)i(:)g(:)g(:)32 b(;)15 b(l)29 b Fq(resp)s(ectiv)m(ely)-8 b(.)50 1313 y(Under)29 b(these)i(assumptions)e(w)m(e)i(sho)m(w)f(that)50 1531 y Fp(\017)h Fq(if)e(the)h(inequalit)m(y)f(\(1.6\))j(is)e(also)g (strict)g(for)g Fo(i)c Fq(=)f Fo(l)r Fq(,)30 b(then)g(\()p Fr(a)2186 1498 y Ft(1)2186 1557 y Fn(l)p Fg(\000)p Ft(1)2307 1531 y Fq(\))g(and)g(\()p Fr(b)2642 1545 y Fn(l)p Fg(\000)p Ft(1)2763 1531 y Fq(\))h(remain)e(true;)50 1749 y Fp(\017)i Fq(if)e(\(1.6\))j(turns)d(in)m(to)i(the)f(equalit)m(y)g(for)g Fo(i)c Fq(=)f Fo(l)r Fq(,)30 b(then)g(\()p Fr(a)2020 1716 y Ft(2)2020 1775 y Fn(l)p Fg(\000)p Ft(1)2141 1749 y Fq(\))h(and)f(\()p Fr(b)2477 1763 y Fn(l)p Fg(\000)p Ft(1)2598 1749 y Fq(\))h(tak)m(e)h(place.)159 1967 y(The)h(assertion)g (\()p Fr(b)825 1981 y Fn(i)855 1967 y Fq(\))h(is)e(the)i(same)g(as)f (\()p Fr(b)1617 1981 y Fn(k)r Fg(\000)p Ft(1)1754 1967 y Fq(\))h(ab)s(o)m(v)m(e)h(with)d Fo(k)25 b Fp(\000)d Fq(1)34 b(replaced)e(b)m(y)i Fo(i)p Fq(.)49 b(The)33 b(assertions)50 2113 y(\()p Fr(a)136 2080 y Ft(1)136 2136 y Fn(i)177 2113 y Fq(\),)e(\()p Fr(a)354 2080 y Ft(2)354 2136 y Fn(i)395 2113 y Fq(\))f(are)h(the)g(follo)m(wing:)50 2331 y(\()p Fr(a)136 2298 y Ft(1)136 2355 y Fn(i)177 2331 y Fq(\))f(F)-8 b(or)32 b(an)m(y)e Fo(C)i Fp(2)25 b Fe(R)39 b Fq(w)m(e)31 b(ha)m(v)m(e)1091 2608 y Fo(P)13 b Fq(\()p Fo(i;)i(y)s Fq(\))26 b Fp(\030)1484 2547 y Fq(\()p Fo(\013)1577 2561 y Fn(i)p Ft(+1)1715 2547 y Fp(\001)15 b(\001)g(\001)h Fo(\013)1894 2561 y Fn(n)1944 2547 y Fq(\))1979 2514 y Fn(N)2048 2547 y Fo(e)2090 2514 y Fg(\000)p Fn(y)2186 2486 y Fj(2)2222 2514 y Fn(=)p Ft(2)p 1484 2587 816 4 v 1780 2606 a Fp(p)p 1856 2606 101 4 v 75 x Fq(2)p Fo(\031)s(y)2311 2608 y Fe(P)p Fq(\()p Fo(A)2469 2571 y Fn(k)2469 2631 y(i)2515 2608 y Fq(\()p Fo(y)s Fq(\)\))50 2886 y(uniformly)27 b(in)i(the)i(in)m(terv)-5 b(al)29 b(of)i Fo(y)d Fp(2)d Fq([)1343 2808 y Ff(p)p 1434 2808 868 4 v 78 x Fq(2)15 b(ln)24 b(\026)-54 b Fo(\013)p Fq(\()p Fo(a)1727 2900 y Fn(i)1778 2886 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2134 2900 y Fn(n)2184 2886 y Fq(\))p Fo(N)30 b Fq(+)20 b Fo(C)q(;)15 b Fp(1)p Fq(\).)50 3104 y(\()p Fr(a)136 3071 y Ft(2)136 3128 y Fn(i)177 3104 y Fq(\))30 b(F)-8 b(or)32 b(an)m(y)e Fo(C)i Fp(2)25 b Fe(R)39 b Fq(w)m(e)31 b(ha)m(v)m(e)265 3382 y Fo(P)13 b Fq(\()p Fo(i;)i(y)s Fq(\))26 b Fp(\030)633 3320 y Fq(\()p Fo(\013)726 3334 y Fn(i)p Ft(+1)864 3320 y Fp(\001)15 b(\001)g(\001)h Fo(\013)1043 3334 y Fn(n)1093 3320 y Fq(\))1128 3287 y Fn(N)1197 3320 y Fo(e)1239 3287 y Fg(\000)p Fn(y)1335 3260 y Fj(2)1371 3287 y Fn(=)p Ft(2)p 633 3361 816 4 v 929 3379 a Fp(p)p 1005 3379 101 4 v 75 x Fq(2)p Fo(\031)s(y)642 3649 y Fp(\002)k Fe(P)788 3548 y Ff(\020)843 3649 y Fo(A)911 3611 y Fn(k)911 3671 y(i)956 3649 y Fq(\()p Fo(y)s Fq(\))1089 3562 y Ff(\\)1206 3548 y(h)1249 3649 y Fo(Z)1311 3663 y Fn(i)1366 3649 y Fo(>)1462 3575 y Ff(\000)1504 3649 y Fo(y)j Fp(\000)1663 3566 y Ff(p)p 1754 3566 868 4 v 83 x Fq(2)15 b(ln)23 b(\026)-54 b Fo(\013)q Fq(\()p Fo(a)2047 3663 y Fn(i)2098 3649 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2454 3663 y Fn(n)2503 3649 y Fq(\))p Fo(N)2622 3575 y Ff(\001)2663 3515 y(r)p 2754 3515 619 4 v 2765 3587 a Fo(a)2813 3601 y Fn(i)p Ft(+1)2956 3587 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)3312 3601 y Fn(n)p 2765 3628 597 4 v 3024 3711 a Fo(a)3072 3725 y Fn(i)3372 3548 y Ff(i\021)3469 3649 y Fo(;)50 3917 y Fq(uniformly)37 b(in)i(the)h(in)m(terv)-5 b(al)39 b(of)i Fo(y)j Fp(2)d Fq([)1424 3840 y Ff(p)p 1515 3840 868 4 v 77 x Fq(2)15 b(ln)24 b(\026)-54 b Fo(\013)p Fq(\()p Fo(a)1808 3931 y Fn(i)1859 3917 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2215 3931 y Fn(n)2265 3917 y Fq(\))p Fo(N)37 b Fq(+)26 b Fo(C)q(;)15 b Fp(1)p Fq(\).)71 b(The)39 b(ev)m(en)m(ts)j Fo(A)3385 3884 y Fn(k)3385 3941 y(i)3430 3917 y Fq(\()p Fo(y)s Fq(\))e(are)50 4063 y(de\014ned)29 b(as:)143 4336 y Fo(A)211 4299 y Fn(k)211 4359 y(i)255 4336 y Fq(\()p Fo(y)s Fq(\))d Fp(\021)495 4181 y Ff( )567 4336 y Fo(Z)629 4350 y Fn(k)r Fg(\000)p Ft(1)792 4336 y Fo(>)888 4181 y Ff(")941 4336 y Fo(y)989 4203 y Ff(r)p 1080 4203 634 4 v 1091 4275 a Fo(a)1139 4289 y Fn(k)r Fg(\000)p Ft(1)1297 4275 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1653 4289 y Fn(n)p 1091 4315 612 4 v 1145 4399 a Fo(a)1193 4413 y Fn(i)1243 4399 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1599 4413 y Fn(n)1734 4336 y Fp(\000)g Fo(Z)1887 4350 y Fn(i)1917 4173 y Ff(s)p 2008 4173 1168 4 v 2211 4275 a Fo(a)2259 4289 y Fn(i)2290 4275 y Fq(\()p Fo(a)2373 4289 y Fn(k)r Fg(\000)p Ft(1)2531 4275 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2887 4289 y Fn(n)2937 4275 y Fq(\))p 2019 4315 1146 4 v 2019 4399 a(\()p Fo(a)2102 4413 y Fn(i)2153 4399 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2509 4413 y Fn(n)2558 4399 y Fq(\)\()p Fo(a)2676 4413 y Fn(i)p Ft(+1)2820 4399 y Fq(+)f Fp(\001)c(\001)g(\001)i Fo(a)3080 4413 y Fn(n)3129 4399 y Fq(\))587 4657 y Fp(\000)j Fo(Z)740 4671 y Fn(i)p Ft(+1)863 4494 y Ff(s)p 954 4494 1356 4 v 1205 4596 a Fo(a)1253 4610 y Fn(i)p Ft(+1)1375 4596 y Fq(\()p Fo(a)1458 4610 y Fn(k)r Fg(\000)p Ft(1)1617 4596 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1973 4610 y Fn(n)2022 4596 y Fq(\))p 965 4636 1334 4 v 965 4719 a(\()p Fo(a)1048 4733 y Fn(i)p Ft(+1)1190 4719 y Fq(+)g Fp(\001)15 b(\001)g(\001)22 b Fq(+)e Fo(a)1547 4733 y Fn(n)1596 4719 y Fq(\)\()p Fo(a)1714 4733 y Fn(i)p Ft(+2)1857 4719 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2213 4733 y Fn(n)2263 4719 y Fq(\))2329 4657 y Fp(\000)g(\001)15 b(\001)g(\001)587 4970 y(\000)20 b Fo(Z)740 4984 y Fn(k)r Fg(\000)p Ft(3)878 4806 y Ff(s)p 969 4806 1387 4 v 1228 4908 a Fo(a)1276 4922 y Fn(k)r Fg(\000)p Ft(3)1414 4908 y Fq(\()p Fo(a)1497 4922 y Fn(k)r Fg(\000)p Ft(1)1655 4908 y Fq(+)g Fp(\001)15 b(\001)g(\001)22 b Fq(+)e Fo(a)2012 4922 y Fn(n)2061 4908 y Fq(\))p 980 4949 1365 4 v 980 5032 a(\()p Fo(a)1063 5046 y Fn(k)r Fg(\000)p Ft(3)1221 5032 y Fq(+)g Fp(\001)15 b(\001)g(\001)22 b Fq(+)d Fo(a)1577 5046 y Fn(n)1627 5032 y Fq(\)\()p Fo(a)1745 5046 y Fn(k)r Fg(\000)p Ft(2)1904 5032 y Fq(+)h Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2260 5046 y Fn(n)2309 5032 y Fq(\))587 5271 y Fp(\000)g Fo(Z)740 5285 y Fn(k)r Fg(\000)p Ft(2)878 5148 y Ff(r)p 969 5148 634 4 v 1193 5209 a Fo(a)1241 5223 y Fn(k)r Fg(\000)p Ft(2)p 980 5250 612 4 v 980 5333 a Fo(a)1028 5347 y Fn(k)r Fg(\000)p Ft(2)1186 5333 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1542 5347 y Fn(n)1623 5271 y Fp(\000)1714 5189 y Ff(p)p 1805 5189 1031 4 v 82 x Fq(2\(ln)j(\026)-54 b Fo(\013)q Fq(\)\()p Fo(a)2153 5285 y Fn(k)r Fg(\000)p Ft(1)2312 5271 y Fq(+)19 b Fp(\001)c(\001)g(\001)22 b Fq(+)e Fo(a)2668 5285 y Fn(n)2717 5271 y Fq(\))p Fo(N)2835 5115 y Ff(#)2888 5141 y(r)p 2979 5141 541 4 v 2990 5209 a Fo(a)3038 5223 y Fn(k)3103 5209 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)3459 5223 y Fn(n)p 2990 5250 519 4 v 3157 5333 a Fo(a)3205 5347 y Fn(k)r Fg(\000)p Ft(1)3520 5115 y Ff(!)3592 5271 y Fo(:)3477 5462 y Fq(\(2)p Fo(:)p Fq(18\))p eop %%Page: 16 16 16 15 bop 50 295 a Fk(16)3270 b Fh(Se)l(ction)27 b Fk(2)159 513 y Fq(By)45 b(virtue)f(of)i(\()p Fr(a)801 480 y Ft(1)801 538 y Fn(l)842 513 y Fq(\),)j(\()p Fr(b)1044 527 y Fn(l)1072 513 y Fq(\),)g(and)c(the)g(inequalities)e(\(2.8\),)50 b(for)45 b(an)m(y)h(giv)m(en)f Fo(\017)k(>)h Fq(0)45 b(there)h(exist)e(a)50 658 y(su\016cien)m(tly)24 b(large)i Fo(x)770 672 y Ft(0)836 658 y Fq(=)f Fo(x)984 672 y Ft(0)1025 658 y Fq(\()p Fo(\017)p Fq(\))h Fp(2)f Fe(R)r Fq(,)33 b(and)26 b Fo(N)10 b Fq(\()p Fo(x)1704 672 y Ft(0)1745 658 y Fo(;)15 b(\017)p Fq(\),)27 b(suc)m(h)f(that,)i(for)d(all)g Fo(N)35 b Fp(\025)25 b Fo(N)10 b Fq(\()p Fo(x)2959 672 y Ft(0)3000 658 y Fo(;)15 b(\017)p Fq(\),)28 b(all)d Fo(y)j Fp(2)d Fe(R)r Fq(,)33 b(and)50 804 y(all)e Fo(t)e Fp(\024)f Fo(t)372 818 y Fn(cr)445 804 y Fq(\()p Fo(y)s Fq(\))h Fp(\021)g Fq(\()p Fo(y)775 742 y Fp(p)p 851 742 595 4 v 62 x Fo(a)899 818 y Fn(l)p Fg(\000)p Ft(1)1040 804 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1396 818 y Fn(n)1467 804 y Fp(\000)h Fo(u)1611 820 y Ft(ln)q(\()p Fn(\013)1747 830 y Fi(l)1774 820 y Fg(\001\001\001)o Fn(\013)1886 829 y Fi(n)1931 820 y Ft(\))p Fn(;N)2050 804 y Fq(\()p Fo(x)2137 818 y Ft(0)2178 804 y Fq(\))2213 738 y Fp(p)p 2289 738 502 4 v 66 x Fo(a)2337 818 y Fn(l)2385 804 y Fq(+)f Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2741 818 y Fn(n)2791 804 y Fq(\))p Fo(=)2871 749 y Fp(p)p 2947 749 170 4 v 55 x Fo(a)2995 818 y Fn(l)p Fg(\000)p Ft(1)3116 804 y Fq(,)34 b(the)e(in)m(tegrand)50 949 y(in)39 b(\(1.6\))j(for)d Fo(i)j Fq(=)f Fo(l)29 b Fp(\000)d Fq(1)41 b(can)f(b)s(e)g(estimated)g(from)g(ab)s(o)m(v)m(e)h(and)f(b)s(elo)m(w)f (b)m(y)h Fo(\013)2839 916 y Fn(N)2839 975 y(l)2908 949 y Fo(P)2979 875 y Ff(\000)3021 949 y Fo(l)r(;)15 b Fp(\001)3115 875 y Ff(\001)3157 949 y Fo(e)3199 916 y Fg(\000)p Fn(t)3283 889 y Fj(2)3319 916 y Fn(=)p Ft(2)3397 949 y Fq(\(\(1)28 b Fp(\006)f Fo(\017)p Fq(\))50 1095 y(times.)61 b(Ha)m(ving)37 b Fo(x)718 1109 y Ft(0)796 1095 y Fq(\014xed,)h(w)m(e)g(again)f(split)f (the)h(in)m(tegral)g(\(1.6\))i(in)m(to)e(t)m(w)m(o)i(terms,)g Fo(I)3071 1109 y Ft(1)3149 1095 y Fq(and)d Fo(I)3372 1109 y Ft(2)3413 1095 y Fq(,)j(where)50 1240 y(the)i(in)m(tegration)g (is)f(from)g Fp(\0001)h Fq(to)h Fo(t)1375 1254 y Fn(cr)1447 1240 y Fq(\()p Fo(y)s Fq(\))g(and)e(from)h Fo(t)2053 1254 y Fn(cr)2125 1240 y Fq(\()p Fo(y)s Fq(\))h(to)f(+)p Fp(1)p Fq(,)j(resp)s(ectiv)m(ely)-8 b(.)72 b(By)41 b(the)g(same)50 1385 y(argumen)m(ts)30 b(as)g(in)f(\(2.17\),)j(the)e(term)g Fo(I)1409 1399 y Ft(2)1479 1385 y Fq(do)s(es)g(not)g(con)m(tribute)g (to)g(\()p Fr(a)2478 1352 y Ft(1)2478 1411 y Fn(l)p Fg(\000)p Ft(1)2599 1385 y Fq(\))h(or)e(\()p Fr(a)2862 1352 y Ft(2)2862 1411 y Fn(l)p Fg(\000)p Ft(1)2983 1385 y Fq(\),)h(and)f(satis\014es)h (the)50 1531 y(b)s(ound)e(of)j(condition)e(\()p Fr(b)926 1545 y Fn(l)p Fg(\000)p Ft(1)1047 1531 y Fq(\).)159 1749 y(T)-8 b(o)36 b(treat)h(the)f(term)g Fo(I)951 1763 y Ft(1)1028 1749 y Fq(w)m(e)g(distinguish)c(the)k(t)m(w)m(o)h(cases)g (when)e(the)h(inequalit)m(y)e(\(1.6\))k(is)c(strict)i(for)50 1895 y Fo(i)25 b Fq(=)g Fo(l)r Fq(,)31 b(and)f(when)f(equalit)m(y)h (holds,)f(resp)s(ectiv)m(ely)-8 b(.)159 2113 y(In)30 b(the)g(second)h(case,)g(the)g(assumed)e(equalit)m(y)h(en)m(tails)866 2357 y Fo(u)918 2373 y Ft(ln)q(\()p Fn(\013)1054 2383 y Fi(l)1081 2373 y Fg(\001\001\001)n Fn(\013)1192 2382 y Fi(n)1237 2373 y Ft(\))p Fn(;N)1356 2357 y Fq(\()p Fo(x)1443 2371 y Ft(0)1484 2357 y Fq(\))c(=)1641 2275 y Ff(p)p 1732 2275 866 4 v 82 x Fq(2)15 b(ln)23 b(\026)-54 b Fo(\013)q Fq(\()p Fo(a)2025 2371 y Fn(l)2073 2357 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2429 2371 y Fn(n)2479 2357 y Fq(\))p Fo(N)30 b Fq(+)20 b Fo(o)p Fq(\(1\))p Fo(:)585 b Fq(\(2)p Fo(:)p Fq(19\))50 2601 y(This)41 b(means)j(that)f(the)h(probabilit)m(y)d Fo(P)13 b Fq(\()p Fo(l)r(;)i Fp(\001)p Fq(\))44 b(en)m(tering)f(the)h(in)m (tegrand)f(of)g Fo(I)2808 2615 y Ft(1)2892 2601 y Fq(has)g(the)h (asymptotics)50 2747 y(claimed)35 b(in)f(\()p Fr(a)587 2714 y Ft(1)587 2772 y Fn(l)627 2747 y Fq(\))j(uniformly)c(in)h(the)i (in)m(terv)-5 b(al)35 b(of)i(in)m(tegration)f Fo(t)e Fp(\024)g Fo(t)2509 2761 y Fn(cr)2582 2747 y Fq(\()p Fo(y)s Fq(\))i(and)g(all)e Fo(y)s Fq(.)58 b(Then)35 b(for)g(an)m(y)50 2892 y Fo(\017)f(>)g Fq(0,)k(one)e(can)g(\014nd)f Fo(N)10 b Fq(\()p Fo(x)1036 2906 y Ft(0)1077 2892 y Fo(;)15 b(\017)p Fq(\))36 b(suc)m(h)g(that,)i(for)e(all)e Fo(N)45 b Fp(\025)34 b Fo(N)10 b Fq(\()p Fo(x)2334 2906 y Ft(0)2375 2892 y Fo(;)15 b(\017)p Fq(\),)38 b(the)e(term)g Fo(I)2975 2906 y Ft(1)3052 2892 y Fq(is)e(b)s(ounded)g(from)50 3037 y(ab)s(o)m(v)m(e)d(and)f(b)s(elo)m(w)g(b)m(y)225 3267 y(\()p Fo(\013)318 3281 y Fn(l)362 3267 y Fp(\001)15 b(\001)g(\001)h Fo(\013)541 3281 y Fn(n)591 3267 y Fq(\))626 3234 y Fn(N)695 3202 y Fp(p)p 771 3202 502 4 v 65 x Fo(a)819 3281 y Fn(l)867 3267 y Fq(+)k Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1223 3281 y Fn(n)p 225 3308 1048 4 v 699 3391 a Fq(2)p Fo(\031)1299 3205 y Ff(Z)1390 3227 y Fn(t)1417 3236 y Fi(cr)1483 3227 y Ft(\()p Fn(y)r Ft(\))1349 3412 y Fg(\0001)1610 3267 y Fo(e)1652 3227 y Fg(\000)p Ft(\()1737 3170 y Fp(p)p 1813 3170 407 4 v 57 x Fn(a)1852 3237 y Fi(l)p Fd(\000)p Fj(1)1959 3227 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)2175 3236 y Fi(n)2220 3227 y Fn(y)r Fg(\000)2316 3190 y(p)p 2375 3190 146 4 v 2375 3227 a Fn(a)2414 3237 y Fi(l)p Fd(\000)p Fj(1)2521 3227 y Fn(t)p Ft(\))2576 3200 y Fj(2)2613 3227 y Fn(=)p Ft(2\()p Fn(a)2753 3237 y Fi(l)2780 3227 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)2996 3236 y Fi(n)3041 3227 y Ft(\))p Fg(\000)p Fn(t)3153 3200 y Fj(2)3189 3227 y Fn(=)p Ft(2)p 1610 3308 1658 4 v 1885 3329 a Fp(p)p 1961 3329 595 4 v 62 x Fo(a)2009 3405 y Fn(l)p Fg(\000)p Ft(1)2150 3391 y Fq(+)g Fp(\001)15 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y Fq(\))887 3846 y Fn(N)956 3879 y Fo(e)998 3846 y Fg(\000)p Fn(y)1094 3818 y Fj(2)1130 3846 y Fn(=)p Ft(2)p 487 3920 722 4 v 773 4003 a Fq(2)p Fo(\031)s(y)1234 3817 y Ff(Z)1325 3839 y Fn(s)1359 3848 y Fi(cr)1426 3839 y Ft(\()p Fn(y)r Ft(\))1285 4023 y Fg(\0001)1731 3879 y Fo(e)1773 3846 y Fg(\000)p Fn(s)1864 3818 y Fj(2)1900 3846 y Fn(=)p Ft(2)p 1552 3920 604 4 v 1552 4025 a Fq(1)k Fp(\000)1812 3937 y Fg(p)p 1872 3937 146 4 v 37 x Fn(a)1911 3984 y Fi(l)p Fd(\000)p Fj(1)2018 3974 y Fn(s)p 1720 4004 426 4 v 1720 4014 a Fg(p)p 1780 4014 327 4 v 48 x Fn(a)1819 4072 y Fi(l)1845 4062 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)2061 4071 y Fi(n)2106 4062 y Fn(y)496 4248 y Fp(\002)f Fe(P)642 4147 y Ff(\020)697 4248 y Fo(A)765 4210 y Fn(k)765 4270 y(l)810 4147 y Ff(\020)864 4248 y Fo(y)912 4118 y Ff(r)p 1003 4118 617 4 v 1060 4186 a Fo(a)1108 4200 y Fn(l)1156 4186 y Fq(+)g Fp(\001)15 b(\001)g(\001)22 b Fq(+)d Fo(a)1512 4200 y Fn(n)p 1014 4227 595 4 v 1014 4310 a Fo(a)1062 4324 y Fn(l)p Fg(\000)p Ft(1)1203 4310 y Fq(+)h Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1559 4324 y Fn(n)1640 4248 y Fp(\000)g Fo(s)1774 4125 y Ff(r)p 1864 4125 617 4 v 2088 4186 a Fo(a)2136 4200 y Fn(l)p Fg(\000)p Ft(1)p 1875 4227 595 4 v 1875 4310 a Fo(a)1923 4324 y Fn(l)p Fg(\000)p Ft(1)2064 4310 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2420 4324 y Fn(n)2481 4147 y Ff(\021\021)2604 4248 y Fq(d)p Fo(s)p Fq(\(1)h Fp(\006)e Fo(\017)p Fq(\))2961 4210 y Ft(2)405 4528 y Fq(=)487 4466 y(\()p Fo(\013)580 4480 y Fn(l)623 4466 y Fp(\001)c(\001)g(\001)i Fo(\013)803 4480 y Fn(n)852 4466 y Fq(\))887 4433 y Fn(N)956 4466 y Fo(e)998 4433 y Fg(\000)p Fn(y)1094 4406 y Fj(2)1130 4433 y Fn(=)p Ft(2)p 487 4507 722 4 v 773 4590 a Fq(2)p Fo(\031)s(y)1234 4404 y Ff(Z)1325 4426 y Fn(s)1359 4435 y Fi(cr)1426 4426 y Ft(\()p Fn(y)r Ft(\))1285 4610 y Fg(\0001)1731 4466 y Fo(e)1773 4433 y Fg(\000)p Fn(s)1864 4406 y Fj(2)1900 4433 y Fn(=)p Ft(2)p 1552 4507 604 4 v 1552 4612 a Fq(1)k Fp(\000)1812 4524 y Fg(p)p 1872 4524 146 4 v 37 x Fn(a)1911 4571 y Fi(l)p Fd(\000)p Fj(1)2018 4561 y Fn(s)p 1720 4592 426 4 v 1720 4601 a Fg(p)p 1780 4601 327 4 v 48 x Fn(a)1819 4659 y Fi(l)1845 4649 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(a)2061 4658 y Fi(n)2106 4649 y Fn(y)2167 4528 y Fe(P)p Fq(\()p Fo(A)2325 4490 y Fn(k)2325 4550 y(l)p Fg(\000)p Ft(1)2447 4528 y Fq(\()p Fo(y)s(;)15 b(s)p Fq(\)\))g(d)p Fo(s)p Fq(\(1)21 b Fp(\006)f Fo(\017)p Fq(\))3056 4490 y Ft(2)3097 4528 y Fo(;)3477 4248 y Fq(\(2)p Fo(:)p Fq(21\))50 4807 y(where)33 b Fo(s)359 4821 y Fn(cr)431 4807 y Fq(\()p Fo(y)s Fq(\))e(=)e(\()p Fo(y)763 4742 y Fp(p)p 839 4742 502 4 v 65 x Fo(a)887 4821 y Fn(l)936 4807 y Fq(+)19 b Fp(\001)c(\001)g(\001)22 b Fq(+)e Fo(a)1292 4821 y Fn(n)1363 4807 y Fp(\000)i Fo(u)1508 4824 y Ft(ln)q(\()p Fn(\013)1644 4834 y Fi(l)1671 4824 y Fg(\001\001\001)o Fn(\013)1783 4833 y Fi(n)1828 4824 y Ft(\))p Fn(;N)1947 4807 y Fq(\()p Fo(x)2034 4821 y Ft(0)2075 4807 y Fq(\))2110 4745 y Fp(p)p 2186 4745 595 4 v 62 x Fo(a)2234 4821 y Fn(l)p Fg(\000)p Ft(1)2375 4807 y Fq(+)e Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2731 4821 y Fn(n)2781 4807 y Fq(\))p Fo(=)2861 4752 y Fp(p)p 2937 4752 170 4 v 55 x Fo(a)2985 4821 y Fn(l)p Fg(\000)p Ft(1)3139 4807 y Fq(and)33 b Fo(A)3387 4774 y Fn(k)3387 4833 y(l)p Fg(\000)p Ft(1)3508 4807 y Fq(\()p Fo(y)s(;)15 b(s)p Fq(\))50 4953 y(is)22 b(the)i(ev)m(en)m(t)h Fo(A)585 4920 y Fn(k)585 4978 y(l)p Fg(\000)p Ft(1)706 4953 y Fq(\()p Fo(y)s Fq(\))f(with)e Fo(Z)1110 4967 y Fn(l)p Fg(\000)p Ft(1)1254 4953 y Fq(replaced)h(b)m(y)g(the)h(v)-5 b(ariable)22 b(of)h(in)m(tegration)h Fo(s)p Fq(.)38 b(Finally)-8 b(,)23 b(w)m(e)h(substitute)50 5098 y(\(2.19\))32 b(in)m(to)f(the)f (upp)s(er)f(limit)f(of)i(in)m(tegration)h Fo(s)1740 5115 y Fn(cr)r Ft(\()p Fn(y)r Ft(\))1938 5098 y Fq(and)f(the)h(assertions)f (\()p Fr(a)2774 5065 y Ft(2)2774 5124 y Fn(l)p Fg(\000)p Ft(1)2895 5098 y Fq(\))g(and)g(\()p Fr(b)3231 5112 y Fn(l)p Fg(\000)p Ft(1)3352 5098 y Fq(\))g(follo)m(w.)159 5316 y(Let)k(us)e(no)m(w)h(turn)f(to)i(the)f(case)h(of)g(the)f (inequalit)m(y)e(\(1.6\))k(strict)e(for)g Fo(i)d Fq(=)g Fo(l)r Fq(.)48 b(Giv)m(en)33 b Fo(\017)d(>)g Fq(0,)k(one)f(can)50 5462 y(c)m(ho)s(ose)f(a)f(negativ)m(e)i(constan)m(t)1163 5439 y(~)1142 5462 y Fo(C)1207 5476 y Fn(l)1235 5462 y Fq(\()p Fo(\017)p Fq(\))f(of)f(su\016cien)m(tly)f(large)h(mo)s(dulus) d(suc)m(h)j(that)h(\()p Fr(a)3014 5429 y Ft(1)3014 5488 y Fn(l)3055 5462 y Fq(\))f(is)f(satis\014ed)g(and)p eop %%Page: 17 17 17 16 bop 50 295 a Fh(GREM,)28 b(p)l(art)f(1)3089 b Fk(17)50 513 y Fq(that)602 658 y Fo(P)13 b Fq(\()p Fo(Z)32 b(<)919 635 y Fq(~)898 658 y Fo(C)963 672 y Fn(l)991 658 y Fq(\()p Fo(\017)p Fq(\))1098 583 y Ff(p)p 1190 583 595 4 v 1190 658 a Fo(a)1238 672 y Fn(l)p Fg(\000)p Ft(1)1379 658 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1735 672 y Fn(n)1785 658 y Fo(=)1830 599 y Fp(p)p 1906 599 170 4 v 59 x Fo(a)1954 672 y Fn(l)p Fg(\000)p Ft(1)2075 658 y Fq(\))25 b Fo(<)g(\017;)107 b Fq(where)29 b Fo(Z)j Fp(2)25 b(N)13 b Fq(\(0)p Fo(;)i Fq(1\))p Fo(:)322 b Fq(\(2)p Fo(:)p Fq(22\))50 866 y(Next,)34 b(let)e(us)g(decomp)s(ose)h Fo(I)1052 880 y Ft(1)1122 866 y Fp(\021)28 b Fo(I)1268 833 y Fg(0)1261 889 y Ft(1)1323 866 y Fq(+)22 b Fo(I)1463 833 y Fg(00)1456 889 y Ft(2)1509 866 y Fq(,)34 b(s.t.)47 b(in)31 b Fo(I)1891 833 y Fg(0)1884 889 y Ft(1)1957 866 y Fq(the)i(in)m(tegration)g(is)e(from)h Fp(\0001)g Fq(to)3200 849 y(~)3199 866 y Fo(t)3232 880 y Fn(cr)3304 866 y Fq(\()p Fo(y)s Fq(\))h(and)f(in)50 1012 y Fo(I)90 1026 y Ft(2)161 1012 y Fq({)f(from)453 995 y(~)452 1012 y Fo(t)485 1026 y Fn(cr)557 1012 y Fq(\()p Fo(y)s Fq(\))g(to)g Fo(t)850 1026 y Fn(cr)923 1012 y Fq(\()p Fo(y)s Fq(\),)g(where)111 1247 y(~)110 1264 y Fo(t)143 1278 y Fn(cr)215 1264 y Fq(\()p Fo(y)s Fq(\))26 b Fp(\021)455 1190 y Ff(\000)497 1264 y Fo(y)545 1189 y Ff(p)p 635 1189 595 4 v 635 1264 a Fo(a)683 1278 y Fn(l)p Fg(\000)p Ft(1)825 1264 y Fq(+)19 b Fp(\001)c(\001)g(\001)22 b Fq(+)e Fo(a)1181 1278 y Fn(n)1250 1264 y Fp(\000)1341 1194 y(p)p 1417 1194 502 4 v 70 x Fo(a)1465 1278 y Fn(l)1513 1264 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1869 1278 y Fn(n)1919 1264 y Fq([)1944 1182 y Ff(p)p 2035 1182 922 4 v 82 x Fq(2\(ln)k(\026)-54 b Fo(\013)p Fq(\))p Fo(N)10 b Fq(\()p Fo(a)2466 1278 y Fn(l)2515 1264 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2871 1278 y Fn(n)2921 1264 y Fq(\))g(+)3088 1241 y(~)3067 1264 y Fo(C)3132 1278 y Fn(l)3160 1264 y Fq(\()p Fo(\017)p Fq(\)])3292 1190 y Ff(\001)3335 1264 y Fo(=)3380 1205 y Fp(p)p 3456 1205 170 4 v 59 x Fo(a)3504 1278 y Fn(l)p Fg(\000)p Ft(1)3625 1264 y Fo(:)50 1516 y Fq(The)33 b(c)m(hoice)i(of)641 1493 y(~)621 1516 y Fo(C)686 1530 y Fn(l)713 1516 y Fq(\()p Fo(\017)p Fq(\))g(enables)e(us)g(to)h(apply)f(to)h(the)g(in)m(tegrand)f (of)h Fo(I)2503 1483 y Fg(00)2496 1539 y Ft(2)2584 1516 y Fq(the)f(estimate)i(\()p Fr(b)3204 1530 y Fn(l)3232 1516 y Fq(\).)51 b(It)34 b(is)f(easy)50 1662 y(to)c(see,)h(c)m(hanging) e(v)-5 b(ariables)27 b(as)i(in)e(\(2.20\)\),)k(that)e Fo(I)1861 1629 y Fg(00)1854 1684 y Ft(2)1936 1662 y Fq(is)f(b)s(ounded) e(b)m(y)i Fo(\017)p Fq(\()p Fo(\013)2649 1676 y Fn(l)2692 1662 y Fp(\001)15 b(\001)g(\001)i Fo(\013)2872 1676 y Fn(n)2921 1662 y Fq(\))2956 1629 y Fn(N)3026 1662 y Fo(e)3068 1629 y Fg(\000)p Fn(y)3164 1601 y Fj(2)3199 1629 y Fn(=)p Ft(2)3277 1662 y Fo(=)3322 1587 y Fp(p)p 3399 1587 101 4 v 3399 1662 a Fq(2)p Fo(\031)s(y)s Fq(,)29 b(for)50 1807 y(all)g Fo(N)40 b Fq(large)31 b(enough,)f(and)g(all)f Fo(y)k Fq(in)c(the)i(in)m(terv)-5 b(als)29 b(of)i(\()p Fr(a)2048 1774 y Ft(1)2048 1833 y Fn(l)p Fg(\000)p Ft(1)2169 1807 y Fq(\))g(and)e(\()p Fr(b)2505 1821 y Fn(l)p Fg(\000)p Ft(1)2626 1807 y Fq(\).)159 2025 y(T)-8 b(o)31 b(consider)f Fo(I)695 1992 y Fg(0)688 2048 y Ft(1)729 2025 y Fq(,)h(w)m(e)g(are)g (en)m(titled)g(to)g(estimate)h(its)e(in)m(tegrand)g(b)m(y)h(\()p Fr(a)2626 1992 y Ft(1)2626 2051 y Fn(l)2667 2025 y Fq(\))g(uniformly)d (in)i(the)h(in)m(terv)-5 b(al)50 2171 y(of)33 b(in)m(tegration.)49 b(Then)32 b(for)g(all)g Fo(N)43 b Fq(large)33 b(enough)g(this)f(term)h (is)f(b)s(ounded)f(from)h(ab)s(o)m(v)m(e)i(and)f(b)s(elo)m(w)f(b)m(y)50 2316 y(the)g(in)m(tegral)f(\(2.20\))j(with)d Fo(t)1046 2330 y Fn(cr)1118 2316 y Fq(\()p Fo(y)s Fq(\))i(replaced)e(b)m(y)1758 2299 y(~)1757 2316 y Fo(t)1790 2330 y Fn(cr)1862 2316 y Fq(\()p Fo(y)s Fq(\).)46 b(Consequen)m(tly)-8 b(,)32 b(it)f(equals)g(\(2.21\))j(with)d Fo(s)3519 2330 y Fn(cr)3591 2316 y Fq(\()p Fo(y)s Fq(\))50 2462 y(replaced)f(b)m(y)108 2714 y(~)-48 b Fo(s)148 2728 y Fn(cr)220 2714 y Fq(\()p Fo(y)s Fq(\))26 b(=)460 2640 y Ff(\000)502 2714 y Fo(y)550 2644 y Fp(p)p 625 2644 502 4 v 625 2714 a Fo(a)673 2728 y Fn(l)721 2714 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)1077 2728 y Fn(n)1147 2714 y Fp(\000)g Fq([)1263 2632 y Ff(p)p 1354 2632 922 4 v 82 x Fq(2\(ln)k(\026)-54 b Fo(\013)p Fq(\))p Fo(N)10 b Fq(\()p Fo(a)1785 2728 y Fn(l)1834 2714 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(a)2190 2728 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b(\026)-44 b Fn(\013;N)850 3437 y Fq(\()p Fo(x)p Fq(\)\))26 b Fp(\030)f Fo(K)7 b Fq(\()p Fo(\013)1306 3451 y Ft(2)1362 3437 y Fp(\001)15 b(\001)g(\001)i Fo(\013)1542 3451 y Fn(n)1591 3437 y Fq(\))1626 3399 y Fn(N)1696 3437 y Fo(e)1738 3399 y Fg(\000)p Fn(u)1838 3372 y Fj(2)1838 3418 y(ln)g(\026)-38 b Fi(\013;N)2022 3399 y Ft(\()p Fn(x)p Ft(\))p Fn(=)p Ft(2)2198 3437 y Fo(=)2243 3357 y Fp(p)p 2319 3357 101 4 v 80 x Fq(2)p Fo(\031)t(u)2472 3451 y Ft(ln)20 b(\026)-43 b Fn(\013;N)2683 3437 y Fq(\()p Fo(x)p Fq(\))26 b Fp(\030)f Fo(\013)2985 3398 y Fg(\000)p Fn(N)2985 3461 y Ft(1)3111 3437 y Fo(K)7 b(e)3237 3399 y Fg(\000)p Fn(x)3340 3437 y Fo(;)50 3689 y Fq(where)30 b Fo(K)37 b Fq(is)29 b(as)i(de\014ned)e (in)g(Theorem)h(1.1.)42 b(Th)m(us)29 b(the)i(probabilit)m(y)d(in)h(the) h(l.h.s.)g(of)g(\(2.1\))j(equals)764 3941 y(\(1)21 b Fp(\000)f Fo(P)13 b Fq(\(1)p Fo(;)i(u)1199 3955 y Ft(ln)22 b(\026)-44 b Fn(\013;N)1411 3941 y Fq(\()p Fo(x)p Fq(\)\)\))1603 3904 y Fn(\013)1650 3876 y Fi(N)1650 3922 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5462 y Fo(\016)1618 5476 y Ft(1)1697 5462 y Fq(su\016cien)m(tly)g(close)i (to)2507 5438 y(\026)2502 5462 y Fo(\016)2542 5476 y Fn(k)r Fg(\000)p Ft(1)2681 5462 y Fq(,)h(w)m(e)f(mak)m(e)h Fo(y)h Fq(=)d Fo(u)3376 5476 y Ft(ln)21 b(\026)-44 b Fn(\013)q(;N)3587 5462 y Fq(\()p Fo(x)p Fq(\))p eop %%Page: 18 18 18 17 bop 50 295 a Fk(18)3270 b Fh(Se)l(ction)27 b Fk(2)50 513 y Fq(b)s(elong)39 b(to)i(the)g(in)m(terv)-5 b(al)39 b(of)i Fo(y)i Fq(where)c(\(2.24\))k(with)c Fo(i)j Fq(=)f(1)g(holds)e (true.)70 b(Th)m(us,)42 b Fo(P)13 b Fq(\(1)p Fo(;)i(u)3227 527 y Ft(ln)22 b(\026)-43 b Fn(\013;N)3439 513 y Fq(\()p Fo(x)p Fq(\)\))43 b(=)50 658 y Fo(o)p Fq(\()p Fo(\013)187 619 y Fg(\000)p Fn(N)187 682 y Ft(1)313 658 y Fo(e)355 625 y Fg(\000)p Fn(x)458 658 y Fq(\))g(and)f(the)g(assertion)g(\(2.2\)) i(follo)m(ws)d(as)i(in)e(\(2.23\).)78 b(This)41 b(completes)h(the)h (pro)s(of)e(of)i(the)50 804 y(lemma.)d Fp(})50 1022 y Fm(Pr)-5 b(o)g(of)31 b(of)f(The)-5 b(or)g(em)31 b(1.1.)40 b Fq(The)26 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Fq(\),)h(\()p Fr(b)2193 5330 y Fn(i)2223 5316 y Fq(\))f(hold)f(uniformly)e(in)i(the)h (corresp)s(onding)50 5462 y(in)m(terv)-5 b(als)29 b(of)i(parameters)f (where)g Fo(y)1296 5476 y Ft(1)1367 5462 y Fq(substitutes)f Fo(y)k Fq(and)d(where)g Fo(z)2388 5476 y Fn(k)2458 5462 y Fo(>)25 b(y)2599 5476 y Fn(k)2669 5462 y Fo(>)g Fp(\001)15 b(\001)g(\001)26 b Fo(>)f(z)3034 5476 y Ft(1)3100 5462 y Fo(>)g(y)3241 5476 y Ft(1)3282 5462 y Fq(.)p eop %%Page: 19 19 19 18 bop 50 295 a Fh(GREM,)28 b(p)l(art)f(1)3089 b Fk(19)159 513 y Fq(The)34 b(pro)s(ofs)g(of)h(all)e(these)i(assertions)f(are)h (completely)g(analogous)g(to)g(those)g(in)e(Lemma)i(2.1,)i(note)50 658 y(only)h(that)h Fo(t)496 672 y Fn(cr)569 658 y Fq(\()p Fo(y)s Fq(\))g(and)912 641 y(~)911 658 y Fo(t)944 672 y Fn(cr)1017 658 y Fq(\()p Fo(y)s Fq(\))g(b)s(ecome)g Fo(t)1541 672 y Fn(cr)1614 658 y Fq(\()p Fo(y)1694 672 y Ft(1)1734 658 y Fq(\))h(and)1995 641 y(~)1994 658 y Fo(t)2027 672 y Fn(cr)2099 658 y Fq(\()p Fo(y)2179 672 y Ft(1)2220 658 y Fq(\))f(resp)s(ectiv)m(ely)-8 b(.)66 b(Finally)-8 b(,)39 b(considering)e(a)50 804 y(particular)29 b(case)i Fo(i)26 b Fq(=)g(1,)31 b Fo(y)962 818 y Fn(l)1015 804 y Fq(=)26 b Fo(u)1164 818 y Ft(ln)20 b(\026)-43 b Fn(\013;N)1375 804 y Fq(\()p Fo(a)1458 818 y Fn(l)1486 804 y Fq(\),)31 b Fo(z)1619 818 y Fn(l)1673 804 y Fq(=)25 b Fo(u)1821 818 y Ft(ln)c(\026)-44 b Fn(\013)q(;N)2032 804 y Fq(\()p Fo(b)2106 818 y Fn(l)2135 804 y Fq(\))30 b(for)h Fo(l)c Fq(=)f(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(k)34 b Fq(as)d(in)e(\(2.23\))k(leads)d(to)50 949 y(\(ii\).)40 b Fp(})50 1167 y Fm(Pr)-5 b(o)g(of)32 b(of)g(The)-5 b(or)g(em)33 b(1.2.)41 b Fq(W)-8 b(e)30 b(pro)s(ceed)e(b)m(y)h(induction.)38 b(In)28 b(the)h(case)h Fo(k)e Fq(=)d(1,)30 b Fp(P)2815 1134 y Ft(\(1\))2938 1167 y Fq(=)25 b Fp(P)3097 1181 y Ft(1)3167 1167 y Fq(is)j(the)h(P)m (oisson)50 1313 y(p)s(oin)m(t)24 b(pro)s(cess)g(on)h Fe(R)33 b Fq(and)24 b(the)h(statemen)m(t)i(of)e(the)f(theorem)i(is)d (already)i(con)m(tained)g(in)e(the)i(assumptions.)159 1531 y(Assume)h(that)h(\(1.8\))h(is)d(pro)m(v)m(ed)i(for)f Fo(k)15 b Fp(\000)d Fq(1.)40 b(T)-8 b(aking)26 b(in)m(to)g(accoun)m(t)i (the)e(iden)m(tical)f(distribution)e(of)k(the)50 1676 y(r.v.)40 b Fo(Y)277 1690 y Fn(\033)319 1699 y Fj(1)355 1690 y Fg(\001\001\001)n Fn(\033)461 1699 y Fi(i)495 1676 y Fq(,)30 b Fo(i)c Fq(=)f(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(k)s Fq(,)31 b(and)f(the)h(indep)s(endence)d(imp)s(osed)g(on)j (their)e(v)m(ectors)j(w)m(e)f(ma)m(y)g(write:)121 1953 y Fo(F)192 1916 y Fn(N)179 1976 y Ft(\001)239 1985 y Fj(1)275 1976 y Fg(\002\001\001\001)n(\002)p Ft(\001)513 1986 y Fi(k)557 1953 y Fq(\()p Fo(z)t Fq(\))c Fp(\021)d Fe(E)11 b Fo(z)896 1832 y Ff(P)998 1927 y Fi(\033)1035 1941 y Fj(1)1088 1893 y Ft(1)-23 b(I)1128 1906 y Fd(f)p Fi(v)1194 1921 y(N)r(;)p Fj(1)1301 1906 y(\()p Fi(Y)1363 1915 y(\033)1400 1929 y Fj(1)1440 1906 y(\))p Fd(2)p Fj(\001)1558 1920 y(1)1593 1906 y Fd(g)1631 1893 y 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Fp(\000)g Fq(1)2863 4070 y Ff(\021i)2986 4171 y Fp(\021)25 b Fo(I)3129 4133 y Fn(N)3122 4193 y Ft(1)3209 4171 y Fq(+)11 b Fo(I)3338 4133 y Fn(N)3331 4193 y Ft(2)3407 4171 y Fo(:)45 b Fq(\(2)p Fo(:)p Fq(25\))50 4440 y(The)31 b(term)h Fo(I)504 4407 y Fn(N)497 4463 y Ft(2)605 4440 y Fq(in)e(this)h(represen)m(tation)h(con)m(v)m(erges)i (to)e(zero)h(as)f Fo(N)37 b Fp(")28 b(1)p Fq(.)45 b(In)31 b(fact,)j(due)d(to)h(the)g(con)m(v)m(er-)50 4586 y(gence)h(\(1.7\))g (and)e Fo(a)743 4600 y Ft(1)812 4586 y Fo(>)c Fp(\0001)p Fq(,)32 b(for)f(an)m(y)i Fo(\017)27 b(>)g Fq(0)32 b(one)g(can)g(\014nd) f(a)h(su\016cien)m(tly)e(large)i Fo(C)7 b Fq(\()p Fo(\017)p Fq(\))28 b Fo(>)f Fq(0)32 b(and)f Fo(N)10 b Fq(\()p Fo(\017)p Fq(\))50 4731 y(suc)m(h)30 b(that)804 4891 y Fo(P)13 b Fq(\()p Fo(\021)955 4905 y Fn(N)s(;k)r Fg(\000)p Ft(1)1200 4891 y Fp(\025)25 b Fo(C)7 b Fq(\()p Fo(\017)p Fq(\)\))26 b Fp(\024)f Fo(P)1703 4790 y Ff(\020)1772 4805 y(X)1799 4996 y Fn(\033)1841 5005 y Fj(1)1919 4891 y Fq(1)-23 b(I)1974 4908 y Fn(v)2010 4917 y Fi(N)r(;)p Fj(1)2118 4908 y Ft(\()p Fn(Y)2189 4917 y Fi(\033)2226 4931 y Fj(1)2266 4908 y Ft(\))p Fg(2)p Ft(\001)2403 4917 y Fj(1)2468 4891 y Fp(\025)25 b Fo(C)7 b Fq(\()p Fo(\017)p Fq(\))2743 4790 y Ff(\021)2823 4891 y Fo(<)25 b(\017)50 5170 y Fq(for)30 b(all)f Fo(N)35 b Fp(\025)25 b Fo(N)10 b Fq(\()p Fo(\017)p Fq(\).)42 b(Then)607 5439 y Fp(j)p Fo(I)679 5402 y Fn(N)672 5462 y Ft(2)748 5439 y Fp(j)25 b(\024)g Fq(2)p Fo(\017)c Fq(+)f Fe(E)1142 5338 y Ff(\020)1203 5439 y Fp(j)p Fo(f)1273 5453 y Fn(k)1317 5439 y Fq(\()p Fo(z)t Fq(\))p Fp(j)1458 5402 y Fn(\021)1495 5412 y Fi(N)r(;k)q Fd(\000)p Fj(1)1693 5335 y Ff(\014)1693 5389 y(\014)1693 5444 y(\014)1723 5365 y(\000)1765 5439 y Fq(1)h(+)e Fo(\020)1961 5453 y Fn(N)2030 5439 y Fq(\()p Fo(z)t Fq(\))2146 5365 y Ff(\001)2189 5384 y Fn(\021)2226 5394 y Fi(N)r(;k)q Fd(\000)p Fj(1)2442 5439 y Fp(\000)h Fq(1)2578 5335 y Ff(\014)2578 5389 y(\014)2578 5444 y(\014)2609 5439 y Fq(1)-23 b(I)2665 5455 y Fn(\021)2702 5465 y Fi(N)r(;k)q Fd(\000)p Fj(1)2894 5455 y Fn()d Fq(2)31 b(is)e(analogous.)42 b(Again)30 b(in)f(view)h(of)h(the)f(iden)m (tical)g(distribution)c(of)31 b(the)g(r.v.)41 b Fo(Y)3030 1705 y Fn(\033)3072 1714 y Fj(1)3107 1705 y Fg(\001\001\001)o Fn(\033)3214 1714 y Fi(i)3247 1691 y Fq(,)31 b(and)f(of)h(the)50 1837 y(indep)s(endence)d(of)i(their)g(v)m(ectors,)i(w)m(e)f(get:)204 2055 y Fo(F)262 2074 y Ft(\001)322 2052 y Fj(1)322 2096 y(1)358 2074 y Fg(\002\001\001\001)n(\002)p Ft(\001)596 2052 y Fj(1)596 2099 y Fi(k)635 2074 y Fg([)p Ft(\001)744 2052 y Fj(2)744 2096 y(1)780 2074 y Fg(\002\001\001\001)n(\002)p Ft(\001)1018 2052 y Fj(2)1018 2099 y Fi(k)1062 2055 y Fq(\()p Fo(z)t Fq(\))26 b(=)49 b(lim)1300 2114 y Fn(N)7 b Fg("1)1490 2055 y Fo(F)1561 2017 y Fn(N)1548 2083 y Ft(\001)1608 2061 y Fj(1)1608 2105 y(1)1643 2083 y Fg(\002\001\001\001) o(\002)p Ft(\001)1882 2061 y Fj(1)1882 2107 y Fi(k)1921 2083 y Fg([)p Ft(\001)2030 2061 y Fj(2)2030 2105 y(1)2065 2083 y Fg(\002\001\001\001)o(\002)p Ft(\001)2304 2061 y Fj(2)2304 2107 y Fi(k)2348 2055 y Fq(\()p Fo(z)t Fq(\))229 2296 y(=)49 b(lim)325 2356 y Fn(N)7 b Fg("1)514 2296 y Fe(E)569 2195 y Ff(h)618 2296 y Fq(\()p Fe(E)k Fo(z)754 2172 y Ff(P)856 2267 y Fi(\033)893 2281 y Fj(2)946 2233 y Ft(1)-23 b(I)985 2256 y Fd(f)p Fi(v)1051 2272 y(N)r(;)p Fj(2)1159 2256 y(\()p Fi(Y)1221 2265 y(\033)1258 2279 y Fj(2)1298 2256 y(\))p Fd(2)p Fj(\001)1416 2240 y(1)1416 2284 y(2)1451 2256 y Fd(g)1489 2233 y Fg(\001\001\001)1566 2172 y Ff(P)1662 2267 y Fi(\033)1699 2276 y(n)1761 2233 y Ft(1)g(I)1800 2256 y Fd(f)p Fi(v)1866 2272 y(N)r(;k)1978 2256 y Fj(\()p Fi(Y)2040 2265 y(\033)2077 2279 y Fj(1)2112 2265 y Fi(:::\033)2212 2281 y(k)2256 2256 y Fj(\))p Fd(2)p Fj(\001)2374 2240 y(1)2374 2287 y Fi(k)2414 2256 y Fd(g)2456 2296 y Fq(\))2491 2174 y Ff(P)2587 2269 y Fi(\033)2624 2283 y Fj(1)2677 2235 y Ft(1)g(I)2717 2258 y Fd(f)p Fi(v)2783 2274 y(N)r(;)p Fj(1)2890 2258 y(\()p Fi(Y)2952 2267 y(\033)2989 2281 y Fj(1)3029 2258 y(\))p Fd(2)p Fj(\001)3147 2243 y(1)3147 2287 y(1)3182 2258 y Fd(g)588 2538 y Fp(\002)20 b Fq(\()p Fe(E)11 b Fo(z)815 2414 y Ff(P)917 2509 y Fi(\033)954 2523 y Fj(2)1006 2475 y Ft(1)-23 b(I)1046 2498 y Fd(f)p Fi(v)1112 2513 y(N)r(;)p Fj(2)1219 2498 y(\()p Fi(Y)1281 2507 y(\033)1318 2521 y Fj(2)1359 2498 y(\))p Fd(2)p Fj(\001)1477 2482 y(2)1477 2526 y(2)1512 2498 y Fd(g)1550 2475 y Fg(\001\001\001)1627 2414 y Ff(P)1723 2509 y Fi(\033)1760 2525 y(k)1816 2475 y Ft(1)g(I)1856 2498 y Fd(f)p Fi(v)1922 2514 y(N)r(;k)2034 2498 y Fj(\()p Fi(Y)2096 2507 y(\033)2133 2521 y Fj(1)2168 2507 y Fi(:::\033)2268 2523 y(k)2312 2498 y Fj(\))p Fd(2)p Fj(\001)2430 2482 y(2)2430 2528 y Fi(k)2469 2498 y Fd(g)2512 2538 y Fq(\))2547 2416 y Ff(P)2643 2511 y Fi(\033)2680 2525 y Fj(1)2733 2477 y Ft(1)g(I)2773 2500 y Fd(f)p Fi(v)2839 2516 y(N)r(;)p Fj(1)2946 2500 y(\()p Fi(Y)3008 2509 y(\033)3045 2523 y Fj(1)3085 2500 y(\))p Fd(2)p Fj(\001)3203 2485 y(2)3203 2528 y(1)3238 2500 y Fd(g)3280 2437 y Ff(i)229 2756 y Fq(=)49 b(lim)325 2815 y Fn(N)7 b Fg("1)514 2756 y Fe(E)569 2655 y Ff(h)618 2756 y Fq(\()p Fo(f)698 2776 y Ft(2)p Fn(;)p Ft(\001)816 2754 y Fj(1)816 2798 y(2)855 2756 y Fq(\()p Fo(f)935 2776 y Ft(3)p Fn(;)p Ft(\001)1053 2754 y Fj(1)1053 2798 y(3)1093 2756 y Fq(\()p Fp(\001)15 b(\001)g(\001)h Fq(\()p Fo(f)1329 2776 y Fn(k)r(;)p Ft(\001)1452 2754 y Fj(1)1452 2800 y Fi(k)1495 2756 y Fq(\()p Fo(z)t Fq(\)\))f Fp(\001)g(\001)g(\001)j Fq(\)\)\))1889 2634 y Ff(P)1986 2729 y Fi(\033)2023 2743 y Fj(1)2075 2695 y Ft(1)-23 b(I)2115 2718 y Fd(f)p Fi(v)2181 2734 y(N)r(;)p Fj(1)2288 2718 y(\()p Fi(Y)2350 2727 y(\033)2387 2741 y Fj(1)2427 2718 y(\))p Fd(2)p Fj(\001)2545 2702 y(1)2545 2746 y(1)2580 2718 y Fd(g)588 2996 y Fp(\002)20 b Fq(\()p Fo(f)759 3016 y Ft(2)p Fn(;)p Ft(\001)877 2994 y Fj(2)877 3037 y(2)916 2996 y Fq(\()p Fo(f)996 3016 y Ft(3)p Fn(;)p Ft(\001)1114 2994 y Fj(2)1114 3037 y(3)1154 2996 y Fq(\()p Fp(\001)15 b(\001)g(\001)h Fq(\()p Fo(f)1390 3016 y Fn(k)r(;)p Ft(\001)1513 2994 y Fj(2)1513 3040 y Fi(k)1556 2996 y Fq(\()p Fo(z)t Fq(\)\))f Fp(\001)g(\001)g(\001)i Fq(\)\)\))1949 2873 y Ff(P)2046 2969 y Fi(\033)2083 2983 y Fj(1)2136 2935 y Ft(1)-23 b(I)2176 2958 y Fd(f)p Fi(v)2242 2973 y(N)r(;)p Fj(1)2349 2958 y(\()p Fi(Y)2411 2967 y(\033)2448 2981 y Fj(1)2488 2958 y(\))p Fd(2)p Fj(\001)2606 2942 y(2)2606 2986 y(1)2641 2958 y Fd(g)2684 2895 y Ff(i)3477 2537 y Fq(\(2)p Fo(:)p Fq(26\))50 3242 y(where)24 b(in)f(the)i(last)g (line)e(w)m(e)i(applied)d(\(1.8\))27 b(with)c Fo(n)9 b Fp(\000)g Fq(1)24 b(lev)m(els)g(together)i(with)d(the)i(argumen)m(ts) g(similar)d(to)50 3388 y(\(2.25\).)42 b(Recall)28 b(that)i(the)e(coun)m (ting)h(v)-5 b(ariables)1714 3319 y Ff(P)1810 3415 y Fn(x)1851 3424 y Fj(1)1906 3388 y Fq(1)-23 b(I)1962 3402 y Fn(x)2003 3411 y Fj(1)2039 3402 y Fg(2)p Ft(\001)2148 3411 y Fj(1)2216 3388 y Fq(of)29 b(the)g(P)m(oisson)f(p)s(oin)m(t)f (pro)s(cess)h Fp(P)3412 3402 y Ft(1)3482 3388 y Fq(corre-)50 3533 y(sp)s(onding)h(to)j(non-in)m(tersecting)g(segmen)m(ts)g(\001)1681 3500 y Ft(1)1681 3556 y(1)1753 3533 y Fq(and)f(\001)2007 3500 y Ft(2)2007 3556 y(1)2080 3533 y Fq(are)h(indep)s(enden)m(t.)42 b(Therefore)31 b(\(2.26\))j(equals)50 3679 y(the)29 b(pro)s(duct)g(of)g (the)h(generating)f(functions)f Fo(f)1683 3698 y Ft(1)p Fn(;)p Ft(\001)1801 3677 y Fj(1)1801 3720 y(1)1840 3679 y Fq(\()p Fp(\001)15 b(\001)g(\001)i Fq(\()p Fo(f)2077 3695 y Fn(n;)p Ft(\001)2204 3677 y Fj(1)2204 3713 y Fi(n)2252 3679 y Fq(\()p Fo(z)t Fq(\)\))e Fp(\001)g(\001)g(\001)j Fq(\))30 b(and)e Fo(f)2826 3698 y Ft(1)p Fn(;)p Ft(\001)2944 3677 y Fj(2)2944 3720 y(1)2984 3679 y Fq(\()p Fp(\001)15 b(\001)g(\001)h Fq(\()p Fo(f)3220 3698 y Fn(k)r(;)p Ft(\001)3343 3677 y Fj(2)3343 3723 y Fi(k)3386 3679 y Fq(\()p Fo(z)t Fq(\)\))f Fp(\001)g(\001)g(\001)i Fq(\))50 3824 y(\014nishing)27 b(the)k(pro)s(of)e(of)i(the)f(prop)s(ert)m(y)g(\(i\).)41 b(The)29 b(pro)s(of)h(of)h(\(ii\))e(is)h(analogous.)41 b Fp(})50 4042 y Fm(Pr)-5 b(o)g(of)28 b(of)e(Pr)-5 b(op)g(osition)29 b(1.4)c Fq(Supp)s(ose)c(that)k(for)e(some)h Fo(l)h Fq(and)e(some)h Fo(k)s Fq(,)i Fo(J)2497 4056 y Fn(l)p Fg(\000)p Ft(1)2625 4042 y Fq(+)7 b(2)25 b Fp(\024)g Fo(k)j Fp(\024)d Fo(J)3090 4056 y Fn(l)3118 4042 y Fq(,)h(the)d(inequalit)m(y)50 4188 y(\(1.10\))34 b(is)d(not)h(true.)44 b(Then)31 b Fo(A)1108 4202 y Fn(J)1148 4212 y Fi(l)p Fd(\000)p Fj(1)1255 4202 y Ft(+1)p Fn(;k)r Fg(\000)p Ft(1)1534 4188 y Fo(>)c(A)1700 4202 y Fn(k)r(;J)1803 4212 y Fi(l)1833 4188 y Fq(.)45 b(In)31 b(view)g(of)h(the)g(de\014nition)e(of)i Fo(J)3053 4202 y Fn(l)3113 4188 y Fq(either)f(for)g(some)50 4333 y Fo(j)87 4347 y Ft(0)155 4333 y Fo(>)c(J)303 4347 y Fn(l)362 4333 y Fo(A)430 4347 y Fn(J)470 4357 y Fi(l)p Fd(\000)p Fj(1)577 4347 y Ft(+1)p Fn(;k)r Fg(\000)p Ft(1)855 4333 y Fp(\024)g Fo(A)1021 4347 y Fn(k)r(;j)1114 4356 y Fj(0)1184 4333 y Fq(or)32 b(there)f(exists)g Fo(p)p Fq(,)h Fo(k)e Fp(\024)d Fo(p)f(<)h(J)2280 4347 y Fn(l)2308 4333 y Fq(,)32 b(suc)m(h)f(that)h Fo(A)2837 4347 y Fn(J)2877 4357 y Fi(l)p Fd(\000)p Fj(1)2984 4347 y Ft(+1)p Fn(;k)r Fg(\000)p Ft(1)3262 4333 y Fp(\024)27 b Fo(A)3428 4347 y Fn(k)r(;p)3563 4333 y Fq(and)50 4479 y(for)36 b(all)f Fo(j)42 b(>)35 b(p)h(A)662 4493 y Fn(J)702 4503 y Fi(l)p Fd(\000)p Fj(1)809 4493 y Ft(+1)p Fn(;k)r Fg(\000)p Ft(1)1096 4479 y Fo(>)f(A)1270 4493 y Fn(k)r(;j)1370 4479 y Fq(.)59 b(Under)36 b(the)g(former)g(condition)f Fo(A)2668 4493 y Fn(J)2708 4503 y Fi(l)p Fd(\000)p Fj(1)2815 4493 y Ft(+1)p Fn(;k)r Fg(\000)p Ft(1)3102 4479 y Fo(<)g(A)3276 4493 y Fn(J)3316 4503 y Fi(l)3343 4493 y Ft(+1)p Fn(;j)3487 4502 y Fj(0)3563 4479 y Fq(and)50 4624 y Fo(A)118 4638 y Fn(k)r(;J)221 4648 y Fi(l)277 4624 y Fo(<)26 b(A)442 4638 y Fn(J)482 4648 y Fi(l)509 4638 y Ft(+1)p Fn(;j)653 4647 y Fj(0)693 4624 y Fq(,)31 b(whence)g Fo(A)1136 4638 y Fn(J)1176 4648 y Fi(l)p Fd(\000)p Fj(1)1283 4638 y Ft(+1)p Fn(;J)1437 4648 y Fi(l)1494 4624 y Fo(<)c(A)1660 4638 y Fn(J)1700 4648 y Fi(l)1726 4638 y Ft(+1)p Fn(;j)1870 4647 y Fj(0)1941 4624 y Fq(con)m(tradicting)k(the)g(de\014nition)e(of)i Fo(J)3211 4638 y Fn(l)3240 4624 y Fq(.)42 b(Under)31 b(the)50 4769 y(latter)37 b(condition)e Fo(A)777 4783 y Fn(k)r(;p)916 4769 y Fo(>)h(A)1091 4783 y Fn(p)p Ft(+1)p Fn(;j)1280 4769 y Fq(,)j Fo(A)1412 4783 y Fn(J)1452 4793 y Fi(l)p Fd(\000)p Fj(1)1559 4783 y Ft(+1)p Fn(;k)r Fg(\000)p Ft(1)1846 4769 y Fo(>)c(A)2020 4783 y Fn(p)p Ft(+1)p Fn(;j)2209 4769 y Fq(,)k(and)d(th)m(us)g Fo(A)2730 4783 y Fn(J)2770 4793 y Fi(l)p Fd(\000)p Fj(1)2877 4783 y Ft(+1)p Fn(;p)3068 4769 y Fo(>)f(A)3242 4783 y Fn(p)p Ft(+1)p Fn(;j)3469 4769 y Fq(for)h(all)50 4915 y Fo(j)31 b(>)25 b(p)p Fq(.)40 b(This)29 b(implies)e Fo(J)891 4929 y Fn(l)945 4915 y Fp(\024)e Fo(p)30 b Fq(con)m(tradicting)g Fo(p)25 b(<)g(J)1887 4929 y Fn(l)1915 4915 y Fq(.)41 b Fp(})50 5244 y Fm(Pr)-5 b(o)g(of)34 b(of)f(The)-5 b(or)g(em)34 b(1.5.)41 b Fq(T)-8 b(o)31 b(pro)m(v)m(e)g(Theorem)f(1.5)i(w)m(e)f (need)f(t)m(w)m(o)i(auxiliary)c(results:)50 5462 y Fc(Prop)s(osition)49 b(2.2:)57 b Fm(Assume)41 b(that)h(the)f(numb)-5 b(ers)41 b Fo(\015)2032 5476 y Ft(1)2073 5462 y Fo(;)15 b(:)g(:)g(:)31 b(;)15 b(\015)2336 5476 y Fn(m)2446 5462 y Fm(ar)-5 b(e)42 b(such)f(that)g Fo(\015)3064 5476 y Ft(1)3145 5462 y Fo(>)e(\015)3302 5476 y Ft(2)3383 5462 y Fo(>)g Fp(\001)15 b(\001)g(\001)41 b Fo(>)p eop %%Page: 21 21 21 20 bop 50 295 a Fh(GREM,)28 b(p)l(art)f(1)3089 b Fk(21)50 513 y Fo(\015)97 527 y Fn(m)191 513 y Fo(>)25 b Fq(1)p Fm(.)42 b(Then)33 b(the)g(limit)404 806 y Fq(lim)345 861 y Fn(x)p Fg(!\0001)603 682 y Ff(Z)998 856 y Fj(\()p Fi(x)1059 870 y Fj(1)1094 856 y Fi(;:::)11 b(;x)1246 865 y(m)1306 856 y Fj(\))p Fd(2)p Fa(R)1403 843 y Fi(m)1467 856 y(;)665 914 y Fd(9)p Fi(i;)p Fj(1)p Fd(\024)p Fi(i)p Fd(\024)p Fi(m)p Fj(:)p Fi(\015)1008 928 y Fj(1)1044 914 y Fi(x)1080 928 y Fj(1)1115 914 y(+)p Fd(\001\001\001)o Fj(+)p Fi(\015)1305 929 y(i)1334 914 y(x)1370 929 y(i)1399 914 y(>)p Fj(\()p Fi(\015)1507 928 y Fj(1)1542 914 y(+)p Fd(\001\001\001)o Fj(+)p Fi(\015)1732 929 y(i)1761 914 y Fj(\))p Fi(x)1852 806 y Fo(e)1894 769 y Fn(\015)1932 778 y Fj(1)1968 769 y Fn(x)2009 778 y Fj(1)2045 769 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(\014)s(\015)2303 778 y Fi(m)2364 769 y Fn(x)2405 778 y Fi(m)2470 806 y Fp(P)2540 769 y Ft(\()p Fn(m)p Ft(\))2667 806 y Fq(\()p Fo(dx)2801 820 y Ft(1)2857 806 y Fo(:)k(:)g(:)h(dx)3077 820 y Fn(m)3147 806 y Fq(\))295 b(\(2)p Fo(:)p Fq(27\))50 1144 y Fm(is)32 b(\014nite)h(a.s.)50 1362 y Fc(Prop)s(osition)49 b(2.3:)57 b Fm(Assume)41 b(that)h(the)f(numb)-5 b(ers)41 b Fo(\015)2032 1376 y Ft(1)2073 1362 y Fo(;)15 b(:)g(:)g(:)31 b(;)15 b(\015)2336 1376 y Fn(m)2446 1362 y Fm(ar)-5 b(e)42 b(such)f(that)g Fo(\015)3064 1376 y Ft(1)3145 1362 y Fo(>)e(\015)3302 1376 y Ft(2)3383 1362 y Fo(>)g Fp(\001)15 b(\001)g(\001)41 b Fo(>)50 1508 y(\015)97 1522 y Fn(m)191 1508 y Fo(>)25 b Fq(1)p Fm(.)42 b(Then)33 b(for)g(any)h Fo(a)25 b Fp(2)g Fo(R)356 1796 y Fe(P)p Fq(\()p Fp(9)93 b Fm(in\014nitely)33 b(many)94 b Fq(\()p Fo(x)1367 1810 y Ft(1)1408 1796 y Fo(;)15 b(:)g(:)g(:)i(;)e(x)1662 1810 y Fn(m)1731 1796 y Fq(\))26 b Fp(2)f(P)1948 1758 y Ft(\()p Fn(m)p Ft(\))2100 1796 y Fq(:)g Fo(\015)2197 1810 y Ft(1)2238 1796 y Fo(x)2290 1810 y Ft(1)2351 1796 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)h Fo(x)2615 1810 y Fn(m)2684 1796 y Fo(\015)2731 1810 y Fn(m)2825 1796 y Fo(>)25 b(a)p Fq(\))h(=)f(0)306 b(\(2)p Fo(:)p Fq(28\))159 2302 y(In)25 b(fact)h(w)m(e)g(will)d(pro)m(v)m(e)j (statemen)m(ts)i(\(i\))d(and)g(\(ii\))g(of)h(Theorem)f(1.5)h(together)h (with)e(\(2.27\))j(and)c(\(2.28\))50 2448 y(b)m(y)30 b(induction.)159 2666 y(F)-8 b(or)37 b Fo(m)e Fq(=)h(1)h(\(i\))f(and)g (\(ii\))g(are)g(immediate)g(from)g(Theorem)g(1.3.)61 b(T)-8 b(o)37 b(see)g(\(2.27\))h(and)e(\(2.28\))j(note)50 2811 y(that)i Fp(P)327 2779 y Ft(\(1\))465 2811 y Fq(has)g(a)f (\014nite)g(n)m(um)m(b)s(er)f(of)h(p)s(oin)m(ts)g(in)f([)p Fo(x;)15 b Fp(1)p Fq(\))41 b(a.s.)71 b(for)40 b(an)m(y)h Fo(x)h Fp(2)g Fe(R)r Fq(.)77 b(Then)39 b(the)i(in)m(tegral)50 2884 y Ff(R)110 2906 y Fg(1)93 2989 y Fn(x)203 2957 y Fo(e)245 2924 y Fn(c)278 2933 y Fj(1)313 2924 y Fn(x)354 2933 y Fj(1)394 2957 y Fp(P)464 2924 y Ft(\(1\))562 2957 y Fq(\()p Fo(dx)p Fq(\))25 b(is)d(\014nite)g(a.s.)39 b(while)22 b Fe(E)1535 2884 y Ff(R)1602 2906 y Fn(x)1584 2989 y Fg(\0001)1733 2957 y Fo(e)1775 2924 y Fn(c)1808 2933 y Fj(1)1844 2924 y Fn(x)1885 2933 y Fj(1)1925 2957 y Fp(P)1995 2924 y Ft(\(1\))2093 2957 y Fq(\()p Fo(dx)p Fq(\))k(=)f Fo(e)2426 2924 y Ft(\()p Fn(c)2487 2933 y Fj(1)2523 2924 y Fg(\000)p Ft(1\))p Fn(x)2690 2957 y Fo(=)p Fq(\()p Fo(c)2809 2971 y Ft(1)2857 2957 y Fp(\000)6 b Fq(1\))26 b Fp(!)f Fq(0)f(as)g Fo(x)h Fp(!)g(\0001)p Fq(.)159 3175 y(Assume)33 b(that)h(\(i\),)h(\(ii\))e(\(2.27\))j(and)d (\(2.28\))j(are)e(true)g(up)f(to)h Fo(m)22 b Fp(\000)g Fq(1.)52 b(First)33 b(of)h(all)e(let)i(us)f(sho)m(w)h(the)50 3321 y(existence)41 b(of)f Fp(E)8 b Fq(.)71 b(W)-8 b(e)41 b(ma)m(y)g(consider)e Fp(E)48 b Fq(as)41 b(the)f(cluster)g(pro)s(cess)g (\(see)h([D)m(V])h(Chapter)e(8\))h(with)e(the)50 3466 y(pro)s(cess)28 b Fo(N)438 3480 y Fn(c)504 3466 y Fq(of)h(the)g(cen)m (ters)h(of)f(clusters)f Fo(\015)1543 3480 y Ft(1)1584 3466 y Fo(x)1636 3480 y Ft(1)1694 3466 y Fq(+)16 b Fp(\001)f(\001)g (\001)k Fq(+)d Fo(\015)2039 3480 y Fn(m)p Fg(\000)p Ft(1)2202 3466 y Fo(x)2254 3480 y Fn(m)p Fg(\000)p Ft(1)2444 3466 y Fq(at)30 b(the)f(pro)5 b(jection)29 b(p)s(oin)m(ts)e(of)i Fp(P)3583 3433 y Ft(\()p Fn(m)p Ft(\))50 3611 y Fq(on)c(the)g(plane)f (of)i(the)f(\014rst)f Fo(m)10 b Fp(\000)g Fq(1)25 b(co)s(ordinates)g (and)f(the)h(indep)s(enden)m(t)e(P)m(oisson)i(p)s(oin)m(t)f(pro)s (cesses)h(of)h(the)50 3757 y(in)m(tensit)m(y)k Fo(e)464 3724 y Fg(\000)p Fn(x)566 3757 y Fo(dx)h Fq(corresp)s(onding)d(to)k (eac)m(h)f(cluster.)40 b(Then)30 b(b)m(y)g(Prop)s(osition)e(8.2.I)s(I)j (in)e([D)m(V])j(it)e(su\016ces)50 3902 y(to)35 b(sho)m(w)g(that)g(for)g (an)m(y)g(b)s(ounded)d Fo(A)h Fp(2)f Fe(R)1585 3829 y Ff(R)1628 3935 y Fn(R)1703 3902 y Fe(P)p Fq(\()p Fo(A)h Fp(j)g Fo(\015)1999 3916 y Ft(1)2040 3902 y Fo(x)2092 3916 y Ft(1)2156 3902 y Fq(+)22 b Fp(\001)15 b(\001)g(\001)25 b Fq(+)d Fo(\015)2519 3916 y Fn(m)p Fg(\000)p Ft(1)2682 3902 y Fo(x)2734 3916 y Fn(m)p Fg(\000)p Ft(1)2896 3902 y Fq(\))p Fo(N)3004 3916 y Fn(c)3041 3902 y Fq(\()p Fo(dx)p Fq(\))36 b(is)d(\014nite)h(a.s.)50 4048 y(where)c Fe(P)p Fq(\()p Fo(A)d Fp(j)g Fo(\015)597 4062 y Ft(1)638 4048 y Fo(x)690 4062 y Ft(1)751 4048 y Fq(+)20 b Fo(:)15 b(:)g(:)h(\015)1010 4062 y Fn(m)p Fg(\000)p Ft(1)1172 4048 y Fo(x)1224 4062 y Fn(m)p Fg(\000)p Ft(1)1387 4048 y Fq(\))31 b(is)f(the)h(probabilit)m (y)d(that)k Fo(A)f Fq(con)m(tains)g(no)f(p)s(oin)m(t)g(of)h(the)g (cluster)50 4193 y(with)i(the)j(cen)m(ter)g Fo(\015)745 4207 y Ft(1)786 4193 y Fo(x)838 4207 y Ft(1)902 4193 y Fq(+)23 b Fo(:)15 b(:)g(:)24 b Fq(+)f Fo(\015)1266 4207 y Fn(m)p Fg(\000)p Ft(1)1428 4193 y Fo(x)1480 4207 y Fn(m)p Fg(\000)p Ft(1)1642 4193 y Fq(.)55 b(It)35 b(su\016ces)f(to)i (pro)m(v)m(e)g(this)e(for)h Fo(A)e Fq(=)f([)p Fo(a;)15 b Fp(1)p Fq(\).)56 b(A)35 b(simple)50 4339 y(calculation)29 b(giv)m(es)i(that)332 4594 y Fe(P)p Fq(\()p Fo(A)c Fp(j)e Fo(\015)614 4608 y Ft(1)655 4594 y Fo(x)707 4608 y Ft(1)768 4594 y Fq(+)20 b Fo(:)15 b(:)g(:)21 b Fq(+)f Fo(\015)1123 4608 y Fn(m)p Fg(\000)p Ft(1)1285 4594 y Fo(x)1337 4608 y Fn(m)p Fg(\000)p Ft(1)1499 4594 y Fq(\))26 b(=)f(1)c Fp(\000)f Fq(exp)o Fp(f\000)p Fo(e)2109 4556 y Fg(\000)p Ft(\()p Fn(a)p Fg(\000)p Fn(\015)2328 4565 y Fj(1)2365 4556 y Fn(x)2406 4565 y Fj(1)2442 4556 y Fg(\000\001\001\001)n(\000)p Fn(\015)2658 4565 y Fi(m)p Fd(\000)p Fj(1)2799 4556 y Fn(x)2840 4565 y Fi(m)p Fd(\000)p Fj(1)2981 4556 y Ft(\))p Fn(=\015)3084 4565 y Fi(m)3149 4594 y Fp(g)1560 4766 y(\024)25 b Fo(e)1698 4729 y Ft(\()p Fg(\000)p Fn(a)p Ft(+)p Fn(\015)1916 4738 y Fj(1)1952 4729 y Fn(x)1993 4738 y Fj(1)2029 4729 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(\015)2244 4738 y Fi(m)p Fd(\000)p Fj(1)2384 4729 y Fn(x)2425 4738 y Fi(m)p Fd(\000)p Fj(1)2566 4729 y Ft(\))p Fn(=\015)2669 4738 y Fi(m)3477 4676 y Fq(\(2)p Fo(:)p Fq(29\))50 5013 y(and)50 5178 y Ff(Z)100 5385 y Fn(R)175 5302 y Fo(e)217 5264 y Ft(\()p Fn(\015)283 5273 y Fj(1)320 5264 y Fn(x)361 5273 y Fj(1)396 5264 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(\015)611 5273 y Fi(m)p Fd(\000)p Fj(1)751 5264 y Fn(x)792 5273 y Fi(m)p Fd(\000)p Fj(1)933 5264 y Ft(\))p Fn(=\015)1036 5273 y Fi(m)1102 5302 y Fo(N)1175 5316 y Fn(c)1212 5302 y Fq(\()p Fo(dx)p Fq(\))h(=)1503 5178 y Ff(Z)1554 5385 y Fn(R)1609 5366 y Fi(m)p Fd(\000)p Fj(1)1769 5302 y Fo(e)1811 5264 y Ft(\()p Fn(\015)1877 5273 y Fj(1)1913 5264 y Fn(x)1954 5273 y Fj(1)1990 5264 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(\015)2205 5273 y Fi(m)p Fd(\000)p Fj(1)2345 5264 y Fn(x)2386 5273 y Fi(m)p Fd(\000)p Fj(1)2527 5264 y Ft(\))p Fn(=\015)2630 5273 y Fi(m)2696 5302 y Fp(P)2766 5264 y Ft(\()p Fn(m)p Fg(\000)p Ft(1\))2985 5302 y Fq(\()p Fo(dx)3119 5316 y Ft(1)3161 5302 y Fo(;)15 b(:)g(:)g(:)h(dx)3421 5316 y Fn(m)p Fg(\000)p Ft(1)3584 5302 y Fq(\))25 b Fo(<)g Fp(1)121 b Fq(a.s.)3477 5462 y(\(2)p Fo(:)p Fq(30\))p eop %%Page: 22 22 22 21 bop 50 295 a Fk(22)3270 b Fh(Se)l(ction)27 b Fk(2)50 513 y Fq(b)m(y)j(\(2.27\))j(for)d Fo(m)20 b Fp(\000)g Fq(1,)31 b(since)f Fo(\015)1140 527 y Ft(1)1181 513 y Fo(=\015)1273 527 y Fn(m)1368 513 y Fo(>)25 b Fp(\001)15 b(\001)g(\001)26 b Fo(>)f(\015)1738 527 y Fn(m)p Fg(\000)p Ft(1)1901 513 y Fo(=\015)1993 527 y Fn(m)2088 513 y Fo(>)g Fq(1.)41 b(Then)29 b Fp(E)39 b Fq(exists)30 b(and)g(has)g(an)g(a.s.)42 b(\014nite)50 658 y(n)m(um)m(b)s(er)26 b(of)j(p)s(oin)m(ts)e(in)f(an)m (y)j(b)s(ounded)c(subset)j(of)g Fp(R)p Fq(.)40 b(F)-8 b(urthermore,)28 b(let)g Fo(y)g Fq(=)d Fo(a=)p Fq(\()p Fo(\015)2935 672 y Ft(1)2993 658 y Fq(+)15 b Fp(\001)g(\001)g(\001)h Fq(+)f Fo(\015)3333 672 y Fn(m)3403 658 y Fq(\).)40 b(Then)144 908 y Fo(P)202 922 y Fn(j)266 908 y Fp(\021)25 b Fe(P)417 807 y Ff(\020)472 908 y Fp(9)p Fq(\()p Fo(x)610 922 y Ft(1)651 908 y Fo(;)15 b(:)g(:)g(:)31 b(;)15 b(x)919 922 y Fn(m)989 908 y Fq(\))25 b Fp(2)g(P)1205 871 y Ft(\()p Fn(m)p Ft(\))1357 908 y Fq(:)h Fp(8)p Fo(i)f Fq(=)g(1)p Fo(;)15 b Fq(2)p Fo(;)g(:)g(:)g(:)33 b(;)15 b(j)26 b Fp(\000)20 b Fq(1)31 b Fo(\015)2236 922 y Ft(1)2276 908 y Fo(x)2328 922 y Ft(1)2389 908 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(\015)2744 922 y Fn(i)2775 908 y Fo(x)2827 922 y Fn(i)2882 908 y Fp(\024)25 b Fq(\()p Fo(\015)3060 922 y Ft(1)3122 908 y Fq(+)19 b Fp(\001)c(\001)g(\001)22 b Fq(+)e Fo(\015)3477 922 y Fn(i)3507 908 y Fq(\))p Fo(y)s(;)968 1108 y(\015)1015 1122 y Ft(1)1056 1108 y Fo(x)1108 1122 y Ft(1)1169 1108 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(\015)1524 1122 y Fn(j)1562 1108 y Fo(x)1614 1122 y Fn(j)1678 1108 y Fo(>)25 b Fq(\()p Fo(\015)1856 1122 y Ft(1)1917 1108 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(\015)2272 1122 y Fn(j)2311 1108 y Fq(\))p Fo(y)2394 1007 y Ff(\021)266 1327 y Fp(\024)362 1204 y Ff(Z)868 1341 y Fj(\()p Fi(x)929 1355 y Fj(1)964 1341 y Fi(;:::)10 b(;x)1115 1357 y(k)1155 1341 y Fj(\))p Fd(2)p Fa(R)1252 1328 y Fi(k)1295 1341 y Fj(:)434 1402 y Fd(8)p Fi(i)p Fj(=1)p Fi(;)p Fj(2)p Fi(;:::)g(;j)s Fd(\000)p Fj(1)30 b Fi(\015)914 1416 y Fj(1)949 1402 y Fi(x)985 1416 y Fj(1)1021 1402 y(+)p Fd(\001\001\001)o Fj(+)p Fi(\015)1211 1417 y(i)1239 1402 y(x)1275 1417 y(i)1304 1402 y Fd(\024)p Fj(\()p Fi(\015)1412 1416 y Fj(1)1448 1402 y(+)p Fd(\001\001\001)o Fj(+)p Fi(\015)1638 1417 y(i)1666 1402 y Fj(\))p Fi(y)r(;)660 1462 y(\015)694 1476 y Fj(1)729 1462 y Fi(x)765 1476 y Fj(1)801 1462 y(+)p Fd(\001\001\001)o Fj(+)p Fi(\015)991 1477 y(j)1025 1462 y(x)1061 1477 y(j)1095 1462 y(>)p Fj(\()p Fi(\015)1203 1476 y Fj(1)1238 1462 y(+)p Fd(\001\001\001)o Fj(+)p Fi(\015)1428 1477 y(j)1462 1462 y Fj(\))p Fi(y)1788 1327 y Fo(e)1830 1290 y Fg(\000)p Fn(x)1928 1299 y Fj(1)1964 1290 y Fg(\000\001\001\001)o(\000)p Fn(x)2184 1299 y Fi(j)2237 1327 y Fo(dx)2336 1341 y Ft(1)2392 1327 y Fp(\001)15 b(\001)g(\001)i Fo(dx)2613 1341 y Fn(j)2676 1327 y Fq(=)25 b Fo(e)2814 1290 y Fg(\000)p Fn(j)t(y)3477 1562 y Fq(\(2)p Fo(:)p Fq(31\))50 1708 y(leading)31 b(to)i(the)f(exp)s(onen)m(tial)f(b) s(ound)g Fe(P)1463 1607 y Ff(\020)1517 1708 y Fp(9)p Fo(x)d Fp(2)g(E)37 b Fq(:)28 b Fo(x)g(>)g(a)2102 1607 y Ff(\021)2185 1708 y Fp(\024)2284 1640 y Ff(P)2380 1662 y Fn(m)2380 1735 y(j)t Ft(=1)2526 1708 y Fo(P)2584 1722 y Fn(j)2651 1708 y Fp(\024)2750 1640 y Ff(P)2846 1662 y Fn(m)2846 1735 y(j)t Ft(=1)2991 1708 y Fo(e)3033 1675 y Fg(\000)p Fn(j)t(a=)p Ft(\()p Fn(\015)3266 1684 y Fj(1)3303 1675 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(\015)3517 1684 y Fi(j)3551 1675 y Ft(\))3584 1708 y Fo(:)33 b Fq(It)50 1853 y(follo)m(ws)i(b)m(y)h(Borel-Can)m(telli)g(lemma)f(that)i Fp(E)44 b Fq(has)36 b(an)h(a.s.)58 b(\014nite)36 b(n)m(um)m(b)s(er)f (of)h(p)s(oin)m(ts)f(in)g(an)m(y)i(in)m(terv)-5 b(al)50 1999 y([)p Fo(a;)15 b Fp(1)p Fq(\))31 b(and)f(\(ii\))f(is)h(pro)m(v)m (en.)159 2217 y(No)m(w)i(\(2.28\))h(is)d(almost)h(immediate.)41 b(F)-8 b(or)32 b(an)m(y)f Fo(\017)26 b(>)g Fq(0)31 b(one)g(can)h (\014nd)d(a)i(large)g Fo(A)p Fq(\()p Fo(\017;)15 b(a)p Fq(\))32 b(suc)m(h)f(that)h(the)50 2362 y(probabilit)m(y)c(\(2.28\))k (is)e(not)g(bigger)g(than)540 2593 y Fe(P)p Fq(\()p Fp(9)91 b Fq(inf.)39 b(man)m(y)91 b(\()p Fo(x)1336 2607 y Ft(1)1377 2593 y Fo(;)15 b(:)g(:)g(:)i(;)e(x)1631 2607 y Fn(m)1700 2593 y Fq(\))26 b Fp(2)f(P)1917 2556 y Ft(\()p Fn(m)p Ft(\))2043 2593 y Fo(\015)2090 2607 y Ft(1)2131 2593 y Fo(x)2183 2607 y Ft(1)2244 2593 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)h Fo(x)2508 2607 y Fn(m)2577 2593 y Fo(\015)2624 2607 y Fn(m)2719 2593 y Fo(>)25 b(a)600 2766 y Fp(8)p Fo(i)g Fq(=)g(1)p Fo(;)15 b(:)g(:)g(:)33 b(;)15 b(m)20 b Fp(\000)g Fq(1)p Fo(;)15 b Fp(j)p Fo(\015)1414 2780 y Ft(1)1456 2766 y Fo(x)1508 2780 y Ft(1)1569 2766 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(\015)1924 2780 y Fn(i)1954 2766 y Fo(x)2006 2780 y Fn(i)2036 2766 y Fp(j)26 b Fo(<)f Fq(\()p Fo(\015)2265 2780 y Ft(1)2326 2766 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(\015)2681 2780 y Fn(i)2712 2766 y Fq(\))p Fo(A)p Fq(\()p Fo(\017;)15 b(a)p Fq(\)\))22 b(+)e Fo(\017:)3477 2676 y Fq(\(2)p Fo(:)p Fq(32\))50 2989 y(This)26 b(follo)m(ws)h(from)g(the)i 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Ft(1)2250 3686 y Fq(+)g Fp(\001)15 b(\001)g(\001)21 b Fq(+)f Fo(\015)2605 3700 y Fn(j)2643 3686 y Fq(\))p Fo(A)p Fq(\()p Fo(a;)15 b(\017)p Fq(\)\))23 b(+)d Fo(\017)50 3903 y Fq(where)36 b(the)g(terms)h(in)e(the)h(sum)g(equal)g(zero)h(b)m (y)f(induction)e(assumption.)57 b(Since)36 b Fo(\017)f(>)g Fq(0)i(is)e(arbitrary)-8 b(,)50 4049 y(\(2.28\))32 b(follo)m(ws.)159 4267 y(T)-8 b(o)31 b(sho)m(w)f(\(2.27\))j(w)m(e)d(consider)343 4412 y Ff(Z)728 4586 y Fj(\()p Fi(x)789 4600 y Fj(1)824 4586 y Fi(;:::)11 b(;x)976 4595 y(m)1036 4586 y Fj(\))p Fd(2)p Fa(R)1133 4573 y Fi(m)1197 4586 y(;)404 4644 y Fd(9)p Fi(i;)p Fj(1)p Fd(\024)p Fi(i)p Fd(\024)p Fi(m)p Fj(:)p Fi(c)743 4658 y Fj(1)778 4644 y Fi(x)814 4658 y Fj(1)850 4644 y(+)p Fd(\001\001\001)o Fj(+)p Fi(c)1036 4659 y(i)1064 4644 y(x)1100 4659 y(i)1129 4644 y(>)p Fj(\()p Fi(c)1233 4658 y Fj(1)1268 4644 y(+)p Fd(\001\001\001)o Fj(+)p Fi(c)1454 4659 y(i)1482 4644 y Fj(\))p Fi(y)1572 4536 y Fo(e)1614 4499 y Fn(c)1647 4508 y Fj(1)1682 4499 y Fn(x)1723 4508 y Fj(1)1759 4499 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(c)1969 4508 y Fi(m)2028 4499 y Fn(x)2069 4508 y Fi(m)2135 4536 y Fp(P)2205 4499 y Ft(\()p Fn(m)p Ft(\))2331 4536 y Fq(\()p Fo(dx)2465 4550 y Ft(1)2522 4536 y Fo(:)k(:)g(:)h(dx) 2742 4550 y Fn(m)2811 4536 y Fq(\))353 4846 y(=)482 4732 y Fn(m)449 4759 y Ff(X)449 4955 y Fn(k)r Ft(=1)596 4722 y Ff(Z)1031 4851 y Fj(\()p Fi(x)1092 4865 y Fj(1)1128 4851 y Fi(;:::)10 b(;x)1279 4860 y(m)1339 4851 y Fj(\))p Fd(2)p Fa(R)1436 4838 y Fi(m)1500 4851 y(;)668 4909 y Fd(8)p Fi(i)p Fj(=1)p Fi(;:::)g(;k)q Fd(\000)p Fj(1:)p Fi(c)1085 4923 y Fj(1)1121 4909 y Fi(x)1157 4923 y Fj(1)1192 4909 y(+)p Fd(\001\001\001)p Fj(+)p Fi(c)1379 4924 y(i)1407 4909 y(x)1443 4924 y(i)1471 4909 y Fd(\024)p Fj(\()p Fi(c)1575 4923 y Fj(1)1610 4909 y(+)p Fd(\001\001\001)o Fj(+)p Fi(c)1796 4924 y(i)1824 4909 y Fj(\))p Fi(y)846 4969 y(c)876 4983 y Fj(1)911 4969 y Fi(x)947 4983 y Fj(1)982 4969 y(+)p Fd(\001\001\001)o Fj(+)p Fi(c)1168 4985 y(k)1207 4969 y(x)1243 4985 y(k)1283 4969 y(>)p Fj(\()p Fi(c)1387 4983 y Fj(1)1422 4969 y(+)p Fd(\001\001\001)o Fj(+)p Fi(c)1608 4985 y(k)1647 4969 y Fj(\))p Fi(y)1925 4846 y Fo(e)1967 4808 y Fn(c)2000 4817 y Fj(1)2036 4808 y Fn(x)2077 4817 y Fj(1)2112 4808 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(c)2322 4817 y Fi(m)2382 4808 y Fn(x)2423 4817 y Fi(m)2488 4846 y Fp(P)2558 4808 y Ft(\()p Fn(m)p Ft(\))2684 4846 y Fq(\()p Fo(dx)2818 4860 y Ft(1)2875 4846 y Fo(:)15 b(:)g(:)h(dx)3095 4860 y Fn(m)3165 4846 y Fq(\))3477 4721 y(\(2)p Fo(:)p Fq(33\))50 5171 y(where)30 b(eac)m(h)h(term)g(is)e (determined)g(b)m(y)i(induction.)38 b(In)30 b(fact,)i(for)e(the)h Fo(k)s Fq(th)f(term,)h(giv)m(en)f(an)m(y)h(realisation)50 5316 y(of)25 b Fp(P)218 5283 y Ft(\()p Fn(m)p Ft(\))370 5316 y Fq(in)f Fe(R)530 5283 y Fn(m)605 5316 y Fq(,)j(tak)m(e)f(its)f (pro)5 b(jection)25 b(on)g(the)g(plane)f(of)h(the)h(\014rst)e Fo(k)k Fq(co)s(ordinates.)39 b(By)25 b(\(2.28\))j(for)d Fo(k)j(<)d(m)50 5462 y Fq(it)g(con)m(tains)h(a)g(\014nite)e(n)m(um)m(b) s(er)h(of)g(p)s(oin)m(ts)g(\()p Fo(x)1567 5476 y Ft(1)1608 5462 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(x)1877 5476 y Fn(k)1922 5462 y Fq(\))26 b(in)e(the)i(domain)e(of)i Fe(R)2713 5429 y Fn(k)2789 5462 y Fq(where)f Fo(c)3086 5476 y Ft(1)3127 5462 y Fo(x)3179 5476 y Ft(1)3231 5462 y Fq(+)11 b Fp(\001)k(\001)g(\001)g Fo(c)3472 5476 y Fn(k)3517 5462 y Fo(x)3569 5476 y Fn(k)3639 5462 y Fo(>)p eop %%Page: 23 23 23 22 bop 50 295 a Fh(GREM,)28 b(p)l(art)f(1)3089 b Fk(23)50 513 y Fq(\()p Fo(c)124 527 y Ft(1)189 513 y Fq(+)24 b Fp(\001)15 b(\001)g(\001)24 b Fq(+)g Fo(c)547 527 y Fn(k)592 513 y Fq(\))p Fo(y)s Fq(.)57 b(Whenev)m(er)36 b(the)g(\014rst)f Fo(k)k Fq(co)s(ordinates)d(of)g(a)g(p)s(oin)m(t)f(of)h Fp(P)2731 480 y Ft(\()p Fn(m)p Ft(\))2893 513 y Fq(in)e Fe(R)3064 480 y Fn(m)3175 513 y Fq(are)i(\014xed,)h(the)50 658 y(remaining)c Fo(m)22 b Fp(\000)h Fo(k)38 b Fq(co)s(ordinates)c (are)h(distributed)d(as)j Fp(P)2059 625 y Ft(\()p Fn(m)p Fg(\000)p Fn(k)r Ft(\))2317 658 y Fq(in)f Fe(R)2487 625 y Fn(m)p Fg(\000)p Fn(k)2659 658 y Fq(.)54 b(Then)34 b(the)g(in)m(tegral)h(of)g(the)50 804 y(function)g Fo(e)454 771 y Fn(c)487 781 y Fi(k)q Fj(+1)604 771 y Fn(x)645 781 y Fi(k)q Fj(+1)763 771 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(c)972 780 y Fi(m)1032 771 y Fn(x)1073 780 y Fi(m)1174 804 y Fq(o)m(v)m(er)i(these)g(co)s(ordinates)f(is)f (de\014ned)g(b)m(y)h(induction)e(and)i(is)f(\014nite)g(a.s.)50 949 y(Th)m(us)28 b(the)i Fo(k)s Fq(th)f(term)h(of)f(\(2.33\))j(is)d (the)g(sum)g(of)g(an)h(a.s.)41 b(\014nite)28 b(n)m(um)m(b)s(er)g(of)i (terms)f(and)g(eac)m(h)i(of)e(them)h(is)50 1095 y(a.s.)41 b(\014nite.)f(Finally)28 b(w)m(e)j(put)445 1156 y Ff(Z)496 1363 y Fb(R)546 1345 y Fi(m)622 1280 y Fo(e)664 1243 y Fn(c)697 1252 y Fj(1)732 1243 y Fn(x)773 1252 y Fj(1)809 1243 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(c)1018 1252 y Fi(m)1078 1243 y Fn(x)1119 1252 y Fi(m)1184 1280 y Fp(P)1254 1243 y Ft(\()p Fn(m)p Ft(\))1381 1280 y Fq(\()p Fo(dx)1515 1294 y Ft(1)1571 1280 y Fo(:)15 b(:)g(:)h(dx)1791 1294 y Fn(m)1861 1280 y Fq(\))455 1523 y Fp(\021)83 b Fq(lim)551 1578 y Fn(x)p Fg(!\0001)809 1399 y Ff(Z)1195 1573 y Fj(\()p Fi(x)1256 1587 y Fj(1)1291 1573 y Fi(;:::)10 b(;x)1442 1582 y(m)1503 1573 y Fj(\))p Fd(2)p Fa(R)1600 1560 y Fi(m)1663 1573 y(;)871 1631 y Fd(9)p Fi(i;)p Fj(1)p Fd(\024)p Fi(i)p Fd(\024)p Fi(m)p Fj(:)p Fi(c)1210 1645 y Fj(1)1245 1631 y Fi(x)1281 1645 y Fj(1)1317 1631 y(+)p Fd(\001\001\001)o Fj(+)p Fi(c)1503 1646 y(i)1531 1631 y(x)1567 1646 y(i)1596 1631 y(>)p Fj(\()p Fi(c)1700 1645 y Fj(1)1735 1631 y(+)p Fd(\001\001\001)o Fj(+)p Fi(c)1921 1646 y(i)1949 1631 y Fj(\))p Fi(y)2039 1523 y Fo(e)2081 1486 y Fn(c)2114 1495 y Fj(1)2149 1486 y Fn(x)2190 1495 y Fj(1)2226 1486 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(c)2435 1495 y Fi(m)2495 1486 y Fn(c)2528 1495 y Fi(m)2592 1523 y Fp(P)2662 1486 y Ft(\()p Fn(m)p Ft(\))2789 1523 y Fq(\()p Fo(dx)2923 1537 y Ft(1)2980 1523 y Fo(:)15 b(:)g(:)h(dx)3200 1537 y Fn(m)3269 1523 y Fq(\))p Fo(:)50 1753 y Fq(This)28 b(limit)h(is)g(\014nite)g(a.s.)42 b(as)30 b(it)g(v)m(eri\014es)g(Cauc)m (h)m(y's)h(criterion.)39 b(Denote)32 b(b)m(y)472 1939 y Fo(I)7 b Fq(\()p Fo(y)s Fq(\))26 b(=)759 1815 y Ff(Z)1144 1989 y Fj(\()p Fi(x)1205 2003 y Fj(1)1240 1989 y Fi(;:::)11 b(;x)1392 1998 y(m)1452 1989 y Fj(\))p Fd(2)p Fa(R)1549 1976 y Fi(m)1613 1989 y(;)820 2047 y Fd(9)p Fi(i;)p Fj(1)p Fd(\024)p Fi(i)p Fd(\024)p Fi(m)p Fj(:)p Fi(c)1159 2061 y Fj(1)1195 2047 y Fi(x)1231 2061 y Fj(1)1266 2047 y(+)p Fd(\001\001\001)o Fj(+)p Fi(c)1452 2062 y(i)1480 2047 y(x)1516 2062 y(i)1545 2047 y(>)p Fj(\()p Fi(c)1649 2061 y Fj(1)1684 2047 y(+)p Fd(\001\001\001)o Fj(+)p Fi(c)1870 2062 y(i)1898 2047 y Fj(\))p Fi(y)1988 1939 y Fo(e)2030 1902 y Fn(c)2063 1911 y Fj(1)2099 1902 y Fn(x)2140 1911 y Fj(1)2175 1902 y Ft(+)p Fg(\001\001\001)o Ft(+)p Fn(c)2385 1911 y Fi(m)2444 1902 y Fn(x)2485 1911 y Fi(m)2551 1939 y Fp(P)2621 1902 y Ft(\()p Fn(m)p Ft(\))2747 1939 y Fq(\()p Fo(dx)2881 1953 y Ft(1)2938 1939 y Fo(:)k(:)g(:)h(dx)3158 1953 y Fn(m)3227 1939 y Fq(\))p Fo(:)50 2174 y Fq(Then)29 b(for)h(an)m(y)h Fo(\017)25 b(>)g Fq(0)31 b(and)f(an)m(y)g Fo(y)s(;)15 b(y)1316 2141 y Fg(0)1367 2174 y Fp(2)25 b Fe(R)s Fq(,)36 b Fo(y)1622 2141 y Fg(0)1673 2174 y Fo(<)25 b(y)33 b Fq(under)c(assumptions)f Fo(c)2660 2188 y Ft(1)2727 2174 y Fo(>)d Fp(\001)15 b(\001)g(\001)26 b Fo(>)f(c)3089 2188 y Fn(m)3184 2174 y Fo(>)g Fq(1)399 2315 y Fo(P)13 b Fq(\()p Fp(j)p Fo(I)7 b Fq(\()p Fo(y)s Fq(\))21 b Fp(\000)f Fo(I)7 b Fq(\()p Fo(y)937 2277 y Fg(0)963 2315 y Fq(\))p Fp(j)26 b Fo(>)f(\017)p Fq(\))g Fp(\024)g Fo(\017)1375 2277 y Fg(\000)p Ft(2)1473 2315 y Fe(E)10 b Fq(\()q Fo(I)d Fq(\()p Fo(y)1693 2277 y Fg(0)1725 2315 y Fq(\))20 b Fp(\000)g Fo(I)7 b Fq(\()p Fo(y)s Fq(\)\))495 2507 y Fp(\024)25 b Fo(\017)628 2469 y Fg(\000)p Ft(2)741 2383 y Ff(Z)1169 2556 y Fj(\()p Fi(x)1230 2570 y Fj(1)1265 2556 y Fi(;:::)11 b(;x)1417 2565 y(m)1477 2556 y Fj(\))p Fd(2)p Fa(R)1574 2543 y Fi(m)1638 2556 y Fj(:)802 2615 y Fd(8)p Fi(i)p Fj(=1)p Fi(;)p Fj(2)p Fi(;:::)f(;m)30 b(c)1224 2629 y Fj(1)1259 2615 y Fi(x)1295 2629 y Fj(1)1330 2615 y(+)p Fd(\001\001\001)p Fj(+)p Fi(c)1517 2630 y(i)1545 2615 y(x)1581 2630 y(i)1609 2615 y Fd(\024)p Fj(\()p Fi(c)1713 2629 y Fj(1)1748 2615 y(+)p Fd(\001\001\001)p Fj(+)p Fi(c)1935 2630 y(i)1963 2615 y Fj(\))p Fi(x)2054 2507 y Fo(e)2096 2469 y Ft(\()p Fn(c)2157 2478 y Fj(1)2192 2469 y Fg(\000)p Ft(1\))p Fn(x)2354 2478 y Fj(1)2391 2469 y Ft(+)p Fg(\001\001\001)n Ft(+\()p Fn(c)2628 2478 y Fi(m)2688 2469 y Fg(\000)p Ft(1\))p Fn(x)2850 2478 y Fi(m)2916 2507 y Fo(dx)3015 2521 y Ft(1)3071 2507 y Fp(\001)15 b(\001)g(\001)h Fo(dx)3291 2521 y Fn(m)495 2751 y Fq(=)25 b Fo(\017)628 2714 y Fg(\000)p Ft(2)726 2751 y Fo(e)768 2714 y Ft(\()p Fn(c)829 2723 y Fj(1)864 2714 y Fg(\000)p Ft(1\))p Fn(y)r Ft(+)p Fg(\001\001\001)p Ft(+\()p Fn(c)1263 2723 y Fi(m)1323 2714 y Fg(\000)p Ft(1\))p Fn(y)1513 2751 y Fp(!)g Fq(0)p Fo(;)198 b(y)28 b Fp(!)d(\0001)50 2891 y Fq(and)30 b(Borel-Can)m(telli)f(lemma)h (yields)e(the)j(result.)159 3109 y(T)-8 b(o)31 b(start)g(with)e(\(i\),) h(let)h(us)e(remark)h(that)87 3256 y Fo(U)159 3217 y Fg(\000)p Ft(1)149 3282 y Fn(J)n(;N)278 3256 y Fq(\()p Fo(X)388 3270 y Fn(\033)438 3256 y Fq(\))113 3428 y(=)25 b Fo(\015)256 3442 y Ft(1)296 3428 y Fo(u)348 3390 y Fg(\000)p Ft(1)348 3456 y(ln)c(\026)-43 b Fn(\013)469 3465 y Fj(1)504 3456 y Fn(;N)595 3428 y Fq(\()656 3405 y(\026)630 3428 y Fo(X)705 3442 y Fn(\033)747 3451 y Fj(1)783 3442 y Fn(;:::)o(\033)912 3451 y Fi(J)947 3465 y Fj(1)991 3428 y Fq(\))21 b(+)f Fo(\015)1185 3442 y Ft(2)1226 3428 y Fo(u)1278 3390 y Fg(\000)p Ft(1)1278 3456 y(ln)g(\026)-43 b Fn(\013)1398 3465 y Fj(2)1434 3456 y Fn(;N)1524 3428 y Fq(\()1586 3405 y(\026)1559 3428 y Fo(X)1641 3374 y Fn(\033)1683 3383 y Fj(1)1720 3374 y Fn(;:::)n(\033)1848 3383 y Fi(J)1883 3397 y Fj(1)1634 3451 y Fn(\033)1676 3460 y Fi(J)1711 3474 y Fj(1)1748 3460 y(+1)1830 3451 y Ft(+)p Fg(\001\001\001)n Ft(+)p Fn(\033)2048 3460 y 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Fn(\013)2171 3812 y Fi(m)2231 3803 y Fn(;N)2322 3789 y Fq(\()p Fo(x)p Fq(\)\))p Fo(=)2524 3712 y Fp(p)p 2601 3712 83 4 v 2601 3789 a Fo(N)10 b Fq(\))33 b Fp(!)f Fe(P)p Fq(\()p Fp(9)p Fo(y)k Fp(2)c(E)40 b Fq(:)33 b Fo(y)i(>)d(x)p Fq(\))h Fp(\024)50 3935 y Fo(e)92 3902 y Fg(\000)p Fn(hx)237 3935 y Fq(.)41 b(Th)m(us)30 b(max\()p Fo(X)813 3949 y Fn(\033)862 3935 y Fo(=)907 3858 y Fp(p)p 984 3858 V 984 3935 a Fo(N)10 b Fq(\))25 b Fp(!)1244 3864 y(p)p 1319 3864 340 4 v 1319 3935 a Fq(2)q(\026)-46 b Fo(a)1412 3949 y Ft(1)1469 3935 y Fq(ln)23 b(\026)-54 b Fo(\013)1618 3949 y Ft(1)1679 3935 y Fq(+)20 b Fp(\001)15 b(\001)g(\001)21 b Fq(+)1987 3864 y Fp(p)p 2063 3864 396 4 v 71 x Fq(2)q(\026)-46 b Fo(a)2156 3949 y Fn(m)2241 3935 y Fq(ln)23 b(\026)-54 b Fo(\013)2390 3949 y Fn(m)2489 3935 y Fq(in)29 b(probabilit)m(y)-8 b(.)39 b(T)-8 b(o)31 b(deduce)f(from)50 4080 y(this)39 b(the)i(con)m(v)m(ergence)i(of)e(the)g(exp)s(ectation,)j(note)d(that)g (the)g(function)f(max\()p Fo(X)2958 4094 y Fn(\033)3008 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Fo(X)1637 4676 y Fn(\033)1687 4662 y Fo(=)1732 4585 y Fp(p)p 1808 4585 V 77 x Fo(N)10 b Fq(\))39 b(con)m(v)m(erges)j (in)c(probabilit)m(y)f(exp)s(onen)m(tially)g(fast.)50 4807 y(Then)29 b(Borel-Can)m(telli)h(lemma)f(pro)m(v)m(es)j(the)e(a.s.) 41 b(con)m(v)m(ergence.)i Fp(})p eop %%Page: 25 25 25 24 bop 50 295 a Fh(GREM,)28 b(p)l(art)f(1)3089 b Fk(25)50 513 y Fu(App)s(endix)50 731 y Fc(Prop)s(osition)38 b(3.1:)p Fm(F)-7 b(or)34 b(any)f Fo(x)25 b(>)g Fq(0)1000 994 y Ff(\020)1068 1033 y Fq(1)p 1065 1074 52 4 v 1065 1157 a Fo(x)1148 1095 y Fp(\000)1273 1033 y Fq(1)p 1250 1074 93 4 v 1250 1157 a Fo(x)1302 1131 y Ft(3)1353 994 y Ff(\021)1408 1095 y Fo(e)1450 1057 y Fg(\000)p Fn(x)1548 1030 y Fj(2)1583 1057 y Fn(=)p Ft(2)1687 1095 y Fp(\024)1783 971 y Ff(Z)1874 993 y Fg(1)1833 1177 y Fn(x)1966 1095 y Fo(e)2008 1057 y Fg(\000)p Fn(t)2092 1030 y Fj(2)2128 1057 y Fn(=)p Ft(2)2206 1095 y Fo(dt)g Fp(\024)2421 1033 y Fq(1)p 2418 1074 52 4 v 2418 1157 a Fo(x)2481 1095 y(e)2523 1057 y Fg(\000)p Fn(x)2621 1030 y Fj(2)2657 1057 y Fn(=)p Ft(2)2735 1095 y Fo(:)159 1503 y Fq(\(see)31 b([F)-8 b(e],)32 b(Lemma)f(2,)g(page)g(175\).)50 1721 y Fc(Prop)s(osition)40 b(3.2:)j Fq(Let)33 b Fo(f)10 b Fq(\()p Fo(x)1199 1735 y Ft(1)1239 1721 y Fo(;)15 b(:)g(:)g(:)i(x)1453 1735 y Fn(M)1534 1721 y Fq(\))32 b(b)s(e)f(a)i(function)d(of)i Fo(M)42 b Fq(v)-5 b(ariables)31 b(with)g(Lipsc)m(hitz)f(constan)m(t)50 1866 y Fo(L)p Fq(.)40 b(and)30 b Fo(X)429 1880 y Ft(1)470 1866 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(X)762 1880 y Fn(M)874 1866 y Fq(indep)s(enden)m(t)28 b(standard)h(Gaussian)h(r.v.)648 2078 y Fe(P)p Fq(\()p Fp(j)p Fo(f)10 b Fq(\()p Fo(X)928 2092 y Ft(1)970 2078 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(X)1262 2092 y Fn(M)1343 2078 y Fq(\))21 b Fp(\000)f Fe(E)10 b Fo(f)g Fq(\()p Fo(X)1709 2092 y Ft(1)1756 2078 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(X)2048 2092 y Fn(M)2129 2078 y Fq(\))p Fp(j)26 b Fo(>)f(\017)p Fq(\))h Fp(\024)f Fq(exp\()p Fp(\000)p Fo(\017)2787 2040 y Ft(2)2827 2078 y Fo(=)p Fq(\(2)p Fo(L)p Fq(\)\))p Fo(:)414 b Fq(\(3)p Fo(:)p Fq(1\))159 2362 y(\(see)31 b(e.g.)42 b([L)-8 b(T],)31 b(page)g(21\).)50 2553 y Fu(References)-47 2771 y Fq([Bo])47 b(A.)i(Bo)m(vier,)55 b(Statistical)48 b(mec)m(hanics)h(of)h(disordered) d(systems,)54 b(MaPh)m(ySto)c(Lecture)f(Notes)i(10,)159 2916 y(Aarh)m(us,)30 b(2001.)-118 3135 y([BK2])47 b(A.)34 b(Bo)m(vier)g(and)f(I.)h(Kurk)m(o)m(v)-5 b(a,)35 b(Derrida's)e (Generalized)g(Random)g(Energy)h(mo)s(dels)e(2.)51 b(Gibbs)32 b(mea-)159 3280 y(sures)d(and)h(probabilit)m(y)e(cascades.)42 b(preprin)m(t)29 b(2002.)-118 3498 y([BK3])47 b(A.)31 b(Bo)m(vier)h(and)f(I.)g(Kurk)m(o)m(v)-5 b(a,)32 b(Derrida's)f (Generalized)f(Random)h(Energy)g(mo)s(dels)f(3.)44 b(Mo)s(dels)30 b(with)159 3644 y(con)m(tin)m(uous)g(hierarc)m(hies.)39 b(in)29 b(preparation)h(\(2002\).)-129 3862 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)h(free)f(energy)g(in)f(the)i (REM)f(and)g(the)159 4007 y Fo(p)p Fq(-spin)e(SK-mo)s(dels,)h(Ann.)40 b(Probab.)g Fr(30)31 b Fq(\(2002\).)-130 4225 y([CCP])45 b(D.)29 b(Cap)s(o)s(caccia,)h(M.)f(Cassandro,)g(and)f(P)-8 b(.)29 b(Picco,)g(On)f(the)h(existence)g(of)g(Thermo)s(dynamics)d(for)j (the)159 4371 y(Generalised)g(Random)h(Energy)g(Mo)s(del.)40 b(J.)31 b(Stat.)41 b(Ph)m(ysics)29 b Fr(46)p Fq(,)j(493{505)h (\(1987\).)-52 4589 y([D1])47 b(B.)36 b(Derrida,)g(Random)f(energy)h (mo)s(del:)50 b(limit)34 b(of)h(a)h(family)e(of)i(disordered)e(mo)s (dels,)72 b(Ph)m(ys.)56 b(Rev.)159 4735 y(Letts.)41 b Fr(45)p Fq(,)31 b(79-82)i(\(1980\).)-52 4953 y([D2])47 b(B.)d(Derrida,)i(Random)e(energy)f(mo)s(del:)66 b(An)44 b(exactly)g(solv)-5 b(able)43 b(mo)s(del)f(of)i(disordered)d(systems,) 159 5098 y(Ph)m(ys.)f(Rev.)h(B)31 b Fr(24)p Fq(,)g(2613-2626)j (\(1981\))-52 5316 y([D3])47 b(B.)21 b(Derrida,)h(A)e(generalization)g (of)h(the)f(random)g(energy)g(mo)s(del)f(that)i(includes)d (correlations)i(b)s(et)m(w)m(een)159 5462 y(the)30 b(energies,)h(J.)f (Ph)m(ysique.)40 b(Lett.)h Fr(46)p Fq(,)31 b(401{407)j(\(1985\).)p eop %%Page: 26 26 26 25 bop 50 295 a Fk(26)3225 b Fh(R)l(efer)l(enc)l(es)-123 513 y Fq([DG1])47 b(B.)30 b(Derrida)e(and)g(E.)i(Gardner,)f(Solution)e (of)i(the)g(generalized)g(random)f(energy)i(mo)s(del,)e(J.)h(Ph)m(ys.) 40 b(C)159 658 y Fr(19)p Fq(,)31 b(2253{2274)j(\(1986\).)-123 876 y([DG2])47 b(B.)e(Derrida)e(and)h(E.)g(Gardner,)k(Magnetic)d(prop)s (erties)e(and)g(function)g Fo(q)s Fq(\()p Fo(x)p Fq(\))i(of)f(the)h (generalised)159 1022 y(random)29 b(energy)i(mo)s(del,)f(J.)g(Ph)m(ys.) 40 b(C)30 b Fr(19)p Fq(,)h(5783-5798)k(\(1986\))-123 1240 y([DG3])47 b(E.)34 b(Gardner)f(and)g(B.)i(Derrida,)f(The)g (probabilit)m(y)d(distribution)f(of)k(the)g(partition)f(function)f(of)j (the)159 1385 y(random)29 b(energy)i(mo)s(del,)f(J.)g(Ph)m(ys.)40 b(A)p Fr(22)p Fq(,)32 b(1975-1981)i(\(1989\))-72 1604 y([D)m(V])47 b(D.J)30 b(Daley)g(and)e(D.)j(V)-8 b(ere-Jones,)31 b(An)e(in)m(tro)s(duction)e(to)j(the)g(theory)g(of)f(p)s(oin)m(t)g(pro) s(cesses,)g(Sp)s(einger,)159 1749 y(New)h(Y)-8 b(ork,)32 b(1988.)-97 1967 y([D)m(W])47 b(T.C.)31 b(Dorlas)g(and)f(J.R.)i(W)-8 b(edagedera,)33 b(Large)f(deviations)e(and)h(the)g(random)f(energy)i (mo)s(del.)84 b(In)m(t.)159 2113 y(J.)30 b(Mo)s(d.)41 b(Ph)m(ys.)f Fr(B)35 b(15)p Fq(,)c(1{15)h(\(2001\).)-24 2331 y([Ei])45 b(TH.)23 b(Eisele,)i(On)d(a)i(third)e(order)h(phase)g (transition,)48 b(Comm)m(un.)38 b(Math.)h(Ph)m(ys.)f Fr(90)p Fq(,)26 b(125-159)g(\(1983\).)-29 2549 y([F)-8 b(e])47 b(W.)32 b(F)-8 b(eller,)31 b(An)f(in)m(tro)s(duction)f(to)j (probabilit)m(y)d(theory)i(and)f(its)h(applications.)40 b(V)-8 b(ol.)43 b(I,)31 b(John)f(Wiley)159 2695 y(&)g(Sons,)g(New)g(Y) -8 b(ork-London-Sydney)g(,)31 b(1966.)-154 2913 y([GMP])47 b(A.)33 b(Galv)m(ez,)i(S.)d(Martinez,)i(and)e(P)-8 b(.)34 b(Picco,)g(Fluctuations)d(in)h(Derrida's)g(random)g(energy)h(and)f (gen-)159 3058 y(eralized)d(random)h(enery)g(mo)s(dels,)60 b(J.)30 b(Stat.)41 b(Ph)m(ys.)g Fr(54)p Fq(,)31 b(515-529)i(\(1989\).) -53 3276 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.)-70 3495 y([KP])46 b(M.F.)e(Kratz)f(and)f(P)-8 b(.)43 b(Picco,)k(On)42 b(a)h(represen)m(tation)g(of)f(Gibbs)g(measure) g(for)h(R.E.M.,)k(preprin)m(t)159 3640 y(Samos)30 b(151)i(\(2002:)-118 3858 y([LLR])46 b(M.)25 b(R.)h(Leadb)s(etter,)g(G.)g(Lindgren,)e(H.)h (Ro)s(otz)m(\023)-43 b(en,)55 b(Extremes)25 b(and)f(Related)h(Prop)s (erties)f(of)h(Random)159 4004 y(Sequences)30 b(and)g(Pro)s(cesses,)h (Springer,)d(Berlin-Heidelb)s(erg-New)g(Y)-8 b(ork,)31 b(1983.)-52 4222 y([L)-8 b(T])46 b(M.)37 b(Ledoux)f(and)g(L.)h(T)-8 b(alagrand,)38 b(Probabilit)m(y)d(in)g(Banac)m(h)j(space.)60 b(Spinger,)36 b(Berlin-Heidelb)s(erg-)159 4367 y(New)30 b(Y)-8 b(ork,)32 b(1991.)-70 4585 y([OP])46 b(E.)22 b(Olivieri)c(and)j (P)-8 b(.)22 b(Picco,)i(On)d(the)h(existence)g(of)g(thermo)s(dynamics)e (for)i(the)g(random)f(energy)h(mo)s(del,)159 4731 y(Comm)m(un.)40 b(Math.)h(Ph)m(ys.)f Fr(96)p Fq(,)31 b(125-144)j(\(1991\).)-44 4949 y([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)g (pro)s(cesses,)i(Springer,)e(New)h(Y)-8 b(ork,)159 5095 y(1987.)-54 5313 y([Ru])45 b(D.)25 b(Ruelle,)g(A)f(mathematical)h (reform)m(ulation)e(of)i(Derrida's)e(REM)i(and)f(GREM.)49 b(Comm)m(un.)38 b(Math.)159 5458 y(Ph)m(ys.)81 b Fr(108)p Fq(,)31 b(225-239)i(\(1987\).)p eop %%Page: 27 27 27 26 bop 50 295 a Fh(GREM,)28 b(p)l(art)f(1)3089 b Fk(27)-58 513 y Fq([SK])45 b(D.)32 b(Sherrington)d(and)i(S.)g(Kirkpatric)m(k,)f (Solv)-5 b(able)30 b(mo)s(del)g(of)h(a)h(spin)d(glass,)63 b(Ph)m(ys.)43 b(Rev.)h(Lett.)g Fr(35)p Fq(,)159 658 y(1792-1796)34 b(\(1972\).)-48 876 y([T1])46 b(M.)30 b(T)-8 b(alagrand,)30 b(Rigorous)f(lo)m(w)g(temp)s(erature)h(results)e(for)i(mean)f(\014eld)g (p-spin)e(in)m(teraction)j(mo)s(dels.)159 1022 y(Probab.)40 b(Theor.)g(Rel.)g(Fields.)g Fr(117)31 b Fq(303-360)i(\(2000\).)-48 1240 y([T2])46 b(M.)33 b(T)-8 b(alagrand,)32 b(Concen)m(tration)h(of)f (measure)h(and)e(isop)s(erimetric)f(inequalities)g(in)h(pro)s(duct)g (space,)159 1385 y(Publ.)39 b(Math.)i(I.H.E.S.,)31 b Fr(81)p Fq(,)g(73-205)i(\(1995\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF