%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: stat.dvi %%Pages: 28 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips stat.dvi -o stat.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.06.05:1453 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 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Fs(and)f(the)h(P)m(oisson)g(la)m(w.)43 b(W)-8 b(e)33 b(pro)m(v)m(e)h(our)e(result)h(for)f(the)h(ab)s(o)m(v)m(e)g(men) m(tioned)f Fm( )t Fs(,)h Fm(\036)f Fs(and)h(strong)0 4071 y(mixing)j(pro)s(cesses,)42 b(eac)m(h)d(one)f(with)g(its)g(o)m(wn) g(error)g(term,)i(and)e(for)f(strings)h(that)g(do)g(not)g(return)0 4187 y(v)m(ery)32 b(fast)f(to)f(themselv)m(es.)44 b(The)32 b(di\013eren)m(t)f(t)m(yp)s(es)h(of)e(b)s(ounds)i(obtained)e(for)g(the) h(di\013eren)m(t)g(t)m(yp)s(es)h(of)0 4303 y(mixture)g(app)s(ears)h(as) g(a)f(c)m(haracterization)f(for)h(the)i(di\013eren)m(t)e(t)m(yp)s(es)i (of)e(mixing.)141 4437 y(The)39 b(error)e(b)s(ound)h(that)f(w)m(e)i (obtain)d(decreases)k(as)e(a)f(factorial)e(of)j Fm(N)2791 4452 y Fh(t)2820 4437 y Fs(,)h(the)f(n)m(um)m(b)s(er)g(of)f(visits)0 4553 y(to)d(the)g(string)f(up)h(to)g(time)e Fm(t)p Fs(.)48 b(This)34 b(con)m(trol)f(on)h(the)g(tail)e(distribution)g(of)h Fm(N)2921 4568 y Fh(t)2985 4553 y Fs(allo)m(w)f(us)i(to)g(obtain)0 4669 y(also)e(and)g(appro)m(ximation)f(for)h(the)h(momen)m(ts)f(of)g Fm(N)1957 4684 y Fh(t)1987 4669 y Fs(.)141 4803 y(W)-8 b(e)46 b(also)f(w)m(an)m(t)i(to)f(remark)f(that)h(the)g(con)m(v)m (ergence)j(of)c(the)i(return)f(time)f(not)g(to)h(a)g(purely)0 4919 y(exp)s(onen)m(tial)28 b(la)m(w)g(but)h(rather)f(to)h(a)f(mixture) g(of)g(an)g(exp)s(onen)m(tial)g(with)g(a)g(degenerated)i(la)m(w)e(pro)m (v)m(en)0 5035 y(in)35 b(Theorem)g(4)h(and)f(the)h(con)m(v)m(ergence)i (of)d(the)h(so)5 b(journ)36 b(time)e(to)h(a)g(geometric)g(la)m(w)g(for) f(recurren)m(t)0 5152 y(strings)h(pro)m(v)m(en)i(in)d(Theorem)i(10)f (also)f(arise)h(as)h(a)f(coun)m(ter-example)g(of)g(the)h(con)m(v)m (ergence)h(to)e(the)0 5268 y(P)m(oisson)42 b(la)m(w)e(of)h(recurren)m (t)h(ev)m(en)m(ts.)72 b(As)42 b(an)f(application)d(of)j(the)g(error)g (terms)g(w)m(e)i(are)e(allo)m(w)e(to)1855 5517 y Fi(3)p eop %%Page: 4 4 4 3 bop 0 399 a Fs(estimate)32 b(the)h(momen)m(ts)f(of)g(the)h(n)m(um)m 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Fh(m)2523 1680 y Fg(g)2573 1695 y Fh(m)p Fr(2)p Fh(Z)-39 b(Z)2795 1680 y Fg(2)48 b Fs(\012)d(and)f Fm(m)k Fg(2)g Fm(Z)-37 b(Z)7 b(;)45 b Fs(let)0 1796 y Fm(X)81 1811 y Fh(m)188 1796 y Fs(:)c(\012)h Fg(!)e(E)49 b Fs(b)s(e)40 b(the)h Fm(m)p Fs(-th)f(co)s(ordinate)g(pro)5 b(jection,)42 b(that)e(is)g Fm(X)2570 1811 y Fh(m)2636 1796 y Fs(\()p Fm(!)t Fs(\))g(=)h Fm(!)2995 1811 y Fh(m)3061 1796 y Fs(.)67 b(W)-8 b(e)41 b(denote)g(b)m(y)0 1912 y Fm(T)g Fs(:)28 b(\012)g Fg(!)g Fs(\012)33 b(the)g(one-step-left)f (shift)g(op)s(erator.)141 2028 y(W)-8 b(e)30 b(denote)g(b)m(y)g Fg(F)38 b Fs(the)30 b Fm(\033)t Fs(-algebra)d(o)m(v)m(er)j(\012)g (generated)g(b)m(y)g(cylinders.)42 b(Moreo)m(v)m(er)31 b(w)m(e)f(denote)g(b)m(y)0 2145 y Fg(F)72 2160 y Fh(I)144 2145 y Fs(the)j Fm(\033)t Fs(-algebra)e(generated)i(b)m(y)h(cylinders)f (with)f(co)s(ordinates)g(in)g Fm(I)8 b Fs(,)32 b Fm(I)k Fg(\022)28 b Fm(Z)-37 b(Z)7 b Fs(.)141 2261 y(F)-8 b(or)32 b(a)g(subset)i Fm(A)28 b Fg(\022)g Fs(\012)33 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b(P)1822 4979 y Fh(A)1895 4964 y Fg(f)p Fm(\034)1987 4979 y Fh(A)2072 4964 y Fm(>)28 b(r)s Fg(g)f Fs(=)g Fm(I)-21 b(P)2502 4979 y Fh(A)2575 4964 y Fg(f)p Fm(\034)2667 4979 y Fh(A)2752 4964 y Fm(>)28 b(n)22 b Fg(\000)h Fs(1)p Fg(g)48 b Fm(:)0 5268 y Fs(Let)33 b(us)g(de\014ne)h Fg(L)p Fs(\()p Fm(t)p Fs(\))e(as)h(follo)m(ws)1855 5517 y Fi(6)p eop %%Page: 7 7 7 6 bop 700 685 a Fg(L)p Fs(\()p Fm(t)p Fs(\))27 b(=)1011 485 y Fd(8)1011 560 y(>)1011 585 y(<)1011 734 y(>)1011 759 y(:)1126 564 y Fs(1)1177 b Fm(t)28 b(<)f(r)1126 681 y(I)-21 b(P)1224 696 y Fh(A)1298 681 y Fg(f)p Fm(\034)1390 696 y Fh(A)1475 681 y Fm(>)27 b(n)c Fg(\000)f Fs(1)p Fg(g)495 b Fm(r)30 b Fg(\024)e Fm(t)g Fg(\024)h Fm(n)22 b Fg(\000)h Fs(1)1126 801 y Fm(I)-21 b(P)1224 816 y Fh(A)1298 801 y Fg(f)p Fm(\034)1390 816 y Fh(A)1475 801 y Fm(>)27 b(n)c Fg(\000)f Fs(1)p Fg(g)16 b Fm(e)1918 765 y Fr(\000)p Fh(\030)2004 776 y Ff(A)2055 765 y Fh(I)-24 b(P)9 b Fr(f)p Fh(A)p Fr(g)p Fh(t)2352 801 y Fm(t)28 b Fg(\025)g Fm(n)3002 685 y(;)0 1084 y Fs(where)k Fm(\030)323 1099 y Fh(A)410 1084 y Fs(is)f(the)g(same)g(of)f(Theorem)h(1.)43 b(Before)31 b(to)f(state)h(the)g(theorem)g(for)f(return)i(time)d(w)m(e)j(need)0 1201 y(to)g(pro)m(v)m(e)i(the)f(follo)m(wing)d(lemma)0 1401 y Fo(Lemma)37 b(3)46 b Fl(L)-5 b(et)75 b Fg(f)p Fm(X)838 1416 y Fh(m)904 1401 y Fg(g)954 1430 y Fh(m)p Fr(2)p Fh(Z)-40 b(Z)1165 1401 y Fl(b)-5 b(e)75 b Fm(\036)p Fl(-mixing)36 b(\(at)i(any)f(r)-5 b(ate\).)53 b(L)-5 b(et)38 b Fm(A)32 b Fg(2)h(C)2801 1416 y Fh(n)2886 1401 y Fl(and)k Fm(M)44 b Fg(2)33 b Fm(I)-22 b(N)10 b Fl(.)53 b(Then,)0 1517 y(the)35 b(fol)5 b(lowing)33 b(ine)-5 b(quality)35 b(holds)201 1877 y Fm(I)-22 b(P)298 1892 y Fh(A)372 1781 y Fd(n)427 1877 y Fm(\034)469 1892 y Fh(A)548 1877 y Fg(\016)22 b Fm(T)691 1836 y Fh(n)p Fr(\000)p Fe(1)856 1877 y Fg(\024)28 b Fm(M)1065 1781 y Fd(o)1149 1877 y Fg(\024)89 b Fs(inf)1254 1936 y Fe(1)p Fr(\024)p Fh(w)r Fr(\024)p Fh(n)1511 1877 y Fm(n)1586 1781 y Fd(h)1625 1877 y Fm(I)-22 b(P)13 b Fg(f)p Fm(A)1845 1836 y Fe(\()p Fh(w)r Fe(\))1957 1877 y Fg(g)22 b Fs(+)g Fm(\036)p Fs(\()p Fm(n)g Fg(\000)g Fm(w)s Fs(\))2513 1781 y Fd(i)2574 1877 y Fs(+)g Fm(M)10 b(I)-22 b(P)14 b Fg(f)p Fm(A)p Fg(g)22 b Fs(+)3195 1769 y Fh(M)3172 1794 y Fd(X)3167 1977 y Fh(j)t Fe(=)p Fh(n)3314 1877 y Fm(\036)p Fs(\()p Fm(j)6 b Fs(\))34 b Fm(:)0 2330 y Fo(Pro)s(of.)50 b Fs(Consider)33 b Fm(M)38 b(>)28 b(n)p Fs(.)43 b(Let)33 b(1)28 b Fg(\024)g Fm(w)i Fg(\024)e Fm(n)p Fs(.)44 b(The)33 b(mixing)e(prop)s(ert)m(y)i (implies)d(that)798 2734 y Fm(I)-21 b(P)912 2637 y Fd(n)967 2734 y Fm(A)33 b Fs(;)49 b Fm(\034)1191 2749 y Fh(A)1271 2734 y Fg(\016)22 b Fm(T)1414 2692 y Fh(n)p Fr(\000)p Fe(1)1578 2734 y Fg(\024)29 b Fm(M)1788 2637 y Fd(o)638 2978 y Fs(=)798 2870 y Fh(n)p Fr(\000)p Fe(1)804 2895 y Fd(X)803 3077 y Fh(j)t Fe(=1)947 2978 y Fm(I)-21 b(P)1062 2882 y Fd(n)1117 2978 y Fm(A)33 b Fs(;)49 b Fm(\034)1341 2993 y Fh(A)1420 2978 y Fg(\016)22 b Fm(T)1563 2937 y Fh(n)p Fr(\000)p Fe(1)1728 2978 y Fs(=)27 b Fm(j)1877 2882 y Fd(o)1955 2978 y Fs(+)2081 2870 y Fh(M)2058 2895 y Fd(X)2053 3077 y Fh(j)t Fe(=)p Fh(n)2200 2978 y Fm(I)-21 b(P)2314 2882 y Fd(n)2370 2978 y Fm(A)32 b Fs(;)49 b Fm(\034)2593 2993 y Fh(A)2673 2978 y Fg(\016)22 b Fm(T)2816 2937 y Fh(n)p Fr(\000)p Fe(1)2980 2978 y Fs(=)28 b Fm(j)3130 2882 y Fd(o)637 3293 y Fg(\024)798 3186 y Fh(n)p Fr(\000)p Fe(1)804 3210 y Fd(X)803 3393 y Fh(j)t Fe(=1)947 3293 y Fm(I)-21 b(P)1062 3197 y Fd(n)1117 3293 y Fm(A)33 b Fs(;)49 b Fm(T)1370 3252 y Fr(\000)p Fe(\()p Fh(n)p Fr(\000)p Fe(1+)p Fh(j)t Fe(\))1704 3293 y Fm(A)1777 3252 y Fe(\()p Fh(w)r Fe(\))1888 3197 y Fd(o)1966 3293 y Fs(+)2092 3186 y Fh(M)2069 3210 y Fd(X)2064 3393 y Fh(j)t Fe(=)p Fh(n)2211 3293 y Fm(I)-21 b(P)2325 3197 y Fd(n)2381 3293 y Fm(A)32 b Fs(;)49 b Fm(T)2633 3252 y Fr(\000)p Fe(\()p Fh(n)p Fr(\000)p Fe(1+)p Fh(j)t Fe(\))2967 3293 y Fm(A)3040 3197 y Fd(o)637 3606 y Fg(\024)798 3498 y Fh(n)p Fr(\000)p Fe(1)804 3523 y Fd(X)803 3705 y Fh(j)t Fe(=1)947 3606 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)1235 3510 y Fd(\020)1284 3606 y Fm(I)-21 b(P)1399 3510 y Fd(n)1454 3606 y Fm(T)1525 3565 y Fr(\000)p Fe(\()p Fh(n)p Fr(\000)p Fe(1+)p Fh(j)t Fe(\))1859 3606 y Fm(A)1932 3565 y Fe(\()p Fh(w)r Fe(\))2044 3510 y Fd(o)2121 3606 y Fs(+)22 b Fm(\036)p Fs(\()p Fm(n)g Fg(\000)h Fm(w)h Fs(+)e Fm(j)6 b Fs(\))2771 3510 y Fd(\021)638 3921 y Fs(+)825 3814 y Fh(M)803 3838 y Fd(X)798 4021 y Fh(j)t Fe(=)p Fh(n)944 3921 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)1232 3825 y Fd(\020)1281 3921 y Fm(I)-21 b(P)1396 3825 y Fd(n)1451 3921 y Fm(T)1522 3880 y Fr(\000)p Fe(\()p Fh(n)p Fr(\000)p Fe(1+)p Fh(j)t Fe(\))1856 3921 y Fm(A)1929 3825 y Fd(o)2007 3921 y Fs(+)22 b Fm(\036)p Fs(\()p Fm(j)6 b Fs(\))2285 3825 y Fd(\021)2383 3921 y Fm(:)0 4307 y Fs(The)33 b(\014rst)f(inequalit)m(y)f(follo)m(ws)f (since)j Fg(f)p Fm(\034)1502 4322 y Fh(A)1586 4307 y Fs(=)28 b Fm(j)6 b Fg(g)27 b(\032)i(f)p Fm(T)2040 4271 y Fr(\000)p Fh(j)2131 4307 y Fm(A)p Fg(g)e(\032)h(f)p Fm(T)2507 4271 y Fr(\000)p Fh(j)2598 4307 y Fm(A)2671 4271 y Fe(\()p Fh(w)r Fe(\))2783 4307 y Fg(g)p Fs(.)43 b(Using)31 b(the)h(stationar-)0 4423 y(it)m(y)g(of)g(the)h(measure)g(w) m(e)h(ha)m(v)m(e)329 4783 y Fm(I)-21 b(P)427 4798 y Fh(A)501 4687 y Fd(n)556 4783 y Fm(\034)598 4798 y Fh(A)678 4783 y Fg(\016)22 b Fm(T)821 4742 y Fh(n)p Fr(\000)p Fe(1)985 4783 y Fg(\024)28 b Fm(M)1194 4687 y Fd(o)1278 4783 y Fg(\024)g Fm(n)1458 4687 y Fd(h)1497 4783 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)1718 4742 y Fe(\()p Fh(w)r Fe(\))1830 4783 y Fg(g)22 b Fs(+)g Fm(\036)p Fs(\()p Fm(n)g Fg(\000)g Fm(w)s Fs(\))2386 4687 y Fd(i)2447 4783 y Fs(+)g Fm(M)10 b(I)-21 b(P)14 b Fg(f)p Fm(A)p Fg(g)22 b Fs(+)3068 4675 y Fh(M)3046 4700 y Fd(X)3041 4882 y Fh(j)t Fe(=)p Fh(n)3188 4783 y Fm(\036)p Fs(\()p Fm(j)6 b Fs(\))32 b Fm(:)0 5152 y Fs(If)i Fm(M)42 b Fg(\024)32 b Fm(n)i Fs(the)h(second)h(term)e (disapp)s(ears.)50 b(The)35 b(b)s(ound)g(is)f(the)h(same)f(in)g(b)s (oth)g(cases.)51 b(This)35 b(ends)0 5268 y(the)e(pro)s(of)f(since)h Fm(w)i Fs(is)d(arbitrary)-8 b(.)1855 5517 y Fi(7)p eop %%Page: 8 8 8 7 bop 0 399 a Fo(Theorem)37 b(4)46 b Fl(L)-5 b(et)31 b(the)f(pr)-5 b(o)g(c)g(ess)61 b Fg(f)p Fm(X)1386 414 y Fh(m)1452 399 y Fg(g)1502 428 y Fh(m)p Fr(2)p Fh(Z)-40 b(Z)1706 399 y Fl(b)-5 b(e)61 b Fm(\036)p Fl(-mixing)30 b(with)g Fm(\036)h Fl(summable.)42 b(Then,)30 b(ther)-5 b(e)31 b(exist)0 515 y(strictly)d(p)-5 b(ositive)26 b(c)-5 b(onstants)26 b Fm(C)1153 530 y Fe(1)1219 515 y Fl(and)h Fm(C)1471 530 y Fe(2)1537 515 y Fl(such)g(that)g(for)f(any)h Fm(n)g Fl(and)g(any)g Fm(A)g Fg(2)i(R)2993 530 y Fh(r)3031 515 y Fs(\()p Fm(n)p Fs(\))p Fl(,)f(the)f(fol)5 b(lowing)0 631 y(ine)-5 b(quality)35 b(holds)1217 945 y Fg(j)p Fm(I)-22 b(P)1342 960 y Fh(A)1416 945 y Fg(f)p Fm(\034)1508 960 y Fh(A)1593 945 y Fm(>)27 b(t)p Fg(g)22 b(\000)h(L)p Fs(\()p Fm(t)p Fs(\))p Fg(j)k(\024)h Fm(e)2288 960 y Fe(2)2328 945 y Fs(\()p Fm(A)p Fs(\))35 b Fm(;)0 1240 y Fl(wher)-5 b(e)849 1717 y Fm(e)894 1732 y Fe(2)934 1717 y Fs(\()p Fm(A)p Fs(\))83 b(:=)g Fm(C)1422 1732 y Fe(1)1598 1717 y Fs(inf)1477 1790 y Fh(n)p Fr(\024)p Fh(`)p Fr(\024)1732 1763 y Fb(1)p 1669 1775 158 4 v 1669 1816 a Ff(I)-26 b(P)8 b Fk(f)p Ff(A)p Fk(g)1853 1546 y Fd(2)1853 1695 y(4)1909 1717 y Fm(`I)-22 b(P)13 b Fg(f)p Fm(A)p Fg(g)22 b Fs(+)2390 1609 y Fh(`)2345 1634 y Fd(X)2340 1816 y Fh(j)t Fe(=)p Fh(n)2486 1717 y Fm(\036)p Fs(\()p Fm(j)6 b Fs(\))2666 1546 y Fd(3)2666 1695 y(5)3631 1717 y Fs(\(4\))1179 1974 y(+)97 b Fm(C)1422 1989 y Fe(2)1496 1974 y Fm(n)112 b Fs(inf)1605 2033 y Fe(1)p Fr(\024)p Fh(w)r Fr(\024)p Fh(n)1862 1878 y Fd(h)1901 1974 y Fm(I)-22 b(P)14 b Fg(f)p Fm(A)2122 1933 y Fe(\()p Fh(w)r Fe(\))2233 1974 y Fg(g)22 b Fs(+)g Fm(\036)p Fs(\()p Fm(n)g Fg(\000)h Fm(w)s Fs(\))2790 1878 y Fd(i)2880 1974 y Fm(:)0 2495 y Fs(W)-8 b(e)35 b(remark)f(that)h(the)g(theorem)f(considers)i(the)f (la)m(w)f(of)g Fm(\034)2194 2510 y Fh(A)2286 2495 y Fs(starting)g(from) f(the)i(measure)g(concen-)0 2611 y(trated)e(in)e Fm(A)p Fs(,)i(that)g(is)f Fm(I)-21 b(P)947 2626 y Fh(A)1004 2611 y Fs(.)43 b(W)-8 b(e)33 b(call)e(this)h(the)h(\014rst)g Fl(r)-5 b(eturn)41 b Fs(time)31 b(\(to)h Fm(A)p Fs(\).)0 2822 y Fo(Pro)s(of.)50 b Fs(F)-8 b(or)31 b Fm(t)d Fg(\024)h Fm(n)22 b Fg(\000)h Fs(1,)32 b(since)h Fm(r)i Fs(is)d(the)h(p)s(erio)s (dicit)m(y)e(of)h Fm(A)p Fs(,)h(w)m(e)g(ha)m(v)m(e)h(immediately)1424 3136 y Fm(I)-21 b(P)1522 3151 y Fh(A)1596 3136 y Fg(f)p Fm(\034)1688 3151 y Fh(A)1773 3136 y Fm(>)27 b(t)p Fg(g)h Fs(=)f Fg(L)p Fs(\()p Fm(t)p Fs(\))33 b Fm(:)141 3430 y Fs(No)m(w)g(let)f Fm(`)h Fs(b)s(e)f(the)h(one)g(that)g(de\014nes)h Fm(e)1634 3445 y Fe(1)1674 3430 y Fs(\()p Fm(A)p Fs(\))e(in)g(\(4\).)43 b(F)-8 b(or)32 b Fm(t)c Fg(\025)g Fm(`)22 b Fs(+)g(2)p Fm(n)32 b Fs(w)m(e)i(ha)m(v)m(e)292 3841 y Fg(j)p Fm(I)-21 b(P)418 3856 y Fh(A)492 3841 y Fg(f)p Fm(\034)584 3856 y Fh(A)669 3841 y Fm(>)27 b(t)p Fg(g)22 b(\000)h(L)p Fs(\()p Fm(t)p Fs(\))p Fg(j)2444 b Fs(\(5\))132 3999 y Fg(\024)292 3899 y Fd(\014)292 3949 y(\014)292 3999 y(\014)p Fm(I)-21 b(P)418 4014 y Fh(A)492 3999 y Fg(f)p Fm(\034)584 4014 y Fh(A)669 3999 y Fm(>)27 b(t)p Fg(g)22 b(\000)h Fm(I)-21 b(P)1077 4014 y Fh(A)1151 3903 y Fd(n)1206 3999 y Fm(A)1279 3958 y Fh(c)1279 4024 y Fe(1)1351 3999 y Fs(;)49 b Fm(\034)1469 4014 y Fh(A)1549 3999 y Fg(\016)22 b Fm(T)1692 3958 y Fh(`)p Fe(+2)p Fh(n)1885 3999 y Fm(>)28 b(t)22 b Fg(\000)h Fs(\()p Fm(`)f Fs(+)g(2)p Fm(n)p Fs(\))2490 3903 y Fd(o)2545 3899 y(\014)2545 3949 y(\014)2545 3999 y(\014)133 4182 y Fs(+)292 4082 y Fd(\014)292 4132 y(\014)292 4182 y(\014)p Fm(I)-21 b(P)418 4197 y Fh(A)492 4085 y Fd(n)547 4182 y Fm(A)620 4141 y Fh(c)620 4206 y Fe(1)692 4182 y Fs(;)49 b Fm(\034)810 4197 y Fh(A)890 4182 y Fg(\016)22 b Fm(T)1033 4141 y Fh(`)p Fe(+2)p Fh(n)1226 4182 y Fm(>)28 b(t)22 b Fg(\000)h Fs(\()p Fm(`)f Fs(+)g(2)p Fm(n)p Fs(\))1831 4085 y Fd(o)1908 4182 y Fg(\000)h Fm(I)-21 b(P)2106 4197 y Fh(A)2163 4182 y Fg(f)p Fm(A)2286 4141 y Fh(c)2286 4206 y Fe(1)2325 4182 y Fg(g)p Fm(I)g(P)2490 4085 y Fd(n)2545 4182 y Fm(\034)2587 4197 y Fh(A)2667 4182 y Fg(\016)21 b Fm(T)2809 4141 y Fh(`)p Fe(+2)p Fh(n)3003 4182 y Fm(>)27 b(t)c Fg(\000)f Fs(\()p Fm(`)g Fs(+)g(2)p Fm(n)p Fs(\))3607 4085 y Fd(o)3663 4082 y(\014)3663 4132 y(\014)3663 4182 y(\014)133 4364 y Fs(+)83 b Fm(I)-21 b(P)390 4379 y Fh(A)447 4364 y Fg(f)p Fm(A)570 4323 y Fh(c)570 4389 y Fe(1)610 4364 y Fg(g)677 4265 y Fd(\014)676 4315 y(\014)676 4364 y(\014)p Fm(I)g(P)819 4268 y Fd(n)874 4364 y Fm(\034)916 4379 y Fh(A)995 4364 y Fg(\016)22 b Fm(T)1138 4323 y Fh(`)p Fe(+2)p Fh(n)1332 4364 y Fm(>)27 b(t)c Fg(\000)f Fs(\()p Fm(`)g Fs(+)g(2)p Fm(n)p Fs(\))1936 4268 y Fd(o)2014 4364 y Fg(\000)g Fm(I)-21 b(P)30 b Fg(f)p Fm(\034)2320 4379 y Fh(A)2405 4364 y Fm(>)d(t)p Fg(g)2593 4265 y Fd(\014)2593 4315 y(\014)2593 4364 y(\014)133 4547 y Fs(+)83 b Fm(I)-21 b(P)390 4562 y Fh(A)447 4547 y Fg(f)p Fm(A)570 4506 y Fh(c)570 4572 y Fe(1)610 4547 y Fg(g)677 4447 y Fd(\014)676 4497 y(\014)676 4547 y(\014)p Fm(I)g(P)30 b Fg(f)o Fm(\034)910 4562 y Fh(A)995 4547 y Fm(>)e(t)p Fg(g)22 b(\000)h Fm(e)1351 4506 y Fr(\000)p 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Fg(g)p Fm(`)32 b(:)0 5152 y Fs(Finally)-8 b(,)36 b(the)j(most)e(left)g (term)g(of)h(the)g(ab)s(o)m(v)m(e)h(b)s(ound)f(is)f(b)s(ounded)i(using) e(again)f(Lemma)h(3.)59 b(The)0 5268 y(most)32 b(righ)m(t)g(term)g (follo)m(ws)f(b)m(y)j(Theorem)e(1.)44 b(This)32 b(ends)i(the)f(pro)s (of)f(of)g(the)h(theorem.)1855 5517 y Fi(9)p eop %%Page: 10 10 10 9 bop 0 399 a Fo(Remark.)64 b Fs(W)-8 b(e)40 b(w)m(an)m(t)h(to)e (emphasize)h(the)g(relationship)e(b)s(et)m(w)m(een)j Fm(\030)2644 414 y Fh(A)2741 399 y Fs(and)e Fm(I)-21 b(P)3035 414 y Fh(A)3109 399 y Fg(f)p Fm(A)3232 362 y Fh(c)3232 423 y Fe(1)3271 399 y Fg(g)p Fs(.)65 b(The)40 b(pa-)0 515 y(rameter)28 b Fm(\030)402 530 y Fh(A)488 515 y Fs(has)h(an)f(explicit)g(form)f(giv)m(en)i(in)f(Abadi)g (\(2001b\).)41 b(It)29 b(is)g(usually)f(di\016cult)f(to)i(compute)0 631 y(it.)63 b(It)39 b(is)g(pro)m(v)m(en)i(in)d(Theorem)i(4)f(of)g (Abadi)f(\(2001a\))g(that)i(when)g(the)g(system)g(is)f(exp)s(onen)m (tially)0 747 y Fm(\036)p Fs(-mixing)24 b(,)29 b 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2011 y(hand)42 b(notation)f(let)g(us)i(write)e Fm(A)1269 2026 y Fh(j)1350 2011 y Fs(=)i Fm(T)1540 1975 y Fr(\000)p Fh(j)t(r)1665 2011 y Fm(A)1738 1975 y Fe(\()p Fh(r)r Fe(\))1873 2011 y Fs(for)f(all)d Fm(j)50 b Fg(2)44 b Fm(Z)-37 b(Z)7 b Fs(.)72 b(W)-8 b(e)42 b(in)m(tro)s(duce)g(the)h(follo) m(wing)0 2128 y(de\014nition)0 2298 y Fo(De\014nition)36 b(9)46 b Fl(Given)34 b Fm(A)28 b Fg(2)g(R)1188 2313 y Fh(r)1226 2298 y Fl(,)35 b(we)g(de\014ne)f(the)h(se)-5 b(quenc)g(e)34 b Fm(r)2322 2313 y Fh(i)2350 2298 y Fs(\()p Fm(A)p Fs(\);)51 b Fm(i)28 b Fs(=)g(1)p Fm(;)17 b Fs(2)p Fm(;)g(:)g(:)g(:)33 b Fl(as)i(fol)5 b(lows)811 2601 y Fm(r)855 2616 y Fh(i)883 2601 y Fs(\()p Fm(A)p Fs(\))28 b(:=)g Fm(I)-22 b(P)1305 2505 y Fd(n)1360 2601 y Fm(A)1433 2616 y Fe(1)1507 2601 y Fg(j)35 b Fm(A)1643 2616 y Fe(0)1682 2601 y Fs(;)17 b Fm(A)1799 2616 y Fr(\000)p Fe(1)1894 2601 y Fs(;)g Fm(:)g(:)g(:)e Fs(;)i Fm(A)2185 2616 y Fr(\000)p Fe(\()p Fh(i)p Fr(\000)p Fe(1\))2413 2601 y Fs(;)g Fm(T)2528 2560 y Fh(i)p Fr(\000)p Fe(1)2646 2601 y Fm(A)2719 2560 y Fe(\()p Fh(s)p Fe(\))2811 2505 y Fd(o)2918 2601 y Fm(:)0 2890 y Fl(That)29 b(is,)i(if)e Fm(A)h Fl(is)f(an)g Fm(n)p Fl(-cylinder)g(with)h(p)-5 b(erio)g(d)29 b Fm(r)j Fl(and)d(r)-5 b(est)30 b Fm(s)p Fl(,)g Fm(r)2332 2905 y Fh(i)2390 2890 y Fl(is)f(the)h(pr)-5 b(ob)g(ability)29 b(that)h(the)g(pr)-5 b(o)g(c)g(ess)0 3006 y(r)g(ep)g(e)g(ats)42 b(the)g(p)-5 b(erio)g(d)41 b(of)h Fm(A)p Fl(,)h(given)f(that)g(it)g (alr)-5 b(e)g(ady)42 b(o)-5 b(c)g(curr)g(e)g(d)41 b Fm(i)i Fl(times)e(and)h(onc)-5 b(e)41 b(the)h(r)-5 b(est)42 b(in)g(the)0 3122 y(imme)-5 b(diate)34 b(p)-5 b(ast,)34 b(al)5 b(l)35 b(of)f(them)h(adjac)-5 b(ently.)0 3387 y Fs(Since)35 b Fm(A)g Fs(is)g(\014xed)h(w)m(e)g(will)c(a)m(v)m(oid)j (to)g(write)f(the)i(dep)s(endence)h(on)e Fm(A)g Fs(in)f Fm(r)2771 3402 y Fh(i)2799 3387 y Fs(\()p Fm(A)p Fs(\))h(writing)f (just)h Fm(r)3561 3402 y Fh(i)3589 3387 y Fs(.)51 b(In)0 3503 y(the)43 b(follo)m(wing)d(theorem,)k(w)m(e)g(assume)f(that)f (there)h(exists)g(the)g(limit)c(of)j(the)h(sequence)i Fg(f)p Fm(r)3506 3518 y Fh(i)3534 3503 y Fg(g)3584 3518 y Fh(i)p Fr(2)p Fh(I)-24 b(N)3729 3503 y Fs(,)0 3620 y(with)32 b(v)m(elo)s(cit)m(y)h Fm(d)634 3635 y Fh(i)661 3620 y Fs(,)g(namely)-8 b(,)32 b(there)h(is)f(a)g(real)g(n)m(um)m(b)s (er)h Fm(r)2096 3635 y Fh(A)2186 3620 y Fs(suc)m(h)h(that)1567 3947 y Fg(j)p Fm(r)1639 3962 y Fh(i)1689 3947 y Fg(\000)23 b Fm(r)1833 3962 y Fh(A)1890 3947 y Fg(j)k(\024)h Fm(d)2101 3962 y Fh(i)2162 3947 y Fm(;)1442 b Fs(\(6\))0 4199 y(for)38 b(all)e Fm(i)i Fg(\025)h Fs(1,)g(where)h Fm(d)937 4214 y Fh(i)1003 4199 y Fs(is)e(a)h(decreasing)g(sequence)i(of)d(real)f(n)m (um)m(b)s(ers)j(going)d(to)h(zero)h(and)g(0)e Fm(<)0 4315 y(r)44 4330 y Fh(A)129 4315 y Fm(<)27 b Fs(1.)0 4507 y Fo(Theorem)37 b(10)46 b Fl(L)-5 b(et)89 b Fg(f)p Fm(X)986 4522 y Fh(m)1052 4507 y Fg(g)1102 4536 y Fh(m)p Fr(2)p Fh(Z)-40 b(Z)1320 4507 y Fl(b)-5 b(e)44 b(a)g(stationary)h(pr)-5 b(o)g(c)g(ess.)72 b(L)-5 b(et)45 b Fm(A)g Fg(2)h(R)2905 4522 y Fh(r)2944 4507 y Fl(.)73 b(Assume)44 b(that)h(the)0 4623 y(limit)38 b(in)g(\(6\))f(exists.)55 b(Then,)38 b(ther)-5 b(e)38 b(is)g(a)g(c)-5 b(onstant)38 b Fm(c)p Fl(,)g(with)g Fs(0)c Fm(<)g(c)f(<)h Fs(1)p Fl(,)39 b(such)f(that)g(the) g(fol)5 b(lowing)0 4739 y(ine)-5 b(qualities)34 b(hold)g(for)h(al)5 b(l)34 b Fm(k)d Fg(2)d Fm(I)-22 b(N)564 5093 y Fd(\014)564 5143 y(\014)564 5193 y(\014)p Fm(I)g(P)689 5208 y Fh(A)763 5193 y Fg(f)p Fm(S)873 5208 y Fh(A)957 5193 y Fs(=)28 b Fm(k)s Fg(g)22 b(\000)g Fs(\(1)g Fg(\000)h Fm(r)1539 5208 y Fh(A)1596 5193 y Fs(\))35 b Fm(r)1716 5152 y Fh(k)1713 5218 y(A)1770 5093 y Fd(\014)1770 5143 y(\014)1770 5193 y(\014)27 b Fg(\024)63 b Fm(c)2007 5152 y Fh(k)2066 5080 y(k)r Fe(+)p Fh(q)2070 5110 y Fd(X)2074 5292 y Fh(i)p Fe(=)p Fh(q)2210 5193 y Fm(d)2261 5208 y Fh(i)2317 5193 y Fg(\024)g Fs(\()p Fm(k)25 b Fs(+)d(1\))34 b Fm(c)2832 5152 y Fh(k)2875 5193 y Fm(d)2926 5208 y Fh(q)2999 5193 y Fm(;)1832 5517 y Fi(12)p eop %%Page: 13 13 13 12 bop 0 399 a Fl(wher)-5 b(e)34 b Fm(q)e Fs(=)27 b([)p Fm(n=r)s Fs(])p Fl(.)0 697 y Fo(Pro)s(of.)47 b Fs(F)-8 b(or)24 b Fm(k)31 b Fs(=)d(0)d(w)m(e)h(just)g(note)f(that)g Fm(I)-21 b(P)1610 712 y Fh(A)1684 697 y Fg(f)p Fm(S)1794 712 y Fh(A)1878 697 y Fs(=)28 b(0)p Fg(g)f Fs(=)h Fm(r)2256 712 y Fh(q)2319 697 y Fs(and)d Fg(j)p Fm(r)2573 712 y Fh(q)2618 697 y Fg(\000)7 b Fm(r)2746 712 y Fh(A)2804 697 y Fg(j)27 b Fs(=)h Fg(j)p Fs(1)7 b Fg(\000)g Fm(r)3175 712 y Fh(q)3221 697 y Fg(\000)g Fs(\(1)g Fg(\000)g Fm(r)3527 712 y Fh(A)3585 697 y Fs(\))p Fg(j)27 b(\024)0 813 y Fm(d)51 828 y Fh(q)89 813 y Fs(.)43 b(Supp)s(ose)34 b Fm(k)d Fg(\025)d Fs(1,)k(w)m(e)i(ha)m(v)m(e)161 1297 y Fm(I)-21 b(P)259 1312 y Fh(A)333 1297 y Fg(f)p Fm(S)443 1312 y Fh(A)528 1297 y Fs(=)27 b Fm(k)s Fg(g)83 b Fs(=)987 1211 y Fm(I)-21 b(P)1102 1115 y Fd(n)1157 1211 y Fm(A)p Fs(;)17 b Fm(A)1347 1226 y Fe(1)1386 1211 y Fs(;)g Fm(:)g(:)g(:)f Fs(;)h Fm(A)1678 1226 y Fh(k)1721 1211 y Fs(;)g Fm(A)1838 1175 y Fh(c)1838 1236 y(k)r Fe(+1)1970 1115 y Fd(o)p 987 1274 1039 4 v 1371 1366 a Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)818 1610 y Fs(=)987 1523 y Fm(I)-21 b(P)1102 1427 y Fd(n)1157 1523 y Fm(A)p Fs(;)17 b Fm(A)1347 1538 y Fe(1)1386 1523 y Fs(;)g Fm(:)g(:)g(:)f Fs(;)h Fm(A)1678 1538 y Fh(k)1721 1523 y Fs(;)g Fm(A)1838 1487 y Fh(c)1838 1548 y(k)r Fe(+1)1970 1427 y Fd(o)p 987 1587 V 1117 1678 a Fm(I)-21 b(P)30 b Fg(f)p Fm(A)p Fs(;)17 b Fm(A)1472 1693 y Fe(1)1511 1678 y Fs(;)g Fm(:)g(:)g(:)f Fs(;)h Fm(A)1803 1693 y Fh(k)1845 1678 y Fg(g)2091 1542 y Fm(I)-21 b(P)29 b Fg(f)p Fm(A)p Fs(;)17 b Fm(A)2445 1557 y Fe(1)2484 1542 y Fs(;)g Fm(:)g(:)g(:)f Fs(;)h Fm(A)2776 1557 y Fh(k)2819 1542 y Fg(g)p 2045 1587 869 4 v 2045 1678 a Fm(I)-21 b(P)30 b Fg(f)p Fm(A)p Fs(;)17 b Fm(A)2400 1693 y Fe(1)2439 1678 y Fs(;)g Fm(:)g(:)g(:)f Fs(;)h Fm(A)2731 1693 y Fh(k)r Fr(\000)p Fe(1)2864 1678 y Fg(g)2940 1610 y Fm(:)g(:)g(:)3081 1542 y(I)-21 b(P)30 b Fg(f)p Fm(A)p Fs(;)17 b Fm(A)3436 1557 y Fe(1)3475 1542 y Fg(g)p 3081 1587 444 4 v 3159 1678 a Fm(I)-21 b(P)30 b Fg(f)p Fm(A)p Fg(g)3567 1610 y Fm(:)0 1954 y Fs(Note)j(that)f(b)m(y)i(stationarit)m (y)85 2361 y Fm(I)-21 b(P)29 b Fg(f)p Fm(A)p Fs(;)17 b Fm(A)439 2376 y Fe(1)478 2361 y Fs(;)g Fm(:)g(:)g(:)f Fs(;)h Fm(A)770 2376 y Fh(k)813 2361 y Fg(g)83 b Fs(=)f Fm(I)-21 b(P)1219 2264 y Fd(n)1274 2361 y Fm(T)1345 2320 y Fh(q)r(r)1417 2361 y Fm(A)1490 2320 y Fe(\()p Fh(s)p Fe(\))1582 2361 y Fs(;)17 b Fm(A)1699 2376 y Fr(\000)p Fe(\()p Fh(q)r Fr(\000)p Fe(1\))1937 2361 y Fs(;)g Fm(:)g(:)g(:)e Fs(;)i Fm(A)2228 2376 y Fe(0)2268 2361 y Fs(;)g Fm(A)2385 2376 y Fe(1)2424 2361 y Fs(;)g Fm(:)g(:)g(:)f Fs(;)h Fm(A)2716 2376 y Fh(k)r Fr(\000)p Fe(1)2848 2361 y Fs(;)g Fm(A)2965 2376 y Fh(k)3008 2264 y Fd(o)946 2543 y Fs(=)82 b Fm(I)-21 b(P)1219 2447 y Fd(n)1274 2543 y Fm(T)1345 2502 y Fe(\()p Fh(k)r Fe(+)p Fh(q)r Fr(\000)p Fe(1\))p Fh(r)1655 2543 y Fm(A)1728 2502 y Fe(\()p Fh(s)p Fe(\))1820 2543 y Fs(;)17 b Fm(A)1937 2559 y Fr(\000)p Fe(\()p Fh(k)r Fe(+)p Fh(q)r Fr(\000)p Fe(2\))2268 2543 y Fs(;)g Fm(:)g(:)g(:)f Fs(;)h Fm(A)2560 2558 y Fr(\000)p Fh(k)r Fe(+1)2748 2543 y Fs(;)g Fm(A)2865 2558 y Fr(\000)p Fh(k)r Fe(+2)3052 2543 y Fs(;)g Fm(:)g(:)g(:)f Fs(;)h Fm(A)3344 2558 y Fe(0)3383 2543 y Fs(;)g Fm(A)3500 2558 y Fe(1)3540 2447 y Fd(o)3644 2543 y Fm(:)0 2844 y Fs(Therefore)1261 3101 y Fm(I)-21 b(P)29 b Fg(f)p Fm(A)p Fs(;)17 b Fm(A)1615 3116 y Fe(1)1655 3101 y Fs(;)g Fm(:)g(:)g(:)e Fs(;)i Fm(A)1946 3116 y Fh(k)1989 3101 y Fg(g)p 1216 3145 869 4 v 1216 3237 a Fm(I)-21 b(P)29 b Fg(f)p Fm(A)p Fs(;)17 b Fm(A)1570 3252 y Fe(1)1610 3237 y Fs(;)g Fm(:)g(:)g(:)e Fs(;)i Fm(A)1901 3252 y Fh(k)r Fr(\000)p Fe(1)2034 3237 y Fg(g)2121 3169 y Fs(=)28 b Fm(r)2269 3184 y Fh(k)r Fe(+)p Fh(q)r Fr(\000)p Fe(1)2523 3169 y Fm(:)0 3512 y Fs(Hence)996 3802 y Fm(I)-21 b(P)1094 3817 y Fh(A)1167 3802 y Fg(f)p Fm(S)1277 3817 y Fh(A)1362 3802 y Fs(=)27 b Fm(k)s Fg(g)h Fs(=)f(\(1)22 b Fg(\000)h Fm(r)1953 3817 y Fh(k)r Fe(+)p Fh(q)2084 3802 y Fs(\))28 b Fm(r)2194 3817 y Fh(k)r Fe(+)p Fh(q)r Fr(\000)p Fe(1)2460 3802 y Fm(:)17 b(:)g(:)44 b(r)2663 3817 y Fh(q)2733 3802 y Fm(:)0 4092 y Fs(By)33 b(Lemma)e(8)463 4465 y Fd(\014)463 4515 y(\014)463 4565 y(\014)p Fm(I)-21 b(P)589 4580 y Fh(A)663 4565 y Fg(f)o Fm(S)772 4580 y Fh(A)857 4565 y Fs(=)28 b Fm(k)s Fg(g)22 b(\000)g Fs(\(1)g Fg(\000)h Fm(r)1439 4580 y Fh(A)1496 4565 y Fs(\))p Fm(r)1581 4524 y Fh(k)1578 4589 y(A)1635 4465 y Fd(\014)1635 4515 y(\014)1635 4565 y(\014)83 b Fs(=)1906 4391 y Fd(\014)1906 4440 y(\014)1906 4490 y(\014)1906 4540 y(\014)1906 4590 y(\014)1906 4640 y(\014)1934 4565 y Fs(\(1)21 b Fg(\000)i Fm(r)2186 4580 y Fh(k)r Fe(+)p Fh(q)2317 4565 y Fs(\))2372 4451 y Fh(k)r Fe(+)p Fh(q)r Fr(\000)p Fe(1)2428 4482 y Fd(Y)2424 4664 y Fh(i)p Fe(=)p Fh(q)2606 4565 y Fm(r)2650 4580 y Fh(i)2700 4565 y Fg(\000)g Fs(\(1)f Fg(\000)g Fm(r)3052 4580 y Fh(A)3109 4565 y Fs(\))p Fm(r)3194 4524 y Fh(k)3191 4589 y(A)3248 4391 y Fd(\014)3248 4440 y(\014)3248 4490 y(\014)3248 4540 y(\014)3248 4590 y(\014)3248 4640 y(\014)1745 4891 y Fg(\024)84 b Fm(c)1948 4849 y Fh(k)2040 4777 y(k)r Fe(+)p Fh(q)2043 4807 y Fd(X)2047 4990 y Fh(i)p Fe(=)p Fh(q)2184 4891 y Fm(d)2235 4906 y Fh(i)2290 4891 y Fg(\024)28 b Fs(\()p Fm(k)d Fs(+)d(1\))33 b Fm(c)2769 4849 y Fh(k)2844 4891 y Fm(d)2895 4906 y Fh(q)2965 4891 y Fm(;)0 5268 y Fs(where)h Fm(c)27 b Fs(=)h(max)p Fg(f)p Fm(r)731 5283 y Fe(0)770 5268 y Fm(;)17 b(r)858 5283 y Fh(A)914 5268 y Fg(g)p Fs(.)44 b(This)32 b(ends)i(the)f(pro)s(of)f(of)g(the)h (Theorem.)1832 5517 y Fi(13)p eop %%Page: 14 14 14 13 bop 0 399 a Fo(Example)37 b(11)45 b Fl(L)-5 b(et)88 b Fg(f)p Fm(X)972 414 y Fh(m)1038 399 y Fg(g)1088 428 y Fh(m)p Fr(2)p Fh(Z)-40 b(Z)1306 399 y Fl(b)-5 b(e)43 b(a)h(irr)-5 b(e)g(ducible)43 b(and)g(ap)-5 b(erio)g(dic)43 b(\014nite)g(state)h(Markov)g(chain.)0 515 y(L)-5 b(et)32 b Fm(A)c Fg(2)g(R)444 530 y Fh(r)482 515 y Fl(.)44 b(Then)31 b(the)h(se)-5 b(quenc)g(e)31 b Fm(r)1403 530 y Fh(i)1463 515 y Fl(is)h(c)-5 b(onstant,)32 b(mor)-5 b(e)31 b(pr)-5 b(e)g(cisely,)32 b(by)g(the)g(Markovian)f(pr)-5 b(op)g(erty)0 631 y(and)34 b(for)h(al)5 b(l)34 b Fm(i)28 b Fg(2)g Fm(I)-22 b(N)515 959 y(r)559 974 y Fh(i)615 959 y Fs(=)27 b Fm(I)-22 b(P)832 863 y Fd(n)888 959 y Fm(A)961 974 y Fe(1)1035 959 y Fg(j)35 b Fm(A)1171 974 y Fe(0)1210 959 y Fs(;)17 b Fm(A)1327 974 y Fr(\000)p Fe(1)1421 959 y Fs(;)g Fm(:)g(:)g(:)f Fs(;)h Fm(A)1713 975 y Fr(\000)p Fe(\()p Fh(i)p Fr(\000)p Fe(1\))1941 959 y Fm(;)g(T)2056 918 y Fh(i)p Fr(\000)p Fe(1)2174 959 y Fm(A)2247 918 y Fe(\()p Fh(s)p Fe(\))2339 863 y Fd(o)2457 959 y Fs(=)27 b Fm(I)-22 b(P)2674 863 y Fd(n)2730 959 y Fm(A)2803 974 y Fe(1)2877 959 y Fg(j)35 b Fm(A)3013 918 y Fe(\(1\))3107 863 y Fd(o)3214 959 y Fm(:)0 1285 y Fl(Thus,)f Fm(r)320 1300 y Fh(A)405 1285 y Fs(=)28 b Fm(r)553 1300 y Fh(i)616 1285 y Fl(for)34 b(al)5 b(l)35 b Fm(i)28 b Fg(2)g Fm(I)-22 b(N)10 b Fl(.)45 b(F)-7 b(urthermor)i(e,)34 b Fm(I)-22 b(P)1940 1300 y Fh(A)1997 1285 y Fg(f)p Fm(A)2120 1249 y Fh(c)2120 1309 y Fe(1)2160 1285 y Fg(g)27 b Fs(=)h(1)22 b Fg(\000)g Fm(r)2555 1300 y Fh(A)2612 1285 y Fl(,)35 b(and)f(mor)-5 b(e)g(over)1253 1619 y Fm(I)-22 b(P)1351 1634 y Fh(A)1424 1619 y Fg(f)p Fm(S)1534 1634 y Fh(A)1619 1619 y Fs(=)27 b Fm(k)s Fg(g)h Fs(=)f(\(1)22 b Fg(\000)h Fm(r)2210 1634 y Fh(A)2267 1619 y Fs(\))35 b Fm(r)2387 1577 y Fh(k)2384 1643 y(A)2476 1619 y Fm(:)0 2036 y Fo(Example)i(12)45 b Fl(F)-7 b(or)38 b(a)i(Gibbs)f(me)-5 b(asur)g(e,)40 b(the)f(c)-5 b(ondition)39 b(\(6\))f(is)i(warr)-5 b(ante)g(d)38 b(by)i(the)f(c)-5 b(ontinuity)40 b(of)0 2152 y(the)28 b(p)-5 b(otential.)42 b(F)-7 b(or)27 b(a)h(chain)f(of)h(in\014nite)g (or)-5 b(der)27 b(\(se)-5 b(e)28 b(v.g.)42 b(F)-7 b(ern\023)-50 b(andez,)27 b(F)-7 b(err)i(ari)27 b(and)h(Galves,)h(2001\))0 2268 y(c)-5 b(ondition)34 b(\(6\))g(is)h(assur)-5 b(e)g(d)34 b(by)h(the)g(nul)5 b(lness)34 b(hyp)-5 b(othesis.)0 2578 y Fn(5)135 b(P)l(oisson)45 b(limit)i(la)l(w)0 2797 y Fs(In)40 b(this)f(section)h(w)m(e)g(pro)m(v)m(e)h(that)f(the)g(n)m(um)m (b)s(er)g(of)f(o)s(ccurrences)i(of)e(the)h(string)f Fm(A)h Fs(in)f(an)g(in)m(terv)-5 b(al)0 2913 y(of)33 b(time)f(of)h(length)g Fm(t)h Fs(has)g(appro)m(ximately)e(P)m(oisson)i(distribution)d(with)j (parameter)f Fm(tI)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)p Fs(.)46 b(The)0 3029 y(result)33 b(holds)f(when)h(the)g(string)f(do)s (es)h(not)g(recur)g(v)m(ery)h(fast.)0 3255 y Fo(De\014nition)i(13)46 b Fl(Given)38 b Fm(A)d Fg(2)h(C)1231 3270 y Fh(n)1278 3255 y Fl(,)k(we)e(de\014ne)g(for)h Fm(j)i Fg(2)35 b Fm(I)-22 b(N)11 b Fl(,)39 b(the)g Fm(j)6 b Fl(-)p Fs(o)s(ccurrence)38 b(time)g Fl(of)g Fm(A)h Fl(as)g(the)0 3371 y(r.v.)44 b Fm(\034)232 3386 y Fh(A)290 3335 y Fe(\()p Fh(j)t Fe(\))409 3371 y Fs(:)27 b(\012)i Fg(!)e Fm(I)-22 b(N)30 b Fg([)20 b(f1g)p Fl(,)34 b(de\014ne)-5 b(d)32 b(on)i(the)g(pr)-5 b(ob)g(ability)33 b(sp)-5 b(ac)g(e)33 b Fs(\(\012)p Fm(;)17 b Fg(F)10 b Fm(;)17 b(I)-22 b(P)13 b Fs(\))34 b Fl(as)f(fol)5 b(lows:)43 b(for)34 b(any)0 3488 y Fm(!)d Fg(2)d Fs(\012)p Fl(,)35 b Fm(\034)363 3503 y Fh(A)421 3452 y Fe(\(1\))515 3488 y Fs(\()p Fm(!)t Fs(\))27 b(=)g Fm(\034)828 3503 y Fh(A)886 3488 y Fs(\()p Fm(!)t Fs(\))34 b Fl(and)g(for)h Fm(j)e Fg(\025)c Fs(2)937 3822 y Fm(\034)979 3837 y Fh(A)1036 3780 y Fe(\()p Fh(j)t Fe(\))1128 3822 y Fs(\()p Fm(!)t Fs(\))e(=)g(inf)6 b Fg(f)p Fm(x)28 b(>)g(\034)1796 3837 y Fh(A)1853 3780 y Fe(\()p Fh(j)t Fr(\000)p Fe(1\))2035 3822 y Fs(\()p Fm(!)t Fs(\))f(:)g Fm(T)2328 3780 y Fh(x)2372 3822 y Fs(\()p Fm(!)t Fs(\))g Fg(2)h Fm(A)p Fg(g)35 b Fm(:)141 4239 y Fs(W)-8 b(e)33 b(in)m(tro)s(duce)g(the)g(follo)m(wing)d (notation.)42 b(F)-8 b(or)31 b(0)d Fg(\024)g Fm(s)g(<)f(t)33 b Fs(in)m(tegers)g(let)1531 4520 y Fm(N)1619 4479 y Fh(t)1609 4544 y(s)1677 4520 y Fs(=)1828 4412 y Fh(t)1781 4437 y Fd(X)1785 4619 y Fh(i)p Fe(=)p Fh(s)1917 4520 y Fo(1)-26 b Fs(I)1982 4537 y Fh(T)2033 4519 y Fk(\000)p Ff(i)2108 4537 y Fh(A)2197 4520 y Fm(:)0 4809 y Fs(So,)65 b Fm(N)283 4773 y Fh(t)273 4834 y(s)346 4809 y Fs(coun)m(ts)34 b(the)f(n)m(um)m(b) s(er)f(of)h(o)s(ccurrences)h(of)e Fm(A)h Fs(b)s(et)m(w)m(een)h Fm(s)f Fs(and)g Fm(t)p Fs(.)0 5035 y Fo(De\014nition)j(14)46 b Fl(L)-5 b(et)40 b Fm(s)c Fg(2)h Fm(I)-22 b(N)10 b Fl(.)59 b(De\014ne)38 b Fg(B)1614 5050 y Fh(n)1698 5035 y Fs(=)e Fg(B)1875 5050 y Fh(n)1923 5035 y Fs(\()p Fm(s)p Fs(\))j Fl(as)g(the)h(set)f(of)h(those)f Fm(A)d Fg(2)h(C)3176 5050 y Fh(n)3263 5035 y Fl(which)i(r)-5 b(e)g(cur)0 5152 y(b)g(efor)g(e)34 b(time)g Fm(n=s)p Fl(,)g(namely,)g Fm(A)28 b Fg(2)g(B)1344 5167 y Fh(n)1426 5152 y Fl(if)34 b(and)g(only)g(if)h(ther)-5 b(e)34 b(is)g(an)g(inte)-5 b(ger)34 b Fs(1)28 b Fg(\024)g Fm(j)34 b Fg(\024)28 b Fm(n=s)34 b Fl(such)h(that)0 5268 y Fm(A)22 b Fg(\\)h Fm(T)255 5232 y Fr(\000)p Fh(j)346 5268 y Fm(A)28 b Fg(6)p Fs(=)f Fg(;)35 b Fl(for)g(some)f Fs(1)27 b Fg(\024)h Fm(j)34 b Fg(\024)28 b Fm(n=s)p Fl(.)1832 5517 y Fi(14)p eop %%Page: 15 15 15 14 bop 141 399 a Fs(The)36 b(follo)m(wing)c(Lemma)i(is)h(the)g(base) h(of)e(the)i(pro)s(of)e(of)g(the)i(P)m(oisson)f(statistics)g(of)f(o)s (ccurrence)0 515 y(times)39 b(of)h(a)g(measurable)f Fm(A)i Fg(62)g(B)1279 530 y Fh(n)1326 515 y Fs(\(2\).)66 b(It)41 b(establish)e(an)h(asymptotic)g(w)m(eak)h(dep)s(endence)i(of)c(the)0 631 y(o)s(ccurrences)c(of)d Fm(A)p Fs(.)0 819 y Fo(Prop)s(osition)k(15) 45 b Fl(L)-5 b(et)76 b Fg(f)p Fm(X)1103 834 y Fh(m)1170 819 y Fg(g)1219 848 y Fh(m)p Fr(2)p Fh(Z)-40 b(Z)1431 819 y Fl(b)-5 b(e)38 b(a)75 b Fm(\036)p Fl(-mixing)37 b(pr)-5 b(o)g(c)g(ess.)53 b(L)-5 b(et)38 b Fm(A)45 b(=)-60 b Fg(2)34 b(B)2916 834 y Fh(n)2963 819 y Fs(\(2\))p Fl(.)53 b(L)-5 b(et)77 b Fm(t)3416 834 y Fe(0)3489 819 y Fs(=)33 b(0)g Fm(<)0 935 y(t)35 950 y Fe(1)102 935 y Fm(<)28 b(t)241 950 y Fe(2)308 935 y Fm(<)g(:)17 b(:)g(:)27 b(<)g(t)692 950 y Fh(k)763 935 y Fg(\024)h Fm(t)g Fs(=)g Fm(t)1070 950 y Fh(k)r Fe(+1)1225 935 y Fg(\000)22 b Fs(1)p Fl(.)45 b(L)-5 b(et)35 b Fm(`)1657 950 y Fh(A)1749 935 y Fl(b)-5 b(e)35 b(the)f Fm(`)h Fl(that)h(de\014nes)d Fm(")2675 950 y Fe(2)2714 935 y Fs(\()p Fm(A)p Fs(\))i Fl(b)-5 b(elow.)44 b(Supp)-5 b(ose)34 b(that)973 1115 y Fs(min)15 b Fg(f)p Fm(t)1237 1130 y Fh(j)1296 1115 y Fg(\000)23 b Fm(t)1431 1130 y Fh(j)t Fr(\000)p Fe(1)1585 1115 y Fs(:)28 b Fm(j)34 b Fs(=)27 b(2)p Fm(;)17 b(:)g(:)g(:)f(;)h(k)s Fg(g)27 b Fm(>)h Fs(2\()p Fm(`)2448 1130 y Fh(A)2526 1115 y Fs(+)23 b Fm(n)p Fs(\))35 b Fm(:)0 1295 y Fl(Then)i(ther)-5 b(e)38 b(exists)g(a)g(p)-5 b(ositive)37 b(c)-5 b(onstant)38 b Fm(C)i(>)34 b Fs(0)k Fl(indep)-5 b(endent)36 b(of)i Fm(A)p Fl(,)h Fm(n)p Fl(,)g Fm(t)f Fl(and)f Fm(k)42 b Fl(such)37 b(that)i(the)0 1411 y(fol)5 b(lowing)33 b(ine)-5 b(quality)35 b(holds)425 1694 y Fd(\014)425 1744 y(\014)425 1794 y(\014)425 1844 y(\014)425 1893 y(\014)425 1943 y(\014)453 1868 y Fm(I)-22 b(P)567 1694 y Fd(8)567 1769 y(<)567 1918 y(:)682 1760 y Fh(k)656 1785 y Fd(\\)640 1968 y Fh(j)t Fe(=1)780 1868 y Fm(\034)822 1883 y Fh(A)879 1827 y Fe(\()p Fh(j)t Fe(\))998 1868 y Fs(=)28 b Fm(t)1137 1883 y Fh(j)1208 1868 y Fs(;)52 b Fm(\034)1329 1883 y Fh(A)1386 1827 y Fe(\()p Fh(k)r Fe(+1\))1602 1868 y Fm(>)27 b(t)1740 1694 y Fd(9)1740 1769 y(=)1740 1918 y(;)1836 1868 y Fg(\000)c Fm(I)-22 b(P)13 b Fg(f)p Fm(A)p Fg(g)2206 1827 y Fh(k)2300 1760 y(k)r Fe(+1)2311 1785 y Fd(Y)2303 1968 y Fh(j)t Fe(=1)2445 1868 y Fm(I)-22 b(P)30 b Fg(f)p Fm(\034)2651 1883 y Fh(A)2736 1868 y Fm(>)e Fs(\001)2921 1883 y Fh(j)2980 1868 y Fg(\000)22 b Fs(2\()p Fm(`)3207 1883 y Fh(A)3286 1868 y Fs(+)g Fm(n)p Fs(\))p Fg(g)3530 1694 y Fd(\014)3530 1744 y(\014)3530 1794 y(\014)3530 1844 y(\014)3530 1893 y(\014)3530 1943 y(\014)264 2167 y Fg(\024)84 b Fm(C)58 b Fs(\()p Fm(I)-22 b(P)14 b Fg(f)p Fm(A)p Fg(g)22 b Fs(+)g Fm(\036)p Fs(\()p Fm(`)1119 2182 y Fh(A)1175 2167 y Fs(\)\))1251 2119 y Fh(k)1345 2167 y Fm(")p Fs(\()p Fm(A)p Fs(\))35 b Fm(;)0 2442 y Fl(wher)-5 b(e)34 b Fm(")p Fs(\()p Fm(A)p Fs(\))28 b(=)f Fm(")647 2457 y Fe(1)686 2442 y Fs(\()p Fm(A)p Fs(\))22 b(+)g Fm(")1001 2457 y Fe(2)1040 2442 y Fs(\()p Fm(A)p Fs(\))p Fl(,)894 2758 y Fm(")940 2773 y Fe(1)979 2758 y Fs(\()p Fm(A)p Fs(\))27 b(:=)125 b(inf)1286 2821 y Fe(1)p Fr(\024)p Fh(w)r Fr(\024)p Fh(n=)p Fe(2)1614 2758 y Fm(n)1689 2636 y Fd(\032)1751 2758 y Fm(I)-22 b(P)13 b Fg(f)p Fm(A)1971 2716 y Fe(\()p Fh(w)r Fe(\))2082 2758 y Fg(g)22 b Fs(+)g Fm(\036)2327 2636 y Fd(\022)2398 2690 y Fm(n)p 2398 2734 59 4 v 2403 2826 a Fs(2)2488 2758 y Fg(\000)h Fm(w)2661 2636 y Fd(\023)o(\033)2835 2758 y Fm(;)0 3082 y Fl(and)759 3475 y Fm(")805 3490 y Fe(2)844 3475 y Fs(\()p Fm(A)p Fs(\))k(:=)187 b(inf)1151 3539 y Fh(n)p Fr(\024)p Fh(`)p Fr(\024)p Fe(1)p Fh(=I)-25 b(P)9 b Fr(f)p Fh(A)p Fr(g)1603 3301 y Fd(8)1603 3375 y(<)1603 3525 y(:)1677 3475 y Fm(`I)-22 b(P)14 b Fg(f)p Fm(A)p Fg(g)21 b Fs(+)2141 3367 y Fe(3)p Fh(`)2114 3392 y Fd(X)2108 3574 y Fh(j)t Fe(=)p Fh(n)2255 3475 y Fm(\036)p Fs(\()p Fm(j)6 b Fs(\))22 b(+)2604 3408 y Fm(\036)17 b Fs(\()p Fm(`)p Fs(\))p 2565 3452 271 4 v 2565 3543 a Fm(I)-22 b(P)13 b Fg(f)p Fm(A)p Fg(g)2845 3301 y Fd(9)2845 3375 y(=)2845 3525 y(;)2970 3475 y Fm(:)634 b Fs(\(7\))0 3984 y(W)-8 b(e)34 b(remark)f(that)h Fm(A)29 b Fg(2)h(C)966 3999 y Fh(n)1047 3984 y Fs(and)k Fm(")1284 3999 y Fe(1)1323 3984 y Fs(\()p Fm(A)p Fs(\))c Fg(!)f Fs(0)k(when)i Fm(n)30 b Fg(!)f(1)p Fs(.)46 b(The)35 b(same)e(is)g(true) h(for)f Fm(")3308 3999 y Fe(2)3347 3984 y Fs(\()p Fm(A)p Fs(\))h(if)e(the)0 4100 y(sequence)j Fg(f)p Fm(\036)p Fs(\()p Fm(l)r Fs(\))p Fg(g)669 4115 y Fh(l)q Fr(2)p Fh(I)-24 b(N)844 4100 y Fs(is)32 b(summable.)0 4311 y Fo(Pro)s(of.)54 b Fs(W)-8 b(e)34 b(pro)m(v)m(e)i(the)e(prop)s(osition)e (b)m(y)j(induction)e(on)g Fm(k)s Fs(.)48 b(First)33 b(w)m(e)i(note)f (that)f(b)m(y)i(stationarit)m(y)0 4427 y(and)e(for)f(an)m(y)h(p)s (ositiv)m(e)f(in)m(teger)g Fm(t)h Fs(\(see)h(v.g.)44 b(pag.)f(16)32 b(in)g(Abadi,)g(2001a\))1118 4754 y Fm(I)-21 b(P)30 b Fg(f)o Fm(\034)1324 4769 y Fh(A)1409 4754 y Fs(=)e Fm(t)p Fg(g)g Fs(=)f Fm(I)-21 b(P)30 b Fg(f)p Fm(A)i Fs(;)50 b Fm(\034)2118 4769 y Fh(A)2203 4754 y Fm(>)27 b(t)c Fg(\000)f Fs(1)p Fg(g)49 b Fm(:)993 b Fs(\(8\))0 5005 y(Let)37 b(us)h(write)e(for)g(short-hand)h(notation)p 1608 4926 41 4 v 36 w Fm(`)1649 5020 y Fh(A)1741 5005 y Fs(=)e(2\()p Fm(`)1980 5020 y Fh(A)2061 5005 y Fs(+)25 b Fm(n)p Fs(\).)57 b(Let)37 b(us)g(write)g(also)f(\001)3185 5020 y Fh(j)3257 5005 y Fs(=)f Fm(t)3403 5020 y Fh(j)3464 5005 y Fg(\000)26 b Fm(t)3602 5020 y Fh(j)t Fr(\000)p Fe(1)3729 5005 y Fs(,)0 5141 y(for)k Fm(j)j Fs(=)28 b(1)p Fm(;)17 b(:)g(:)g(:)f(;)h(k)j Fs(+)d(1.)43 b(F)-8 b(or)29 b(eac)m(h)i(non)g(negativ)m(e)f(in)m(teger)h Fm(i)f Fs(denote)h Fg(P)2599 5156 y Fh(i)2655 5141 y Fs(=)d Fm(I)-21 b(P)2873 5044 y Fd(n)2929 5141 y Fm(\034)2971 5156 y Fh(A)3056 5141 y Fm(>)27 b Fs(\001)3240 5156 y Fh(i)3291 5141 y Fg(\000)p 3391 5061 V 23 w Fm(`)3432 5156 y Fh(A)3489 5044 y Fd(o)3544 5141 y Fs(.)43 b(F)-8 b(or)0 5268 y Fm(k)31 b Fs(=)c(1)1832 5517 y Fi(15)p eop %%Page: 16 16 16 15 bop 855 452 a Fd(\014)855 502 y(\014)855 552 y(\014)855 601 y(\014)855 651 y(\014)855 701 y(\014)883 626 y Fm(I)-21 b(P)998 530 y Fd(n)1053 626 y Fm(\034)1095 641 y Fh(A)1180 626 y Fs(=)28 b Fm(t)1319 641 y Fe(1)1391 626 y Fs(;)49 b Fm(\034)1509 641 y Fh(A)1566 585 y Fe(\(2\))1688 626 y Fm(>)28 b(t)1827 530 y Fd(o)1905 626 y Fg(\000)22 b Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)2335 518 y Fe(2)2300 543 y Fd(Y)2292 726 y Fh(j)t Fe(=1)2431 626 y Fg(P)2500 641 y Fh(j)2537 452 y Fd(\014)2537 502 y(\014)2537 552 y(\014)2537 601 y(\014)2537 651 y(\014)2537 701 y(\014)3631 626 y Fs(\(9\))695 884 y Fg(\024)855 784 y Fd(\014)855 834 y(\014)855 884 y(\014)p Fm(I)-21 b(P)998 787 y Fd(n)1053 884 y Fm(\034)1095 899 y Fh(A)1180 884 y Fs(=)28 b Fm(t)1319 899 y Fe(1)1391 884 y Fs(;)49 b Fm(\034)1509 899 y Fh(A)1566 842 y Fe(\(2\))1688 884 y Fm(>)28 b(t)1827 787 y Fd(o)1905 884 y Fg(\000)22 b Fm(I)-21 b(P)2119 787 y Fd(n)2174 884 y Fm(\034)2216 899 y Fh(A)2301 884 y Fs(=)28 b Fm(t)2440 899 y Fe(1)2512 884 y Fs(;)49 b Fm(N)2676 842 y Fh(t)2666 919 y(t)2691 928 y Fb(1)2726 919 y Fe(+)p 2781 862 29 4 v Fh(`)2810 930 y Ff(A)2892 884 y Fs(=)28 b(0)3045 787 y Fd(o)3100 784 y(\014)3100 834 y(\014)3100 884 y(\014)696 1066 y Fs(+)855 967 y Fd(\014)855 1016 y(\014)855 1066 y(\014)p Fm(I)-21 b(P)998 970 y Fd(n)1053 1066 y Fm(\034)1095 1081 y Fh(A)1180 1066 y Fs(=)28 b Fm(t)1319 1081 y Fe(1)1391 1066 y Fs(;)49 b Fm(N)1555 1025 y Fh(t)1545 1101 y(t)1570 1110 y Fb(1)1605 1101 y Fe(+)p 1660 1045 V Fh(`)1689 1112 y Ff(A)1771 1066 y Fs(=)27 b(0)1923 970 y Fd(o)2001 1066 y Fg(\000)22 b Fm(I)-21 b(P)30 b Fg(f)p Fm(\034)2307 1081 y Fh(A)2392 1066 y Fs(=)d Fm(t)2530 1081 y Fe(1)2570 1066 y Fg(g)16 b(P)2705 1081 y Fe(2)2745 967 y Fd(\014)2745 1016 y(\014)2745 1066 y(\014)696 1249 y Fs(+)855 1149 y Fd(\014)855 1199 y(\014)855 1249 y(\014)p Fm(I)-21 b(P)30 b Fg(f)o Fm(A)j Fs(;)49 b Fm(\034)1271 1264 y Fh(A)1356 1249 y Fm(>)28 b(t)1495 1264 y Fe(1)1557 1249 y Fg(\000)22 b Fs(1)p Fg(g)g(\000)h Fm(I)-21 b(P)1991 1153 y Fd(n)2047 1249 y Fm(A)32 b Fs(;)50 b Fm(N)2317 1205 y Fh(t)2342 1214 y Fb(1)2377 1205 y Fr(\000)p Fe(1)p 2307 1235 V 2307 1291 a Fh(`)2335 1302 y Ff(A)2499 1249 y Fs(=)28 b(0)2652 1153 y Fd(o)2707 1149 y(\014)2707 1199 y(\014)2707 1249 y(\014)49 b Fg(P)2853 1264 y Fe(2)696 1509 y Fs(+)855 1335 y Fd(\014)855 1385 y(\014)855 1435 y(\014)855 1484 y(\014)855 1534 y(\014)855 1584 y(\014)883 1509 y Fm(I)-21 b(P)998 1413 y Fd(n)1053 1509 y Fm(A)33 b Fs(;)49 b Fm(N)1323 1466 y Fh(t)1348 1475 y Fb(1)1383 1466 y Fr(\000)p Fe(1)p 1313 1495 V 1313 1551 a Fh(`)1342 1562 y Ff(A)1505 1509 y Fs(=)28 b(0)1658 1413 y Fd(o)1730 1509 y Fg(P)1799 1524 y Fe(2)1861 1509 y Fg(\000)22 b Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)2291 1401 y Fe(2)2256 1426 y Fd(Y)2248 1609 y Fh(j)t Fe(=1)2387 1509 y Fg(P)2456 1524 y Fh(j)2525 1335 y Fd(\014)2525 1385 y(\014)2525 1435 y(\014)2525 1484 y(\014)2525 1534 y(\014)2525 1584 y(\014)2602 1509 y Fm(:)0 1856 y Fs(In)35 b(the)g(third)f(term)g(in)g (the)h(righ)m(t)e(hand)i(side)g(of)f(\(9\))g(w)m(e)i(used)g(\(8\).)49 b(The)35 b(\014rst)g(term)f(in)g(the)h(ab)s(o)m(v)m(e)0 1973 y(b)s(ound)e(is)f(equal)g(to)301 2420 y Fm(I)-21 b(P)416 2245 y Fd(8)416 2320 y(<)416 2469 y(:)490 2420 y Fm(\034)532 2435 y Fh(A)617 2420 y Fs(=)27 b Fm(t)755 2435 y Fe(1)827 2420 y Fs(;)904 2308 y Fh(t)929 2317 y Fb(1)964 2308 y Fe(+)p 1019 2252 V Fh(`)1047 2319 y Ff(A)1098 2308 y Fr(\000)p Fe(1)999 2337 y Fd([)931 2519 y Fh(i)p Fe(=)p Fh(t)1035 2528 y Fb(1)1070 2519 y Fe(+1)1204 2420 y Fm(T)1275 2378 y Fr(\000)p Fh(i)1358 2420 y Fm(A)33 b Fs(;)49 b Fm(N)1628 2378 y Fh(t)1618 2454 y(t)1643 2463 y Fb(1)1678 2454 y Fe(+)p 1733 2398 V Fh(`)1762 2465 y Ff(A)1844 2420 y Fs(=)27 b(0)1996 2245 y Fd(9)1996 2320 y(=)1996 2469 y(;)2153 2420 y Fg(\024)2313 2308 y Fh(t)2338 2317 y Fb(1)2374 2308 y Fe(+)p 2429 2252 V Fh(`)2457 2319 y Ff(A)2507 2308 y Fr(\000)p Fe(1)2396 2337 y Fd(X)2341 2519 y Fh(i)p Fe(=)p Fh(t)2445 2528 y Fb(1)2480 2519 y Fe(+1)2614 2420 y Fm(I)-21 b(P)2729 2323 y Fd(n)2784 2420 y Fm(T)2855 2378 y Fr(\000)p Fh(t)2935 2387 y Fb(1)2974 2420 y Fm(A)33 b Fs(;)49 b Fm(T)3227 2378 y Fr(\000)p Fh(i)3310 2420 y Fm(A)3383 2323 y Fd(o)3582 2420 y Fs(\(10\))2154 2760 y(=)p 2313 2592 V 2313 2648 a Fh(`)2342 2659 y Ff(A)2393 2648 y Fr(\000)p Fe(1)2338 2677 y Fd(X)2341 2859 y Fh(i)p Fe(=1)2499 2760 y Fm(I)-21 b(P)2614 2664 y Fd(n)2669 2760 y Fm(A)33 b Fs(;)49 b Fm(T)2922 2719 y Fr(\000)p Fh(i)3005 2760 y Fm(A)3078 2664 y Fd(o)3182 2760 y Fm(:)0 3090 y Fs(W)-8 b(e)37 b(divide)f(the)g(ab)s(o)m(v)m(e)i(sum)e(in)g(those)h(terms)f(with)g(1)e Fg(\024)h Fm(i)f Fg(\024)h Fm(n=)p Fs(2,)i Fm(n=)p Fs(2)d Fm(<)g(i)h Fg(\024)f Fs(2)p Fm(n)p Fs(,)k(and)e Fm(i)f(>)f Fs(2)p Fm(n)p Fs(.)0 3206 y(F)-8 b(or)32 b(1)27 b Fg(\024)h Fm(i)g Fg(\024)g Fm(n=)p Fs(2,)33 b(the)g(leading)e(term)h(of)g(the)h (ab)s(o)m(v)m(e)g(sum)g(is)f(zero)h(since)g Fm(A)28 b Fg(62)g(B)3040 3221 y Fh(n)3087 3206 y Fs(\(2\).)43 b(Moreo)m(v)m(er) 533 3522 y Fe(2)p Fh(n)512 3547 y Fd(X)403 3735 y Fh(i)p Fe(=\()p Fh(n=)p Fe(2\)+1)757 3630 y Fm(I)-21 b(P)872 3533 y Fd(n)927 3630 y Fm(A)33 b Fs(;)49 b Fm(T)1180 3588 y Fr(\000)p Fh(i)1263 3630 y Fm(A)1336 3533 y Fd(o)1474 3630 y Fg(\024)1765 3522 y Fe(2)p Fh(n)1744 3547 y Fd(X)1635 3735 y Fh(i)p Fe(=\()p Fh(n=)p Fe(2\)+1)1989 3630 y Fm(I)-21 b(P)2104 3533 y Fd(n)2159 3630 y Fm(A)33 b Fs(;)49 b Fm(T)2412 3588 y Fr(\000)p Fh(i)2495 3630 y Fm(A)2568 3588 y Fe(\()p Fh(w)r Fe(\))2679 3533 y Fd(o)1474 3952 y Fg(\024)1765 3844 y Fe(2)p Fh(n)1744 3869 y Fd(X)1635 4057 y Fh(i)p Fe(=\()p Fh(n=)p Fe(2\)+1)1989 3952 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)2277 3856 y Fd(\020)2326 3952 y Fm(I)-21 b(P)2441 3856 y Fd(n)2496 3952 y Fm(A)2569 3911 y Fe(\()p Fh(w)r Fe(\))2680 3856 y Fd(o)2758 3952 y Fs(+)22 b Fm(\036)17 b Fs(\()o Fm(i)23 b Fg(\000)f Fm(w)s Fs(\))3234 3856 y Fd(\021)1474 4240 y Fg(\024)84 b Fm(C)39 b(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)33 b Fm(n)2155 4119 y Fd(\022)2216 4240 y Fm(I)-21 b(P)2331 4143 y Fd(n)2386 4240 y Fm(A)2459 4199 y Fe(\()p Fh(w)r Fe(\))2571 4143 y Fd(o)2648 4240 y Fs(+)22 b Fm(\036)2821 4119 y Fd(\022)2892 4172 y Fm(n)p 2892 4217 59 4 v 2897 4308 a Fs(2)2982 4240 y Fg(\000)h Fm(w)3155 4119 y Fd(\023)o(\023)3325 4240 y Fm(;)0 4548 y Fs(for)44 b(an)m(y)h(1)i Fg(\024)h Fm(w)i Fg(\024)e Fm(n=)p Fs(2.)78 b(W)-8 b(e)45 b(still)c(recall)i(the) i(de\014nition)e(of)h Fm(A)2558 4512 y Fe(\()p Fh(w)r Fe(\))2714 4548 y Fs(at)f(the)i(end)g(of)f(section)g(2.)0 4665 y(Similarly)p 640 5038 29 4 v 640 5094 a Fh(`)669 5105 y Ff(A)719 5094 y Fr(\000)p Fe(1)665 5123 y Fd(X)646 5305 y Fh(i)p Fe(=2)p Fh(n)826 5206 y Fm(I)-21 b(P)940 5110 y Fd(n)996 5206 y Fm(A)32 b Fs(;)50 b Fm(T)1249 5165 y Fr(\000)p Fh(i)1331 5206 y Fm(A)1404 5110 y Fd(o)1543 5206 y Fg(\024)p 1703 5038 V 1703 5094 a Fh(`)1732 5105 y Ff(A)1782 5094 y Fr(\000)p Fh(n)p Fr(\000)p Fe(1)1777 5123 y Fd(X)1776 5305 y Fh(i)p Fe(=)p Fh(n)1987 5206 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)k Fs(\()o Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)22 b Fs(+)g Fm(\036)p Fs(\()p Fm(i)p Fs(\)\))1832 5517 y Fi(16)p eop %%Page: 17 17 17 16 bop 1543 482 a Fg(\024)83 b Fm(C)40 b(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)2101 311 y Fd(0)2100 460 y(@)2173 482 y Fm(`)2214 497 y Fh(A)2271 482 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)22 b Fs(+)2666 370 y Fe(3)p Fh(`)2730 381 y Ff(A)2663 399 y Fd(X)2662 581 y Fh(i)p Fe(=)p Fh(n)2800 482 y Fm(\036)p Fs(\()p Fm(i)p Fs(\))2967 311 y Fd(1)2967 460 y(A)3089 482 y Fm(:)141 842 y Fs(The)32 b(second)h(term)d(in)h(the) g(righ)m(t)g(hand)g(side)g(of)g(inequalit)m(y)f(\(9\))h(is)g(b)s (ounded)h(using)e(the)i(mixing)0 958 y(prop)s(ert)m(y)h(b)m(y)1033 1246 y Fm(\036)p Fs(\()p 1129 1167 41 4 v Fm(`)1169 1261 y Fh(A)1248 1246 y Fg(\000)23 b Fm(n)p Fs(\))33 b Fm(I)-21 b(P)29 b Fg(f)p Fm(\034)1683 1261 y Fh(A)1768 1246 y Fs(=)f Fm(t)1907 1261 y Fe(1)1946 1246 y Fg(g)g(\024)g Fm(\036)p Fs(\()p Fm(`)2266 1261 y Fh(A)2323 1246 y Fs(\))k Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)32 b Fm(:)0 1515 y Fs(Analogous)e(computations)f(are)i(used)g(to)f(b)s(ound)h(the)g (third)f(and)g(fourth)h(terms)f(in)g(the)h(righ)m(t)f(hand)0 1631 y(side)j(of)f(inequalit)m(y)f(\(9\).)43 b(This)33 b(ends)h(the)f(pro)s(of)f(for)g Fm(k)e Fs(=)e(1.)141 1842 y(No)m(w)45 b(let)f(us)h(supp)s(ose)g(that)g(the)f(theorem)g (holds)h(for)e Fm(k)34 b Fg(\000)c Fs(1)44 b(and)h(pro)m(v)m(e)g(for)f Fm(k)s Fs(.)79 b(W)-8 b(e)45 b(use)g(a)0 1958 y(triangular)36 b(inequalit)m(y)h(where)j(the)f(terms)f(in)m(v)m(olv)m(ed)h(are)f (de\014ned)i(b)s(elo)m(w.)61 b(W)-8 b(e)39 b(brie\015y)f(commen)m(t)0 2074 y(the)33 b(idea)f(b)s(ehind)g(eac)m(h)i(term.)43 b(F)-8 b(or)32 b(eac)m(h)h(non)g(negativ)m(e)g Fm(i)f Fs(denote)i Fg(S)2596 2089 y Fh(i)2652 2074 y Fs(=)27 b Fg(f)p Fm(\034)2847 2089 y Fh(A)2932 2074 y Fs(=)h Fm(t)3071 2089 y Fh(i)3099 2074 y Fg(g)p Fs(.)343 2260 y Fd(\014)343 2310 y(\014)343 2360 y(\014)343 2410 y(\014)343 2459 y(\014)343 2509 y(\014)371 2435 y Fm(I)-21 b(P)485 2260 y Fd(8)485 2335 y(<)485 2484 y(:)601 2327 y Fh(k)574 2352 y Fd(\\)559 2534 y Fh(j)t Fe(=1)698 2435 y Fg(S)758 2450 y Fh(j)860 2435 y Fs(;)49 b Fm(\034)978 2450 y Fh(A)1036 2393 y Fe(\()p Fh(k)r Fe(+1\))1251 2435 y Fm(>)28 b(t)1390 2260 y Fd(9)1390 2335 y(=)1390 2484 y(;)1486 2435 y Fg(\000)22 b Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)1856 2393 y Fh(k)1915 2327 y(k)r Fe(+1)1927 2352 y Fd(Y)1919 2534 y Fh(j)t Fe(=1)2061 2435 y Fg(P)2130 2450 y Fh(j)2199 2260 y Fd(\014)2199 2310 y(\014)2199 2360 y(\014)2199 2410 y(\014)2199 2459 y(\014)2199 2509 y(\014)2254 2435 y Fg(\024)29 b Fm(I)h Fs(+)22 b Fm(I)8 b(I)29 b Fs(+)22 b Fm(I)8 b(I)g(I)30 b Fs(+)22 b Fm(I)8 b(V)44 b Fs(+)22 b Fm(V)54 b(:)141 2806 y Fs(In)34 b Fm(I)42 b Fs(w)m(e)35 b(in)m(tro)s(duce)f(a)g(gap)f (of)h(length)p 1609 2727 V 33 w Fm(`)1650 2821 y Fh(A)1741 2806 y Fs(at)g(the)g(left)f(of)g(the)i Fm(k)s Fs(-th)e(o)s(ccurrence)j (of)d Fm(A)p Fs(,)i(namely)-8 b(,)0 2922 y(b)s(et)m(w)m(een)35 b(co)s(ordinates)d Fm(t)930 2937 y Fh(k)995 2922 y Fg(\000)p 1094 2843 V 22 w Fm(`)1135 2937 y Fh(A)1214 2922 y Fs(+)22 b(1)33 b(and)f Fm(t)1618 2937 y Fh(k)1683 2922 y Fg(\000)23 b Fs(1)55 3374 y Fm(I)91 b Fs(:=)375 3199 y Fd(\014)375 3249 y(\014)375 3299 y(\014)375 3349 y(\014)375 3398 y(\014)375 3448 y(\014)402 3374 y Fm(I)-21 b(P)517 3199 y Fd(8)517 3274 y(<)517 3423 y(:)633 3266 y Fh(k)606 3291 y Fd(\\)591 3473 y Fh(j)t Fe(=1)730 3374 y Fg(S)790 3389 y Fh(j)892 3374 y Fs(;)49 b Fm(\034)1010 3389 y Fh(A)1067 3332 y Fe(\()p Fh(k)r Fe(+1\))1283 3374 y Fm(>)27 b(t)1421 3199 y Fd(9)1421 3274 y(=)1421 3423 y(;)1517 3374 y Fg(\000)c Fm(I)-21 b(P)1732 3199 y Fd(8)1732 3274 y(<)1732 3423 y(:)1805 3266 y Fh(k)r Fr(\000)p Fe(1)1824 3291 y Fd(\\)1808 3473 y Fh(j)t Fe(=1)1951 3374 y Fg(S)2011 3389 y Fh(j)2113 3374 y Fs(;)49 b Fm(N)2277 3328 y Fh(t)2302 3340 y Ff(k)2341 3328 y Fr(\000)p 2396 3271 29 4 v Fh(`)2425 3339 y Ff(A)2267 3395 y Fh(t)2292 3407 y Ff(k)q Fk(\000)p Fb(1)2409 3395 y Fe(+1)2531 3374 y Fs(=)27 b(0)33 b(;)49 b Fm(T)2863 3332 y Fr(\000)p Fh(t)2943 3344 y Ff(k)2985 3374 y Fm(A)33 b Fs(;)49 b Fm(N)3255 3332 y Fh(t)3245 3398 y(t)3270 3410 y Ff(k)3309 3398 y Fe(+1)3431 3374 y Fs(=)27 b(0)3583 3199 y Fd(9)3583 3274 y(=)3583 3423 y(;)3657 3199 y(\014)3657 3249 y(\014)3657 3299 y(\014)3657 3349 y(\014)3657 3398 y(\014)3657 3448 y(\014)202 3706 y Fs(=)97 b Fm(I)-21 b(P)489 3531 y Fd(8)489 3606 y(<)489 3755 y(:)563 3598 y Fh(k)r Fr(\000)p Fe(1)581 3623 y Fd(\\)566 3805 y Fh(j)t Fe(=1)708 3706 y Fg(S)768 3721 y Fh(j)870 3706 y Fs(;)49 b Fm(N)1034 3660 y Fh(t)1059 3672 y Ff(k)1098 3660 y Fr(\000)p 1153 3604 V Fh(`)1182 3671 y Ff(A)1024 3727 y Fh(t)1049 3739 y Ff(k)q Fk(\000)p Fb(1)1166 3727 y Fe(+1)1288 3706 y Fs(=)28 b(0)k(;)1656 3593 y Fh(t)1681 3605 y Ff(k)1719 3593 y Fr(\000)p Fe(1)1687 3623 y Fd([)1549 3824 y Fh(i)p Fe(=)p Fh(t)1653 3836 y Ff(k)1692 3824 y Fr(\000)p 1747 3768 V Fh(`)1776 3835 y Ff(A)1826 3824 y Fe(+1)1933 3706 y Fm(T)2004 3664 y Fr(\000)p Fh(i)2086 3706 y Fm(A)h Fs(;)49 b Fm(T)2339 3664 y Fr(\000)p Fh(t)2419 3676 y Ff(k)2462 3706 y Fm(A)32 b Fs(;)50 b Fm(N)2732 3664 y Fh(t)2722 3730 y(t)2747 3742 y Ff(k)2785 3730 y Fe(+1)2907 3706 y Fs(=)28 b(0)3060 3531 y Fd(9)3060 3606 y(=)3060 3755 y(;)201 4051 y Fg(\024)481 3939 y Fh(t)506 3951 y Ff(k)545 3939 y Fr(\000)p Fe(1)498 3968 y Fd(X)375 4169 y Fh(i)p Fe(=)p Fh(t)479 4181 y Ff(k)517 4169 y Fr(\000)p 572 4113 V Fh(`)601 4180 y Ff(A)651 4169 y Fe(+1)758 4051 y Fm(I)-21 b(P)872 3877 y Fd(8)872 3952 y(<)872 4101 y(:)946 3943 y Fh(k)r Fr(\000)p Fe(1)964 3968 y Fd(\\)949 4151 y Fh(j)t Fe(=1)1091 4051 y Fm(T)1162 4010 y Fr(\000)p Fh(t)1242 4020 y Ff(j)1279 4051 y Fm(A)33 b Fs(;)49 b Fm(T)1532 4010 y Fr(\000)p Fh(i)1615 4051 y Fm(A)32 b Fs(;)50 b Fm(T)1868 4010 y Fr(\000)p Fh(t)1948 4022 y Ff(k)1990 4051 y Fm(A)2063 3877 y Fd(9)2063 3952 y(=)2063 4101 y(;)2186 4051 y Fm(:)0 4491 y Fs(As)32 b(in)e(\(10\))h(w)m(e)i(split)d(the)h(ab)s(o)m(v)m(e)i (sum)e(in)f(terms)i(with)f Fm(t)2105 4506 y Fh(k)2167 4491 y Fg(\000)p 2264 4412 41 4 v 20 w Fm(`)2305 4506 y Fh(A)2382 4491 y Fs(+)19 b(1)28 b Fg(\024)g Fm(i)g Fg(\024)g Fm(t)2860 4506 y Fh(k)2923 4491 y Fg(\000)20 b Fs(2)p Fm(n)p Fs(,)31 b Fm(t)3220 4506 y Fh(k)3283 4491 y Fg(\000)20 b Fs(2)p Fm(n)f Fs(+)h(1)27 b Fg(\024)0 4608 y Fm(i)i Fg(\024)g Fm(t)203 4623 y Fh(k)268 4608 y Fg(\000)23 b Fm(n=)p Fs(2)g Fg(\000)g Fs(1)32 b(and)i Fm(t)954 4623 y Fh(k)1019 4608 y Fg(\000)23 b Fm(n=)p Fs(2)28 b Fg(\024)i Fm(i)e Fg(\024)h Fm(t)1612 4623 y Fh(k)1678 4608 y Fg(\000)23 b Fs(1.)45 b(The)34 b(b)s(ound)f(is)g (obtained)f(iterating)f(the)j(mixing)0 4724 y(prop)s(ert)m(y)-8 b(.)44 b(W)-8 b(e)33 b(recall)e(that)i(b)m(y)g(h)m(yp)s(othesis)h(\001) 1765 4739 y Fh(j)1829 4724 y Fm(>)p 1933 4645 V 28 w(`)1974 4739 y Fh(A)2063 4724 y Fs(for)e(all)f Fm(i)d Fs(=)f(1)p Fm(;)17 b(:)g(:)g(:)f(;)h(k)s Fs(.)945 5063 y Fh(t)970 5075 y Ff(k)1009 5063 y Fr(\000)p Fe(2)p Fh(n)983 5092 y Fd(X)860 5293 y Fh(i)p Fe(=)p Fh(t)964 5305 y Ff(k)1002 5293 y Fr(\000)p 1057 5237 29 4 v Fh(`)1086 5304 y Ff(A)1136 5293 y Fe(+1)1243 5175 y Fm(I)-21 b(P)1358 5001 y Fd(8)1358 5076 y(<)1358 5225 y(:)1432 5067 y Fh(k)r Fr(\000)p Fe(1)1450 5092 y Fd(\\)1435 5275 y Fh(j)t Fe(=1)1577 5175 y Fm(T)1648 5134 y Fr(\000)p Fh(t)1728 5144 y Ff(j)1765 5175 y Fm(A)32 b Fs(;)49 b Fm(T)2017 5134 y Fr(\000)p Fh(i)2100 5175 y Fm(A)33 b Fs(;)49 b Fm(T)2353 5134 y Fr(\000)p Fh(t)2433 5146 y Ff(k)2475 5175 y Fm(A)2548 5001 y Fd(9)2548 5076 y(=)2548 5225 y(;)1832 5517 y Fi(17)p eop %%Page: 18 18 18 17 bop 699 490 a Fg(\024)p 860 322 29 4 v 860 378 a Fh(`)889 389 y Ff(A)939 378 y Fr(\000)p Fh(n)888 407 y Fd(X)887 589 y Fh(i)p Fe(=)p Fh(n)1086 490 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)1406 394 y Fd(\020)1455 490 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)22 b Fs(+)g Fm(\036)p Fs(\()p 1942 411 41 4 v Fm(`)1983 505 y Fh(A)2062 490 y Fg(\000)h Fm(n)p Fs(\))2258 394 y Fd(\021)2307 417 y Fh(k)r Fr(\000)p Fe(1)2489 490 y Fs(\()p Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)22 b Fs(+)g Fm(\036)p Fs(\()p Fm(i)p Fs(\)\))699 805 y Fg(\024)84 b Fs(\()p Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)22 b Fs(+)g Fm(\036)p Fs(\()p Fm(`)1426 820 y Fh(A)1482 805 y Fs(\)\))1558 757 y Fh(k)1650 634 y Fd(0)1650 783 y(@)1723 805 y Fm(C)39 b(`)1873 820 y Fh(A)1930 805 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)22 b Fs(+)2325 693 y Fe(3)p Fh(`)2389 704 y Ff(A)2322 722 y Fd(X)2321 904 y Fh(i)p Fe(=)p Fh(n)2459 805 y Fm(\036)p Fs(\()p Fm(i)p Fs(\))2626 634 y Fd(1)2626 783 y(A)2748 805 y Fm(:)0 1165 y Fs(F)-8 b(urthermore)593 1506 y Fh(t)618 1518 y Ff(k)657 1506 y Fr(\000)p Fe(\()p Fh(n=)p Fe(2\))p Fr(\000)p Fe(1)721 1540 y Fd(X)599 1722 y Fh(i)p Fe(=)p Fh(t)703 1734 y Ff(k)741 1722 y Fr(\000)p Fe(2)p Fh(n)p Fe(+1)987 1623 y Fm(I)-21 b(P)1101 1448 y Fd(8)1101 1523 y(<)1101 1672 y(:)1175 1515 y Fh(k)r Fr(\000)p Fe(1)1193 1540 y Fd(\\)1178 1722 y Fh(j)t Fe(=1)1320 1623 y Fm(T)1391 1582 y Fr(\000)p Fh(t)1471 1592 y Ff(j)1508 1623 y Fm(A)33 b Fs(;)49 b Fm(T)1761 1582 y Fr(\000)p Fh(i)1844 1623 y Fm(A)32 b Fs(;)50 b Fm(T)2097 1582 y Fr(\000)p Fh(t)2177 1594 y Ff(k)2219 1623 y Fm(A)2292 1448 y Fd(9)2292 1523 y(=)2292 1672 y(;)432 1956 y Fg(\024)593 1839 y Fh(t)618 1851 y Ff(k)657 1839 y Fr(\000)p Fe(\()p Fh(n=)p Fe(2\))p Fr(\000)p Fe(1)721 1873 y Fd(X)599 2055 y Fh(i)p Fe(=)p Fh(t)703 2067 y Ff(k)741 2055 y Fr(\000)p Fe(2)p Fh(n)p Fe(+1)987 1956 y Fm(I)-21 b(P)1101 1781 y Fd(8)1101 1856 y(<)1101 2005 y(:)1175 1848 y Fh(k)r Fr(\000)p Fe(1)1193 1873 y Fd(\\)1178 2055 y Fh(j)t Fe(=1)1320 1956 y Fm(T)1391 1915 y Fr(\000)p Fh(t)1471 1925 y Ff(j)1508 1956 y Fm(A)33 b Fs(;)49 b Fm(T)1761 1915 y Fr(\000)p Fh(i)1844 1956 y Fm(A)32 b Fs(;)50 b Fm(T)2097 1915 y Fr(\000)p Fh(t)2177 1927 y Ff(k)2219 1956 y Fm(A)2292 1915 y Fe(\()p Fh(w)r Fe(\))2403 1781 y Fd(9)2403 1856 y(=)2403 2005 y(;)432 2272 y Fg(\024)678 2165 y Fe(2)p Fh(n)p Fr(\000)p Fe(1)702 2189 y Fd(X)593 2378 y Fh(i)p Fe(=\()p Fh(n=)p Fe(2\)+1)979 2272 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)1299 2176 y Fd(\020)1349 2272 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)22 b Fs(+)g Fm(\036)p Fs(\()p 1836 2193 V Fm(`)1876 2287 y Fh(A)1956 2272 y Fg(\000)g Fm(n)p Fs(\))2151 2176 y Fd(\021)2201 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b(\\\014ll-up")c(the)j (gap)f(w)m(e)i(created)f(in)f Fm(I)1832 5517 y Fi(18)p eop %%Page: 19 19 19 18 bop 10 633 a Fm(I)8 b(I)g(I)91 b Fs(:=)432 458 y Fd(\014)432 508 y(\014)432 558 y(\014)432 608 y(\014)432 657 y(\014)432 707 y(\014)459 633 y Fm(I)-21 b(P)574 458 y Fd(8)574 533 y(<)574 682 y(:)648 525 y Fh(k)r Fr(\000)p Fe(1)666 549 y Fd(\\)651 732 y Fh(j)t Fe(=1)793 633 y Fg(S)853 648 y Fh(j)955 633 y Fs(;)49 b Fm(N)1119 586 y Fh(t)1144 598 y Ff(k)1183 586 y Fr(\000)p 1238 530 29 4 v Fh(`)1267 597 y Ff(A)1109 654 y Fh(t)1134 666 y Ff(k)q Fk(\000)p Fb(1)1251 654 y Fe(+1)1373 633 y Fs(=)28 b(0)1526 458 y Fd(9)1526 533 y(=)1526 682 y(;)1621 633 y Fg(\000)23 b Fm(I)-21 b(P)1836 458 y Fd(8)1836 533 y(<)1836 682 y(:)1910 525 y Fh(k)r Fr(\000)p Fe(1)1928 549 y Fd(\\)1913 732 y Fh(j)t Fe(=1)2055 633 y Fg(S)2115 648 y Fh(j)2217 633 y Fs(;)49 b Fm(N)2381 586 y Fh(t)2406 598 y Ff(k)2445 586 y Fr(\000)p Fe(1)2371 654 y Fh(t)2396 666 y Ff(k)q Fk(\000)p Fb(1)2513 654 y Fe(+1)2635 633 y Fs(=)28 b(0)2788 458 y Fd(9)2788 533 y(=)2788 682 y(;)2861 458 y(\014)2861 508 y(\014)2861 558 y(\014)2861 608 y(\014)2861 657 y(\014)2861 707 y(\014)2905 633 y Fm(I)-21 b(P)3020 536 y Fd(n)3075 633 y Fm(A)33 b Fs(;)49 b Fm(N)3345 586 y Fh(t)p Fr(\000)p Fh(t)3450 598 y Ff(k)3335 654 y Fe(1)3522 633 y Fs(=)27 b(0)3674 536 y Fd(o)258 965 y Fg(\024)97 b Fm(I)-21 b(P)546 790 y Fd(8)546 865 y(<)546 1014 y(:)620 857 y Fh(k)r Fr(\000)p Fe(1)638 882 y Fd(\\)623 1064 y Fh(j)t Fe(=1)765 965 y Fg(S)825 980 y Fh(j)927 965 y Fs(;)49 b Fm(N)1091 919 y Fh(t)1116 931 y Ff(k)1156 919 y Fr(\000)p 1211 862 V Fh(`)1239 930 y Ff(A)1081 986 y Fh(t)1106 998 y Ff(k)q Fk(\000)p Fb(1)1224 986 y Fe(+1)1346 965 y Fs(=)27 b(0)32 b(;)1674 852 y Fh(t)1699 864 y Ff(k)1737 852 y Fr(\000)p Fe(1)1704 882 y Fd([)1607 1083 y Fh(t)1632 1095 y Ff(k)1670 1083 y Fr(\000)p 1725 1027 V Fh(`)1754 1094 y Ff(A)1804 1083 y Fe(+1)1911 965 y Fm(T)1982 923 y Fr(\000)p Fh(i)2065 965 y Fm(A)2138 790 y Fd(9)2138 865 y(=)2138 1014 y(;)2261 965 y 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Fg(\024)96 b Fm(C)24 b Fs(\()o Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)22 b Fs(+)g Fm(\036)p Fs(\()p Fm(`)1284 2550 y Fh(A)1341 2535 y Fs(\)\))1417 2487 y Fh(k)r Fr(\000)p Fe(1)1599 2535 y Fm(")p Fs(\()p Fm(A)p Fs(\))32 b Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)32 b Fm(:)141 2797 y Fs(In)h Fm(V)54 b Fs(w)m(e)34 b(use)f(again)e(the)i(inductiv)m(e)g(h)m(yp)s (othesis)h(with)e Fm(k)f Fs(=)c(1)602 3230 y Fm(V)104 b Fs(:=)83 b Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)1220 3189 y Fh(k)r Fr(\000)p Fe(1)1412 3122 y Fh(k)1378 3147 y Fd(Y)1370 3329 y Fh(j)t Fe(=1)1509 3230 y Fg(P)1578 3245 y Fh(j)1664 3130 y Fd(\014)1664 3180 y(\014)1664 3230 y(\014)o Fm(I)-21 b(P)1806 3134 y Fd(n)1861 3230 y Fm(A)33 b Fs(;)49 b Fm(N)2131 3184 y Fh(t)p Fr(\000)p Fh(t)2236 3196 y Ff(k)2121 3252 y Fe(1)2307 3230 y Fs(=)28 b(0)2460 3134 y Fd(o)2537 3230 y Fg(\000)23 b Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(gP)2977 3245 y Fh(k)r Fe(+1)3110 3130 y Fd(\014)3110 3180 y(\014)3110 3230 y(\014)776 3546 y Fg(\024)96 b Fm(I)-21 b(P)13 b 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y(\014)0 4915 y(\014)0 4965 y(\014)0 5015 y(\014)28 4965 y Fm(I)-22 b(P)142 4844 y Fd(\032)204 4965 y Fm(N)359 4961 y Ff(t)p 292 4973 158 4 v 292 5014 a(I)c(P)8 b Fk(f)p Ff(A)p Fk(g)492 4965 y Fs(=)27 b Fm(k)649 4844 y Fd(\033)734 4965 y Fg(\000)843 4897 y Fm(e)888 4861 y Fr(\000)p Fh(t)973 4897 y Fm(t)1008 4861 y Fh(k)p 843 4942 208 4 v 906 5033 a Fm(k)s Fs(!)1061 4815 y Fd(\014)1061 4865 y(\014)1061 4915 y(\014)1061 4965 y(\014)1061 5015 y(\014)1116 4965 y Fg(\024)1221 4566 y Fd(8)1221 4641 y(>)1221 4666 y(>)1221 4691 y(>)1221 4716 y(>)1221 4741 y(>)1221 4766 y(>)1221 4790 y(>)1221 4815 y(>)1221 4840 y(>)1221 4865 y(<)1221 5015 y(>)1221 5039 y(>)1221 5064 y(>)1221 5089 y(>)1221 5114 y(>)1221 5139 y(>)1221 5164 y(>)1221 5189 y(>)1221 5214 y(>)1221 5239 y(:)1337 4666 y Fm(C)41 b(")p Fs(\()p Fm(A)p Fs(\))1754 4626 y Fh(e)1787 4603 y Ff(k)p 1688 4642 204 4 v 1688 4700 a Fe(\()p Fh(k)r Fr(\000)p Fe(1\)!)1936 4666 y Fm(\025)1993 4681 y Fh(M)2924 4666 y Fs(1)83 b Fg(\024)28 b Fm(k)86 b Fg(\024)3466 4626 y Fe(1)p 3414 4642 141 4 v 3414 4700 a Fh(")p Fe(\()p Fh(A)p Fe(\))1337 4930 y Fm(C)41 b(")p Fs(\()p Fm(A)p Fs(\))2031 4891 y Fh(e)2064 4867 y Ff(k)p 1688 4907 757 4 v 1688 5019 a Fs(\()1782 4984 y Fb(1)p 1736 4996 124 4 v 1736 5037 a Ff(")p Fb(\()p Ff(A)p Fb(\))1869 5019 y Fs(\))1907 5011 y Fe(!)1961 5019 y Fs(\()2056 4984 y Fb(1)p 2009 4996 V 2009 5037 a Ff(")p Fb(\()p Ff(A)p Fb(\))2143 5019 y Fs(\))2180 4962 y Ff(k)q Fk(\000)2319 4942 y Fb(1)p 2272 4951 V 2272 4992 a Ff(")p Fb(\()p Ff(A)p Fb(\))2489 4930 y Fm(\025)2546 4945 y Fh(M)2875 4891 y Fe(1)p 2823 4907 141 4 v 2823 4964 a Fh(")p Fe(\()p Fh(A)p Fe(\))3056 4930 y Fm(<)28 b(k)87 b Fg(\024)3497 4891 y Fe(2)p Fh(t)p 3414 4907 227 4 v 3414 4964 a(nI)-25 b(P)9 b Fr(f)p Fh(A)p Fr(g)1337 5260 y Fm(C)41 b(nI)-22 b(P)14 b Fg(f)p Fm(A)p Fg(g)1899 5221 y Fh(e)1932 5198 y Fk(\000)p Ff(t)p 1822 5237 264 4 v 1822 5295 a Fe(\()p Fh(k)r Fr(\000)p Fe(1\)!)35 b Fh(t)2820 5221 y Fe(2)p Fh(t)p 2737 5237 227 4 v 2737 5295 a(nI)-25 b(P)9 b Fr(f)p Fh(A)p Fr(g)3056 5260 y Fm(<)28 b(k)3768 4965 y(:)1832 5517 y Fi(19)p eop %%Page: 20 20 20 19 bop 0 399 a Fl(wher)-5 b(e)69 b Fm(")p Fs(\()p Fm(A)p Fs(\))27 b(=)h Fm(")682 414 y Fe(1)721 399 y Fs(\()p Fm(A)p Fs(\))22 b(+)g Fm(")1036 414 y Fe(2)1075 399 y Fs(\()p Fm(A)p Fs(\))p Fm(;)52 b(")1349 414 y Fe(1)1388 399 y Fs(\()p Fm(A)p Fs(\))35 b Fl(and)f Fm(")1807 414 y Fe(2)1846 399 y Fs(\()p Fm(A)p Fs(\))h Fl(wer)-5 b(e)34 b(de\014ne)-5 b(d)34 b(in)h(Pr)-5 b(op)g(osition)34 b(15;)1082 702 y Fm(\025)1139 717 y Fh(M)1246 702 y Fs(=)27 b(max)1547 528 y Fd(8)1547 603 y(<)1547 752 y(:)1621 702 y Fs(1)35 b Fm(;)1783 556 y Fd( )1849 702 y Fm(t)1936 556 y Fd(")1984 702 y Fs(1)22 b(+)2182 635 y Fm(\036)p Fs(\()p Fm(`)2319 650 y Fh(A)2376 635 y Fs(\))p 2163 679 271 4 v 2163 771 a Fm(I)-22 b(P)13 b Fg(f)p Fm(A)p Fg(g)2443 556 y Fd(#)q(!)2557 579 y Fh(k)2600 528 y Fd(9)2600 603 y(=)2600 752 y(;)0 1006 y Fl(and)34 b Fm(`)230 1021 y Fh(A)322 1006 y Fl(is)h(the)g Fm(`)f 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Fe(\()p Fh(j)t Fe(\))1598 2180 y Fs(=)27 b Fm(t)1736 2195 y Fh(j)1805 2180 y Fs(;)50 b Fm(\034)1924 2195 y Fh(A)1981 2139 y Fe(\()p Fh(k)r Fe(+1\))2196 2180 y Fm(>)28 b(t)2335 2005 y Fd(9)2335 2080 y(=)2335 2230 y(;)2431 2180 y Fg(\000)23 b Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)2802 2139 y Fh(k)2861 2072 y(k)r Fe(+1)2872 2097 y Fd(Y)2864 2279 y Fh(j)t Fe(=1)3006 2180 y Fg(P)3075 2195 y Fh(j)3112 2005 y Fd(\014)3112 2055 y(\014)3112 2105 y(\014)3112 2155 y(\014)3112 2205 y(\014)3112 2254 y(\014)3582 2180 y Fs(\(13\))683 2546 y(+)865 2463 y Fd(X)843 2640 y Fe(\026)842 2652 y Fh(t)p Fr(2)p Fh(G)969 2664 y Ff(k)1024 2546 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)1295 2505 y Fh(k)1354 2372 y Fd(\014)1354 2422 y(\014)1354 2471 y(\014)1354 2521 y(\014)1354 2571 y(\014)1354 2621 y(\014)1382 2438 y Fh(k)r Fe(+1)1393 2463 y Fd(Y)1385 2645 y Fh(j)t Fe(=1)1527 2546 y Fg(P)1596 2561 y Fh(j)1655 2546 y Fg(\000)1754 2438 y Fh(k)r Fe(+1)1766 2463 y Fd(Y)1758 2645 y Fh(j)t Fe(=1)1900 2546 y Fm(e)1945 2505 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b(\()p Fm(t)22 b Fg(\000)h Fs(3)e Fg(\000)i Fs(2)p Fm(`)p Fs(\))49 b Fm(:)17 b(:)g(:)f Fs(\()p Fm(t)22 b Fg(\000)h Fm(k)i Fs(+)d(1)g Fg(\000)h Fs(2)p 3264 3887 V Fm(`)3304 3981 y Fh(A)3361 3966 y Fs(\))p 769 4010 2664 4 v 2060 4102 a Fm(k)s Fs(!)598 4285 y Fg(\024)759 4139 y Fd( )875 4225 y Fm(t)866 4341 y(k)961 4139 y Fd(!)1055 4285 y Fg(\024)1172 4218 y Fm(t)1207 4181 y Fh(k)p 1170 4262 82 4 v 1170 4353 a Fm(k)s Fs(!)1294 4285 y Fm(:)0 4661 y Fs(Moreo)m(v)m(er,)34 b(the)f(leading)e(term)g(in) h(\(13\))g(is)g(b)s(ounded)h(using)f(Prop)s(osition)f(15.)42 b(Then)34 b(w)m(e)f(ha)m(v)m(e)h(that)0 4777 y(\(13\))e(is)g(b)s (ounded)h(b)m(y)1159 5245 y Fm(C)1281 5178 y(t)1316 5142 y Fh(k)p 1279 5222 V 1279 5313 a Fm(k)s Fs(!)1403 5245 y(\()p Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)21 b Fs(+)h Fm(\036)p Fs(\()p Fm(`)1968 5260 y Fh(A)2025 5245 y Fs(\)\))2101 5204 y Fh(k)2176 5245 y Fm(")p Fs(\()p Fm(A)p Fs(\))32 b Fm(:)1832 5517 y Fi(23)p eop %%Page: 24 24 24 23 bop 141 609 a Fs(The)46 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y(closer)i(and)f(longer)g(o)s (ccurrences)i(of)e Fm(A)p Fs(.)44 b(Then,)34 b(it)d(is)h(b)s(ounded)i (b)m(y)-8 b(,)1832 5517 y Fi(24)p eop %%Page: 25 25 25 24 bop 389 484 a Fd(0)389 634 y(@)462 545 y Fh(t)487 554 y Fb(1)522 545 y Fe(+)p Fh(n)p Fr(\000)p Fe(1)526 572 y Fd(X)471 755 y Fh(i)p Fe(=)p Fh(t)575 764 y Fb(1)610 755 y Fe(+1)726 655 y Fs(+)819 544 y Fh(t)844 553 y Fb(1)879 544 y Fe(+)p 934 487 29 4 v Fh(`)962 555 y Ff(A)1013 544 y Fr(\000)p Fe(1)901 572 y Fd(X)842 755 y Fh(i)p Fe(=)p Fh(t)946 764 y Fb(1)981 755 y Fe(+)p Fh(n)1103 484 y Fd(1)1103 634 y(A)1192 655 y Fm(I)-21 b(P)1307 559 y Fd(n)1362 655 y Fm(N)1450 614 y Fh(t)1475 623 y Fb(1)1511 614 y Fr(\000)p Fe(1)1440 680 y(1)1632 655 y Fs(=)28 b(0)k(;)50 b Fm(T)1965 614 y Fr(\000)p Fh(t)2045 623 y Fb(1)2083 655 y Fm(A)33 b Fs(;)49 b Fm(T)2336 614 y Fr(\000)p Fh(i)2419 655 y Fm(A)33 b Fs(;)49 b Fm(N)2689 614 y Fh(t)2679 690 y(t)2704 699 y Fb(1)2739 690 y Fe(+)p 2794 634 V Fh(`)2823 701 y Ff(A)2905 655 y Fs(=)27 b(0)3057 559 y Fd(o)228 974 y Fg(\024)491 864 y Fh(t)516 873 y Fb(1)551 864 y Fe(+)p Fh(n)p Fr(\000)p Fe(1)555 891 y Fd(X)389 1079 y Fh(i)p Fe(=)p Fh(t)493 1088 y Fb(1)528 1079 y Fe(+\()p Fh(n=)p Fe(2\)+1)858 974 y Fm(I)-21 b(P)972 878 y Fd(n)1028 974 y Fm(N)1116 933 y Fh(t)1141 942 y Fb(1)1176 933 y Fr(\000)p Fe(1)1106 999 y(1)1298 974 y Fs(=)28 b(0)1451 878 y Fd(o)1555 974 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)32 b Fs(\(1)22 b(+)g Fm( )t Fs(\(0\)\))2295 933 y Fe(3)2367 974 y Fm(I)-21 b(P)2481 878 y Fd(n)2537 974 y Fm(A)2610 933 y Fe(\()p Fh(i)p Fr(\000)p Fh(t)2741 942 y Fb(1)2776 933 y Fe(\))2808 878 y Fd(o)2912 974 y Fm(I)g(P)3027 878 y Fd(n)3082 974 y Fm(N)3170 933 y Fh(t)3160 1009 y(t)3185 1018 y Fb(1)3221 1009 y Fe(+)p 3276 953 V Fh(`)3304 1020 y Ff(A)3386 974 y Fs(=)28 b(0)3539 878 y Fd(o)389 1320 y Fs(+)482 1208 y Fh(t)507 1217 y Fb(1)541 1208 y Fe(+)p 596 1152 V Fh(`)625 1219 y Ff(A)675 1208 y Fr(\000)p Fe(1)563 1237 y Fd(X)505 1419 y Fh(i)p Fe(=)p Fh(t)609 1428 y Fb(1)644 1419 y Fe(+)p Fh(n)782 1320 y Fm(I)-21 b(P)897 1224 y Fd(n)952 1320 y Fm(N)1040 1279 y Fh(t)1065 1288 y Fb(1)1101 1279 y Fr(\000)p Fe(1)1030 1345 y(1)1223 1320 y Fs(=)27 b(0)1375 1224 y Fd(o)1479 1320 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)33 b Fs(\(1)22 b(+)g Fm( )t Fs(\(0\)\))2220 1279 y Fe(3)2291 1320 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)32 b Fm(I)-21 b(P)2709 1224 y Fd(n)2764 1320 y Fm(N)2852 1279 y Fh(t)2842 1355 y(t)2867 1364 y Fb(1)2903 1355 y Fe(+)p 2958 1299 V Fh(`)2986 1366 y Ff(A)3068 1320 y Fs(=)28 b(0)3221 1224 y Fd(o)228 1629 y Fg(\024)84 b Fs(\(1)22 b(+)g Fm( )t Fs(\(0\)\))826 1588 y Fe(3)865 1629 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)1153 1533 y Fd(\020)1202 1629 y Fm(n)32 b(I)-21 b(P)1407 1533 y Fd(n)1462 1629 y Fm(A)1535 1588 y Fe(\()p Fh(n=)p Fe(2\))1708 1533 y Fd(o)1785 1629 y Fs(+)p 1883 1550 41 4 v 22 w Fm(`)1924 1644 y Fh(A)1981 1629 y Fm(I)g(P)13 b Fg(f)p Fm(A)p Fg(g)2252 1533 y Fd(\021)2362 1521 y Fe(2)2327 1546 y Fd(Y)2318 1728 y Fh(j)t Fe(=1)2458 1629 y Fm(I)-21 b(P)2572 1533 y Fd(n)2627 1629 y Fm(\034)2669 1644 y Fh(A)2755 1629 y Fm(>)27 b Fs(\001)2939 1644 y Fh(i)2990 1629 y Fg(\000)p 3089 1550 V 22 w Fm(`)3130 1644 y Fh(A)3187 1533 y Fd(o)228 1942 y Fg(\024)84 b Fm(C)515 1845 y Fd(h)554 1942 y Fs(\(1)22 b(+)g Fm( )t Fs(\(0\)\))991 1901 y Fe(2)1063 1942 y Fm(I)-21 b(P)13 b Fg(f)p Fm(A)p Fg(g)1334 1845 y Fd(i)1373 1868 y Fh(k)1432 1942 y Fm(")p Fs(\()p Fm(A)p Fs(\))1719 1834 y Fe(2)1684 1859 y Fd(Y)1676 2041 y Fh(j)t Fe(=1)1815 1942 y Fm(e)1860 1901 y Fr(\000)p Fh(t)1940 1911 y Ff(i)1967 1901 y Fh(I)-24 b(P)9 b Fr(f)p Fh(A)p Fr(g)2172 1942 y Fs(\(1)22 b(+)g Fm(e)2424 1957 y Fe(1)2463 1942 y Fs(\()p Fm(A)p Fs(\)\))228 2190 y Fg(\024)84 b Fm(C)56 b Fs([)p Fm(C)39 b(I)-21 b(P)14 b Fg(f)p Fm(A)p Fg(g)p Fs(])949 2141 y Fh(k)1009 2190 y Fm(")p Fs(\()p Fm(A)p Fs(\))32 b Fm(e)1281 2149 y Fr(\000)p Fh(tI)-24 b(P)9 b Fr(f)p Fh(A)p Fr(g)1583 2190 y Fm(:)0 2466 y Fs(F)-8 b(or)36 b(the)i(\014rst)g(inequalit)m(y)e (w)m(e)i(iterate)e(the)i(mixing)d(prop)s(ert)m(y)j(together)f(with)g (the)h(fact)f(that)g Fm(A)e Fg(62)0 2583 y(B)65 2598 y Fh(n)112 2583 y Fs(\(2\).)50 b(In)35 b(the)g(third)f(inequalit)m(y)g (w)m(e)i(applied)d(Theorem)i(1.)50 b(The)35 b(same)g(argumen)m(t)f(is)h (carried)f(on)0 2699 y(the)f(rest)g(of)f(the)h(pro)s(ofs.)43 b(Then)34 b(w)m(e)g(get)e(the)h(follo)m(wing)d(theorem)0 2889 y Fo(Theorem)37 b(17)46 b Fl(L)-5 b(et)85 b Fg(f)p Fm(X)982 2904 y Fh(m)1048 2889 y Fg(g)1098 2918 y Fh(m)p Fr(2)p Fh(Z)-40 b(Z)1314 2889 y Fl(b)-5 b(e)43 b(a)85 b Fm( )t Fl(-mixing)41 b(or)h(a)85 b Fm(\013)q Fl(-mixing)41 b(pr)-5 b(o)g(c)g(ess.)67 b(L)-5 b(et)43 b Fm(A)53 b(=)-60 b Fg(2)42 b(B)3554 2904 y Fh(n)3601 2889 y Fs(\(2\))p Fl(.)0 3005 y(Then,)34 b(the)h(fol)5 b(lowing)33 b(ine)-5 b(quality)35 b(holds)44 3442 y Fd(\014)44 3492 y(\014)44 3542 y(\014)44 3592 y(\014)44 3641 y(\014)71 3592 y Fm(I)-22 b(P)185 3470 y Fd(\032)248 3592 y Fm(N)403 3588 y Ff(t)p 336 3600 158 4 v 336 3641 a(I)c(P)8 b Fk(f)p Ff(A)p Fk(g)535 3592 y Fs(=)28 b Fm(k)693 3470 y Fd(\033)777 3592 y Fg(\000)887 3524 y Fm(e)932 3488 y Fr(\000)p Fh(t)1017 3524 y Fm(t)1052 3488 y Fh(k)p 887 3568 208 4 v 950 3660 a Fm(k)s Fs(!)1105 3442 y Fd(\014)1105 3492 y(\014)1105 3542 y(\014)1105 3592 y(\014)1105 3641 y(\014)1160 3592 y Fg(\024)1265 3193 y Fd(8)1265 3268 y(>)1265 3293 y(>)1265 3318 y(>)1265 3343 y(>)1265 3368 y(>)1265 3392 y(>)1265 3417 y(>)1265 3442 y(>)1265 3467 y(>)1265 3492 y(<)1265 3641 y(>)1265 3666 y(>)1265 3691 y(>)1265 3716 y(>)1265 3741 y(>)1265 3766 y(>)1265 3791 y(>)1265 3816 y(>)1265 3841 y(>)1265 3866 y(:)1380 3292 y Fm(")p Fs(\()p Fm(A)p Fs(\))1669 3253 y Fh(e)1702 3230 y Ff(k)p 1620 3269 204 4 v 1620 3327 a Fe(\()p Fh(k)r Fr(\000)p Fe(1\)!)1868 3292 y Fm(\025)1925 3307 y Fh(M)2821 3292 y Fs(1)83 b Fg(\024)28 b Fm(k)87 b Fg(\024)3363 3253 y Fe(1)p 3311 3269 141 4 v 3311 3327 a Fh(")p Fe(\()p Fh(A)p Fe(\))1380 3557 y Fm(")p Fs(\()p Fm(A)p Fs(\))1963 3518 y Fh(e)1996 3494 y Ff(k)p 1620 3534 757 4 v 1620 3646 a Fs(\()1714 3611 y Fb(1)p 1668 3623 124 4 v 1668 3664 a Ff(")p Fb(\()p Ff(A)p Fb(\))1801 3646 y Fs(\))1839 3638 y Fe(!)1893 3646 y Fs(\()1988 3611 y Fb(1)p 1941 3623 V 1941 3664 a Ff(")p Fb(\()p Ff(A)p Fb(\))2074 3646 y Fs(\))2112 3589 y Ff(k)q Fk(\000)2251 3569 y Fb(1)p 2204 3578 V 2204 3619 a Ff(")p Fb(\()p Ff(A)p Fb(\))2386 3557 y Fm(\025)2443 3572 y Fh(M)2772 3518 y Fe(1)p 2720 3534 141 4 v 2720 3591 a Fh(")p Fe(\()p Fh(A)p Fe(\))2953 3557 y Fm(<)28 b(k)88 b Fg(\024)3394 3518 y Fe(2)p Fh(t)p 3311 3534 227 4 v 3311 3591 a(nI)-25 b(P)9 b Fr(f)p Fh(A)p Fr(g)1380 3887 y Fm(nI)-22 b(P)14 b Fg(f)p Fm(A)p Fg(g)1831 3848 y Fh(e)1864 3825 y Fk(\000)p Ff(t)p 1754 3864 264 4 v 1754 3922 a Fe(\()p Fh(k)r Fr(\000)p Fe(1\)!)35 b Fh(t)2717 3848 y Fe(2)p Fh(t)p 2634 3864 227 4 v 2634 3922 a(nI)-25 b(P)9 b Fr(f)p Fh(A)p Fr(g)2953 3887 y Fm(<)28 b(k)3685 3592 y(:)0 4184 y Fl(wher)-5 b(e)37 b Fm(")p Fs(\()p Fm(A)p Fs(\))c(=)g Fm(")661 4199 y Fe(1)701 4184 y Fs(\()p Fm(A)p Fs(\))24 b(+)g Fm(")1020 4199 y Fe(2)1059 4184 y Fs(\()p Fm(A)p Fs(\))p Fl(,)39 b(with)f Fm(")1538 4199 y Fe(1)1577 4184 y Fs(\()p Fm(A)p Fs(\))g Fl(de\014ne)-5 b(d)36 b(in)i(Pr)-5 b(op)g(osition)37 b(15)h(\(r)-5 b(eplacing)36 b Fm(\036)i Fl(by)g Fm( )k Fl(or)0 4300 y Fm(\013)q Fl(\))34 b(and)h(for)69 b Fm( )t Fl(-mixing)34 b(pr)-5 b(o)g(c)g(esses,)467 4668 y Fm(")513 4683 y Fe(2)552 4668 y Fs(\()p Fm(A)p Fs(\))28 b(=)f(\(1)22 b(+)g Fm( )t Fs(\(0\)\))1269 4619 y Fe(2)p Fh(k)r Fe(+1)1488 4668 y Fs(inf)1533 4728 y Fh(`)1623 4493 y Fd(8)1623 4568 y(<)1623 4717 y(:)1697 4668 y Fm(I)-22 b(P)13 b Fg(f)p Fm(A)p Fg(g)1984 4497 y Fd(0)1983 4646 y(@)2056 4668 y Fs(1)22 b(+)2230 4585 y Fd(X)2225 4767 y Fh(j)t Fr(\025)p Fh(n)2372 4668 y Fm( )t Fs(\()p Fm(j)6 b Fs(\))2561 4497 y Fd(1)2561 4646 y(A)2655 4668 y Fs(+)22 b Fm( )t Fs(\()p Fm(`)p Fs(\))2937 4493 y Fd(9)2937 4568 y(=)2937 4717 y(;)3062 4668 y Fm(e)3107 4626 y Fr(\000)p Fh(t)3262 4668 y Fm(:)0 5035 y Fl(So,)41 b(if)f Fg(f)p Fl(A)398 5050 y Fh(n)445 5035 y Fg(g)495 5050 y Fh(n)p Fr(2)p Fh(I)-25 b(N)699 5035 y Fl(is)40 b(such)g(that)h(A)1313 5050 y Fh(n)1398 5035 y Fg(62)d(B)1567 5050 y Fh(n)1614 5035 y Fs(\(2\))i Fl(for)g(al)5 b(l)40 b Fm(n)e Fg(2)h Fm(I)-22 b(N)10 b Fl(,)42 b(then)e(the)g(law)g(of)g(the)g(numb)-5 b(er)40 b(of)0 5152 y(o)-5 b(c)g(curr)g(enc)g(es)38 b(of)g Fm(A)713 5167 y Fh(n)799 5152 y Fl(up)h(to)f(the)h(r)-5 b(e-sc)g(ale)g(d)38 b(time)g Fm(t=I)-22 b(P)14 b Fg(f)p Fm(A)2155 5167 y Fh(n)2201 5152 y Fg(g)39 b Fl(c)-5 b(onver)g(ges)37 b(we)-5 b(akly)38 b(to)h(a)g(Poisson)e(law)0 5268 y(as)e Fm(n)27 b Fg(!)h(1)34 b Fl(\(with)h(the)g(ab)-5 b(ove)34 b(r)-5 b(ate\).)44 b(F)-7 b(or)69 b Fm(\013)q Fl(-mixing)34 b(pr)-5 b(o)g(c)g(esses,)1832 5517 y Fi(25)p eop %%Page: 26 26 26 25 bop 822 672 a Fm(")868 687 y Fe(2)907 672 y Fs(\()p Fm(A)p Fs(\))28 b(=)g(inf)1232 732 y Fh(`)1322 497 y Fd(8)1322 572 y(<)1322 721 y(:)1396 672 y Fm(I)-22 b(P)14 b Fg(f)p Fm(A)p Fg(g)1684 501 y Fd(0)1683 650 y(@)1756 672 y Fs(1)21 b(+)1950 564 y Fe(2)p Fh(n)1930 589 y Fd(X)1924 771 y Fh(j)t Fe(=)p Fh(n)2071 672 y Fm(\013)q Fs(\()p Fm(j)6 b Fs(\))2256 501 y Fd(1)2256 650 y(A)2350 672 y Fs(+)2516 604 y Fm(\013)q Fs(\()p Fm(n)p Fs(\))p 2458 648 313 4 v 2458 740 a Fm(I)-22 b(P)14 b Fg(f)p Fm(A)p Fg(g)2729 711 y Fh(k)2781 497 y Fd(9)2781 572 y(=)2781 721 y(;)2906 672 y Fm(:)0 1064 y Fl(The)33 b(law)h(of)g(the)g(numb)-5 b(er)34 b(of)f(o)-5 b(c)g(curr)g(enc)g(es)34 b(of)f Fm(A)1811 1079 y Fh(n)1893 1064 y Fl(up)h(to)g(the)g(r)-5 b(e-sc)g(ale)g(d)33 b(time)h Fm(t=I)-22 b(P)13 b Fg(f)p Fm(A)3225 1079 y Fh(n)3272 1064 y Fg(g)34 b Fl(c)-5 b(onver)g(ges)0 1180 y(to)35 b(a)g(Poisson)f(pr)-5 b(o)g(c)g(ess)34 b(if)g Fm(\013)q Fs(\()p Fm(n)p Fs(\))p Fm(=I)-22 b(P)13 b Fg(f)p Fm(A)1457 1195 y Fh(n)1504 1180 y Fg(g)1554 1144 y Fh(k)1624 1180 y Fg(!)28 b Fs(0)34 b Fl(as)h Fm(n)28 b Fg(!)f(1)35 b Fl(for)f(al)5 b(l)35 b Fm(k)s Fl(.)0 1395 y Fo(Remark.)45 b Fs(The)34 b(condition)e Fm(\013)q Fs(\()p Fm(n)p Fs(\))p Fm(=I)-21 b(P)13 b Fg(f)p Fm(A)1561 1410 y Fh(n)1608 1395 y Fg(g)1658 1359 y Fh(k)1729 1395 y Fg(!)29 b Fs(0)k(as)g Fm(n)c Fg(!)g(1)k Fs(for)g(all)e Fm(k)36 b Fs(means)e(basically)d(that) i Fm(\013)0 1511 y Fs(m)m(ust)g(deca)m(y)h(sup)s(er-exp)s(onen)m (tially)e(fast.)0 1722 y Fo(Remark.)83 b Fs(In)46 b(Theorem)g(16)f(and) h(Theorem)g(17)f(and)h(according)f(to)g(De\014nition)f(14)i(w)m(e)g(c)m (hose)0 1838 y Fm(s)39 b Fs(=)h(2.)63 b(This)40 b(v)-5 b(alue)39 b(of)g Fm(s)g Fs(is)g(purely)h(formal)d(and)j(can)f(b)s(e)h (replaced)g(b)m(y)g(an)m(y)g(other)g(\014xed)g(v)-5 b(alue.)0 1955 y(The)35 b(corresp)s(onding)f(mo)s(di\014cation)d(m)m(ust)k(b)s(e) f(done)g(in)g(the)g(de\014nition)f(of)h Fm(")2883 1970 y Fe(1)2922 1955 y Fs(\()p Fm(A)p Fs(\).)48 b(W)-8 b(e)34 b(recall)f(that)0 2071 y(in)44 b(Collet,)k(Galv)m(es)d(and)g(Sc)m (hmitt)f(\(1999\))h(it)f(w)m(as)i(pro)m(v)m(ed)h(that)d(for)h(exp)s (onen)m(tially)89 b Fm( )t Fs(-mixing)0 2187 y(pro)s(cesses)32 b(there)e(exist)g(p)s(ositiv)m(e)f(constan)m(ts)i Fm(C)37 b Fs(and)30 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