Content-Type: multipart/mixed; boundary="-------------9912020954103" This is a multi-part message in MIME format. ---------------9912020954103 Content-Type: text/plain; name="99-460.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-460.keywords" unimodal maps, nonuniformly hyperbolic systems, rates of mixing, random perturbations, ---------------9912020954103 Content-Type: application/postscript; name="bbm.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bbm.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: bbm.dvi %%Pages: 53 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips bbm -o bbm.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.12.02:1126 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 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y Fq(j)2385 5088 y Fh(k)2428 5077 y Fq(j)2461 5088 y Fh(k)q Fi(+1)2590 5077 y Fq(;\033)2663 5057 y Fh(k)2706 5077 y Fq(!)2764 4916 y Fo(\023)2854 5056 y Fs(h)2911 5071 y Fq(!)2969 5056 y Fv(\()p Fs(j)3049 5071 y Fr(0)3093 5056 y Fv(\))17 b Fs(;)1869 5316 y Ft(8)p eop %%Page: 9 9 9 8 bop 0 104 a Fl(ther)-5 b(e)43 b(ar)-5 b(e)43 b Fv(0)e Fs(<)h(u)g(<)g(u)918 68 y Fp(0)989 104 y Fl(and)h Fs(C)1268 52 y Fr(\(4\))1261 116 y Fq(!)1417 104 y Fk(\025)f Fv(1)h Fl(such)g(that)g(for)g(e)-5 b(ach)43 b Fs(')g Fl(and)h Fs( )i Fl(in)d Fs(`)3066 68 y Fp(1)3150 104 y Fv(\()p Ff(Z)3258 119 y Fr(+)3321 104 y Fv(\))p Fl(,)i(the)e(p)-5 b(ast)0 224 y(r)g(andom)36 b(c)-5 b(orr)g(elations)35 b(satisfy)80 294 y Fo(\014)80 354 y(\014)80 414 y(\014)114 409 y Ff(E)174 328 y Fo(\002)222 409 y Fs(')p Fv(\()p Fs(X)417 357 y Fr(\()p Fq(n)p Fr(\))409 444 y Fq(\033)458 424 y Fg(\000)p Fh(n)559 444 y Fq(!)617 409 y Fv(\))p Fs( )t Fv(\()p Fs(X)855 357 y Fr(\(0\))847 444 y Fq(\033)896 424 y Fg(\000)p Fh(n)997 444 y 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Fv(\(1\))49 b(Ob)m(viously)f(one)g(ma)m(y)e(form)m(ulate)h(the)h(main)f(corollary)f (for)i(future)h(correlations,)i(ap-)329 1292 y(proac)m(h)33 b(to)e(equilibrium,)g(etc.,)h(for)g(i.i.d.)43 b(coun)m(table)33 b(Mark)m(o)m(v)f(c)m(hains.)44 b(The)33 b(main)e(corol-)329 1412 y(lary)36 b(can)i(b)s(e)f(also)g(expressed)i(as)e(a)h(result)f(on) h(sp)s(eed)g(of)g(con)m(v)m(ergence)h(to)d(the)i(maximal)329 1531 y(eigen)m(v)m(ector)23 b(of)h(random)e(pro)s(ducts)i(of)f(sto)s(c) m(hastic)h(matrices)e(ha)m(ving)h(a)g(\\to)m(w)m(er)g(structure")329 1651 y(as)35 b(in)h(\(2.6\))o(.)50 b(The)36 b(sligh)m(tly)f(cum)m(b)s (ersome)g(exercise)h(is)f(left)h(to)f(the)g(reader.)52 b(W)-8 b(e)36 b(refer)g(to)329 1771 y(the)i(pap)s(ers)h(of)f(Hennion)h ([He])e(and)i(the)f(b)s(o)s(ok)f(of)i(Bougerol-Lacroix)e([BoL,)g(esp)s (ecially)329 1890 y(Chapter)28 b(A.I)s(I)s(I])e(for)i(references)i(on)e (the)g(classical)f(w)m(ork)g(of)h(F)-8 b(ursten)m(b)s(erg,)30 b(Kesten,)g(Guiv-)329 2010 y(arc'h,)j(Ledrappier,)i(and)f(others,)g(on) g(applications)f(of)h(the)g(Oseledec)h(theorem)e(yielding)329 2129 y(exp)s(onen)m(tial)h(b)s(ounds)h(for)f(the)h(sp)s(eed)g(of)f(con) m(v)m(ergence)i(to)e(the)g(maximal)d(eigen)m(v)m(ector)j(of)329 2249 y(random)g(pro)s(ducts)h(of)g Fl(\014nite)f Fv(sto)s(c)m(hastic)h (matrices,)f(under)h(assumptions)g(guaran)m(teeing)329 2368 y(that)e(the)g(maximal)d(Ly)m(apuno)m(v)k(exp)s(onen)m(t)g(is)f (simple.)152 2488 y(\(2\))49 b(Adapting)23 b(Sections)i(7)e(and)h(8)g (similarly)d(as)i(the)h(corresp)s(onding)h(pro)s(ofs)f(of)g(Theorem)g (2\(I)s(I\))329 2607 y(of)32 b([Y)-8 b(o2],)31 b(w)m(e)i(ma)m(y)e(also) h(obtain)g(exp)s(onen)m(tial)g(\(resp)s(ectiv)m(ely)g(p)s(olynomial\))e (estimates)i(in)329 2727 y(\(2.7\))g(if)h(w)m(e)h(c)m(hange)g(the)g (assumptions)f(accordingly)-8 b(.)0 2910 y Fl(Op)j(en)36 b(questions.)152 3051 y Fv(\(1\))49 b(As)24 b(men)m(tioned)g(in)g(the)g (in)m(tro)s(duction,)i(b)m(y)e(adapting)g(Kifer's)g(metho)s(ds)g(in)g 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y(\000)1077 5168 y Fs(f)1137 5127 y Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\))1126 5193 y Fq(\033)s(!)1346 5088 y Fo(\001)1392 5108 y Fp(0)1420 5168 y Fv(\()p Fs(y)t Fv(\))1551 5084 y Fo(\014)1551 5143 y(\014)1611 5168 y Fk(\025)1742 5101 y Fv(1)p 1728 5145 79 4 v 1728 5237 a Fs(C)1835 5168 y Fv(max)n(\()p Fs(e)2105 5127 y Fr(\(2)p Fp(\000)p Fr(2)p Fq(\021)s Fr(\))p Fq(k)2402 5168 y Fs(;)17 b(\017)2487 5127 y Fp(\000)p Fr(1+)p Fq(\021)2698 5168 y Fv(\))g Fs(:)1847 5316 y Ft(10)p eop %%Page: 11 11 11 10 bop 0 100 a Fl(Pr)-5 b(o)g(of)31 b(of)h(Sublemma)i(3.2.)50 b Fv(This)29 b(is)h(an)g(adaptation)f(of)h(the)g(usual)g(\\b)s(ound)g (p)s(erio)s(d)g(estimates")e(of)0 219 y([BC1,)34 b(BC2].)47 b(The)35 b(starting)f(p)s(oin)m(t)g(is)h(the)g(claim)e(that)h(there)h (is)f Fs(C)k(>)30 b Fv(1,)k(indep)s(enden)m(t)j(of)e Fs(\017)g Fv(and)0 339 y Fs(\016)t Fv(,)e(and)h(suc)m(h)g(that)f(for)h (ev)m(ery)f Fs(y)t(;)22 b Fv(~)-55 b Fs(y)30 b Fk(2)e Fs(J)1485 354 y Fq(k)r(;\017)1591 339 y Fv(,)33 b(all)f Fs(!)t(;)24 b Fv(~)-57 b Fs(!)30 b Fk(2)e Fv(\012,)33 b(and)h(all)e(1)c Fk(\024)g Fs(j)33 b Fk(\024)c Fs(p)p Fv(\()p Fs(k)s(;)17 b(\017)p Fv(\))k(+)i(1)755 480 y Fo(\014)755 540 y(\014)788 565 y Fs(f)848 524 y Fq(j)837 589 y(\033)s(!)943 565 y Fv(\()p Fs(y)t Fv(\))e Fk(\000)i Fs(f)1255 517 y Fq(j)1244 593 y(\033)9 b Fr(~)-46 b Fq(!)1350 565 y Fv(\()7 b(~)-57 b Fs(y)s Fv(\))1480 480 y Fo(\014)1480 540 y(\014)1541 565 y Fk(\024)28 b Fs(C)1724 480 y Fo(\014)1724 540 y(\014)1757 565 y Fv(\()p Fs(f)1856 524 y Fq(j)t Fp(\000)p Fr(1)1999 565 y Fv(\))2038 524 y Fp(0)2066 565 y Fv(\()p Fs(f)11 b Fv(\(0\)\))2332 480 y Fo(\014)2332 540 y(\014)2386 565 y Fk(\001)22 b Fv(max)n(\()p Fs(e)2706 524 y Fp(\000)p Fr(2)p Fq(k)2857 565 y Fs(;)17 b(\017)p Fv(\))g Fs(:)549 b Fv(\(3.2\))0 778 y(T)-8 b(o)33 b(c)m(hec)m(k)h (\(3.2\))o(,)f(w)m(e)g(\014rst)h(v)m(erify)f(inductiv)m(ely)g(that)0 916 y Fo(\014)0 975 y(\014)33 1000 y Fs(f)93 959 y Fq(j)82 1025 y(\033)s(!)188 1000 y Fv(\()p Fs(y)t Fv(\))16 b Fk(\000)g Fs(f)488 952 y Fq(j)477 1029 y(\033)9 b Fr(~)-46 b Fq(!)583 1000 y Fv(\()7 b(~)-57 b Fs(y)s Fv(\))713 916 y Fo(\014)713 975 y(\014)774 1000 y Fk(\024)879 919 y Fo(\002)920 1000 y Fs(d)972 1015 y Fq(j)1014 1000 y Fv(\()p Fk(\001)17 b(\001)g(\001)e Fv(\()p Fs(d)1277 1015 y Fr(2)1321 1000 y Fv(\()p Fs(d)1412 1015 y Fr(1)1472 1000 y Fv(+)h(1\))h Fk(\001)g(\001)g(\001)e Fv(\))h(+)g(1)2002 919 y Fo(\003)2060 1000 y Fs(C)41 b Fv(max)n(\()p Fs(e)2442 959 y Fp(\000)p Fr(2)p Fq(k)2593 1000 y Fs(;)17 b(\017)p Fv(\))27 b(=:)h([)p Fs(m)2992 1015 y Fq(j)3034 1000 y Fv(])17 b Fs(C)40 b Fv(max)o(\()p Fs(e)3461 959 y Fp(\000)p Fr(2)p Fq(k)3612 1000 y Fs(;)17 b(\017)p Fv(\))g Fs(;)0 1227 y Fv(where)34 b Fs(d)340 1242 y Fq(i)401 1227 y Fv(=)28 b Fk(j)p Fs(f)594 1191 y Fp(0)583 1260 y Fq(\033)632 1240 y Fh(i)663 1260 y Fq(!)720 1227 y Fv(\()p Fs(x)816 1242 y Fq(i)849 1227 y Fv(\))p Fk(j)f Fv(=)h Fk(j)p Fs(f)1136 1191 y Fp(0)1164 1227 y Fv(\()p Fs(x)1260 1242 y Fq(i)1292 1227 y Fv(\))p Fk(j)33 b Fv(for)g(some)g Fs(x)1851 1242 y Fq(i)1912 1227 y Fk(2)28 b Fv([)p Fs(f)2094 1191 y Fq(i)p Fp(\000)p Fr(1)2083 1252 y Fq(\033)s(!)2228 1227 y Fv(\()p Fs(y)t Fv(\))p Fs(;)17 b(f)2464 1185 y Fq(i)p Fp(\000)p Fr(1)2453 1256 y Fq(\033)9 b Fr(~)-46 b Fq(!)2597 1227 y Fv(\()7 b(~)-57 b Fs(y)s Fv(\)].)120 1366 y(Then,)36 b(to)f(estimate)f Fs(m)1022 1381 y Fq(j)1064 1366 y Fv(,)h(w)m(e)h(let) 1439 1339 y(^)1422 1366 y Fs(d)1474 1381 y Fq(i)1538 1366 y Fv(=)31 b Fk(j)p Fs(f)1734 1329 y Fp(0)1761 1366 y Fv(\()p Fs(f)1860 1329 y Fq(i)1893 1366 y Fv(\(0\)\))p Fk(j)p Fv(,)j(and)h(w)m(e)h(note)f(that)g(since)h Fk(j)p Fs(f)3269 1329 y Fq(i)3302 1366 y Fv(\(0\))22 b Fk(\000)i Fs(x)3610 1381 y Fq(i)3643 1366 y Fk(j)31 b Fs(<)4 1473 y Fv(~)0 1500 y Fs(\025)58 1463 y Fp(\000)p Fq(i\021)240 1500 y Fv(for)43 b(1)g Fk(\024)h Fs(i)g Fk(\024)g Fs(p)29 b Fv(+)g(1,)44 b(b)m(y)f(de\014nition)h(of)f Fs(p)p Fv(,)i(and)e Fk(j)2217 1473 y Fv(^)2200 1500 y Fs(d)2252 1515 y Fq(i)2284 1500 y Fk(j)h(\025)g Fs(e)2523 1463 y Fp(\000)p Fq(\013i)2670 1500 y Fs(=C)50 b Fv(b)m(y)43 b(\(H1\)\(ii\),)g(standard)0 1636 y(argumen)m(ts)29 b(in)m(v)m(olving)e(\(H2\))h(and)h(using)h Fs(e)1625 1600 y Fp(\000)p Fq(j)t(\021)16 b Fr(log)1902 1581 y(~)1898 1600 y Fq(\025)1978 1636 y Fs(<)28 b(e)2129 1600 y Fp(\000)p Fr(2)p Fq(\013j)2354 1636 y Fv(\(see)h([BeY,)f(Lemma)f (1.3]\))h(giv)m(e)g(that)0 1755 y(there)34 b(is)f Fs(C)i(>)28 b Fv(1)33 b(with)959 2028 y Fs(C)1037 1987 y Fp(\000)p Fr(1)1172 2028 y Fk(\024)1289 1884 y Fo(Q)1383 1909 y Fq(j)1383 1989 y(i)p Fr(=1)1534 1959 y Fs(d)1586 1974 y Fq(i)p 1289 2005 330 4 v 1289 2038 a Fo(Q)1383 2063 y Fq(j)1383 2143 y(i)p Fr(=1)1552 2087 y Fv(^)1534 2113 y Fs(d)1586 2128 y Fq(i)1659 2028 y Fk(\024)28 b Fs(C)c(;)17 b Fk(8)g Fv(1)26 b Fk(\024)j Fs(j)k Fk(\024)28 b Fs(p)p Fv(\()p Fs(k)s(;)17 b(\017)p Fv(\))22 b(+)g(1)17 b Fs(:)754 b Fv(\(3.3\))0 2321 y(In)45 b(fact,)j(the)d(pro)s(of)f(of)h(\(3.3\))f (also)g(giv)m(es)h(assertion)g(\(1\))f(of)h(the)g(sublemma.)77 b(\(Note)44 b(that)g(the)0 2441 y(pro)s(of)d(of)g([BeY,)f(Lemma)f(1.3]) h(ma)m(y)f(require)i(taking)f(a)h(smaller)e(v)-6 b(alue)41 b(of)g Fs(\016)k Fv(in)c(\(H2\),)g(in)g(order)0 2560 y(to)d(guaran)m(tee)g(that)g Fk(j)p Fs(f)897 2524 y Fq(j)938 2560 y Fv(\(0\))p Fk(j)d Fs(>)h(\016)42 b Fv(for)c Fs(j)k Fk(\024)36 b Fs(M)1778 2575 y Fr(0)1822 2560 y Fv(,)j(where)g Fs(M)2279 2575 y Fr(0)2361 2560 y Fv(is)f(a)g(large)g(in)m(teger,)h (making)d(use)j(of)0 2680 y(\(H1\)\(ii\).\))j(No)m(w,)32 b(b)m(y)i(de\014nition)f(and)h(\(H1\)\(i\))905 2901 y Fs(m)992 2916 y Fq(j)1062 2901 y Fv(=)29 b Fs(d)1220 2916 y Fq(j)t Fp(\000)p Fr(1)1363 2901 y Fs(m)1450 2916 y Fq(j)t Fp(\000)p Fr(1)1617 2901 y Fv(+)23 b(1)k Fk(\024)h Fs(d)1951 2916 y Fq(j)t Fp(\000)p Fr(1)2095 2901 y Fs(m)2182 2916 y Fq(j)t Fp(\000)p Fr(1)2326 2820 y Fo(\000)2372 2901 y Fv(1)22 b(+)h Fs(C)2626 2875 y Fv(~)2622 2901 y Fs(\025)2680 2860 y Fp(\000)p Fq(j)2784 2820 y Fo(\001)2847 2901 y Fs(;)0 3114 y Fv(so)33 b(that)646 3309 y Fs(m)733 3324 y Fq(j)803 3309 y Fk(\024)908 3169 y Fo(\022)981 3180 y Fq(j)t Fp(\000)p Fr(1)987 3215 y Fo(Y)986 3427 y Fq(i)p Fr(=1)1137 3309 y Fs(d)1189 3324 y Fq(j)1231 3169 y Fo(\023)1320 3180 y Fq(j)t Fp(\000)p Fr(1)1326 3215 y Fo(Y)1325 3427 y Fq(i)p Fr(=1)1459 3309 y Fv(\(1)22 b(+)h Fs(C)1752 3283 y Fv(~)1748 3309 y Fs(\025)1806 3268 y Fp(\000)p Fq(i)1902 3309 y Fv(\))17 b Fs(;)49 b Fv(sho)m(wing)33 b(our)h(claim)d(\(3.2\))o(.)120 3580 y(W)-8 b(e)43 b(ma)m(y)g(no)m(w)g(pro)m(v)m(e)h(assertions)h(\(2\))d (and)i(\(3\))f(of)h(the)g(sublemma.)74 b(Assumption)43 b(\(H1\)\(i\),)0 3699 y(together)38 b(with)h(\(1\),)f(that)g(w)m(e)h (already)f(pro)m(v)m(ed,)i(and)f(the)g(fact)g(that)f Fk(j)p Fs(J)2795 3714 y Fq(k)r(;\017)2901 3699 y Fk(j)e(\025)h Fv(max)n(\()p Fs(e)3349 3663 y Fp(\000)p Fr(2)p Fq(k)3500 3699 y Fs(;)17 b(\017)p Fv(\))p Fs(=C)7 b Fv(,)0 3819 y(yield)1392 3941 y(max)o(\()p Fs(e)1663 3905 y Fp(\000)p Fr(2)p Fq(k)1814 3941 y Fs(;)17 b(\017)p Fv(\))1958 3915 y(~)1955 3941 y Fs(\025)2013 3905 y Fq(p)p Fp(\000)p Fr(1)p 1392 3986 768 4 v 1737 4077 a Fs(C)2200 4008 y Fk(\024)28 b Fv(1)17 b Fs(;)0 4214 y Fv(so)33 b(that)1032 4356 y Fs(p)p Fv(\()p Fs(k)s(;)17 b(\017)p Fv(\))27 b Fk(\024)i Fv(1)21 b(+)i(log)q(\()p Fs(C)g Fv(min\()p Fs(e)2118 4315 y Fr(2)p Fq(k)2206 4356 y Fs(;)17 b(\017)2291 4315 y Fp(\000)p Fr(1)2398 4356 y Fv(\)\))2564 4288 y(1)p 2488 4333 204 4 v 2488 4438 a(log)2637 4411 y(~)2633 4438 y Fs(\025)2719 4356 y(;)828 b Fv(\(3.4\))0 4610 y(sho)m(wing)29 b(the)g(upp)s(er)h(b)s(ound)g(in)f(\(2\).)41 b(F)-8 b(or)29 b(the)g(lo)m(w)m(er)g(b)s(ound,)h(use)g(\(H1\))e Fk(j)p Fs(J)2846 4625 y Fq(k)r(;\017)2951 4610 y Fk(j)g(\024)g Fs(C)c Fv(max)n(\()p Fs(e)3477 4573 y Fp(\000)p Fr(2)p Fq(k)3628 4610 y Fs(;)17 b(\017)p Fv(\),)0 4741 y(the)34 b(de\014nition)f(of)h Fs(p)p Fv(\()p Fs(k)s(;)17 b(\017)p Fv(\))32 b(and)1226 4715 y(~)1222 4741 y Fs(\025)1280 4705 y Fq(K)1361 4715 y Fv(~)1357 4741 y Fs(\025)1415 4705 y Fq(\021)1491 4741 y Fs(<)1601 4715 y Fv(~)1596 4741 y Fs(\025)1654 4705 y Fq(K)5 b Fr(+1)1832 4741 y Fv(.)120 4860 y(F)-8 b(or)43 b(\(3\),)j(letting)e(1)i Fk(\024)g Fs(j)52 b Fk(\024)46 b Fs(p)p Fv(\()p Fs(k)s(;)17 b(\017)p Fv(\))29 b(+)h(1)44 b(it)g(follo)m(ws)g(from)g(\(3.2\))f(that) h(for)g Fs(y)t(;)23 b Fv(~)-56 b Fs(y)48 b Fk(2)f Fs(J)3469 4875 y Fq(k)r(;\017)3619 4860 y Fv(and)0 4980 y(arbitrary)32 b Fs(!)t(;)24 b Fv(~)-57 b Fs(!)30 b Fk(2)e Fv(\012,)204 5197 y Fk(j)p Fs(f)292 5156 y Fq(j)281 5221 y(\033)s(!)387 5197 y Fv(\()p Fs(y)t Fv(\))21 b Fk(\000)i Fs(f)699 5149 y Fq(j)688 5225 y(\033)9 b Fr(~)-46 b Fq(!)794 5197 y Fv(\()7 b(~)-57 b Fs(y)s Fv(\))p Fk(j)27 b(\024)h Fs(C)7 b Fk(j)p Fv(\()p Fs(f)1289 5156 y Fq(j)1331 5197 y Fv(\))1370 5156 y Fp(0)1397 5197 y Fv(\()p Fs(f)k Fv(\(0\)\))p Fk(j)32 b Fv(max)n(\()p Fs(e)1993 5156 y Fp(\000)p Fr(2)p Fq(k)2144 5197 y Fs(;)17 b(\017)p Fv(\))27 b Fk(\024)h Fs(C)2478 5156 y Fr(2)2523 5197 y Fk(j)p Fv(\()p Fs(f)2650 5156 y Fq(j)2639 5221 y(\033)s(!)2745 5197 y Fv(\))2784 5156 y Fp(0)2812 5197 y Fv(\()p Fs(y)t Fv(\))p Fk(j)17 b Fv(max)l(\()p Fs(e)3256 5156 y Fp(\000)p Fr(2)p Fq(k)3407 5197 y Fs(;)g(\017)p Fv(\))g Fs(:)1847 5316 y Ft(11)p eop %%Page: 12 12 12 11 bop 0 100 a Fv(Th)m(us,)34 b(the)g(de\014nition)f(of)h Fs(p)p Fv(\()p Fs(k)s(;)17 b(\017)p Fv(\))32 b(giv)m(es)833 339 y Fs(C)911 298 y Fr(2)956 339 y Fk(j)p Fv(\()p Fs(f)1083 298 y Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\)+1)1072 363 y Fq(\033)s(!)1392 339 y Fv(\))1431 298 y Fp(0)1459 339 y Fv(\()p Fs(y)t Fv(\))p Fk(j)20 b(\001)i Fv(max)o(\()p Fs(e)1959 298 y Fp(\000)p Fr(2)p Fq(k)2110 339 y Fs(;)17 b(\017)p Fv(\))27 b Fk(\025)2370 313 y Fv(~)2366 339 y Fs(\025)2424 298 y Fp(\000)p Fq(\021)s Fr(\()p Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\)+1\))2919 339 y Fs(:)0 569 y Fv(Finally)32 b(\(3.4\))g(implies)368 841 y(~)364 867 y Fs(\025)422 826 y Fp(\000)p Fq(\021)s Fr(\()p Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\)+1\))928 867 y Fk(\025)c Fs(e)1079 821 y Fp(\000)p Fr(\()p Fq(\021)17 b Fr(log)1350 803 y(~)1346 821 y Fq(\025)q Fr(\))1425 741 y Fo(\002)1466 821 y Fq(C)5 b Fr(+log)1706 741 y Fo(\000)1765 821 y Fr(min)1911 741 y Fo(\000)1957 821 y Fq(e)1995 791 y Fi(2)p Fh(k)2072 821 y Fq(;\017)2129 791 y Fg(\000)p Fi(1)2222 741 y Fo(\001)o(\001\003)2354 821 y Fq(=)14 b Fr(log)2529 803 y(~)2526 821 y Fq(\025)2605 867 y Fk(\025)2723 800 y Fv(max)n(\()p Fs(e)2993 763 y Fp(\000)p Fr(2)p Fq(\021)s(k)3188 800 y Fs(;)j(\017)3273 763 y Fq(\021)3321 800 y Fv(\))p 2723 844 638 4 v 3002 935 a Fs(C)3388 867 y(;)0 1125 y Fv(and)34 b(w)m(e)f(conclude)i(that)914 1378 y Fk(j)p Fv(\()p Fs(f)1041 1337 y Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\))1030 1402 y Fq(\033)s(!)1249 1378 y Fv(\))1288 1337 y Fp(0)1316 1378 y Fv(\()p Fs(y)t Fv(\))p Fk(j)26 b(\025)1633 1310 y Fv(1)p 1618 1355 79 4 v 1618 1446 a Fs(C)1725 1378 y Fv(max)o(\()p Fs(e)1996 1337 y Fr(\(1)p Fp(\000)p Fq(\021)s Fr(\)2)p Fq(k)2292 1378 y Fs(;)17 b(\017)2377 1337 y Fp(\000)p Fr(1+)p Fq(\021)2589 1378 y Fv(\))g Fs(:)115 b Fe(\003)0 1699 y Fl(Pr)-5 b(o)g(of)35 b(of)h(L)-5 b(emma)38 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Fv(\()p Fs(!)t Fv(\))27 b(=)h Fk(f)p Fs(x)g Fk(2)g Fs(I)35 b Fk(j)27 b Fs(m)34 b Fv(is)f(the)h(\014rst)g(h)m (yp)s(erb)s(olic)f(time)59 b Fk(\025)29 b Fs(p)2713 4713 y Fr(0)2791 4698 y Fv(for)k(\()p Fs(!)t(;)17 b(x)p Fv(\))p Fk(g)420 4858 y Fs(H)511 4816 y Fp(\003)503 4882 y Fq(m)578 4858 y Fv(\()p Fs(!)t Fv(\))27 b(=)h Fk(f)p Fs(x)g Fk(2)g Fs(I)35 b Fk(j)27 b Fs(m)34 b Fv(is)f(the)h(\014rst)g(h)m(yp)s(erb)s (olic)f(return)62 b Fk(\025)28 b Fs(p)2791 4873 y Fr(0)2869 4858 y Fv(for)33 b(\()p Fs(!)t(;)17 b(x)p Fv(\))p Fk(g)g Fs(:)0 5095 y Fv(Finally)-8 b(,)31 b(w)m(e)j(set)g Fs(E)741 5059 y Fp(\003)735 5120 y Fq(m)810 5095 y Fv(\()p Fs(!)t Fv(\))27 b(=)h Fs(I)h Fk(n)1231 5020 y Fo(S)1314 5045 y Fq(m)1314 5125 y(k)r Fr(=)p Fq(p)1460 5135 y Fi(0)1521 5095 y Fs(H)1612 5059 y Fp(\003)1604 5123 y Fq(k)1658 5095 y Fv(\()p Fs(!)t Fv(\).)1847 5316 y Ft(17)p eop %%Page: 18 18 18 17 bop 0 100 a Fx(Lemma)61 b(3.8)f(\(Hyp)s(erb)s(olic)g(return)f (estimates\).)c Fl(L)-5 b(et)54 b Fv(0)60 b Fs(<)g(\020)7 b Fv(\()p Fs(\017)p Fv(\))60 b Fk(\024)h Fs(\015)6 b Fv(\()p Fs(\017)p Fv(\))p Fs(=)p Fv(2)52 b Fl(b)-5 b(e)53 b(as)g(in)0 219 y(L)-5 b(emma)37 b(3.7.)46 b(Ther)-5 b(e)35 b(is)h Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))28 b Fs(>)h Fv(1)p Fl(,)35 b(such)h(that)g(for)f(al)5 b(l)37 b Fs(!)32 b Fk(2)c Fv(\012)2396 234 y Fr(0)2477 219 y Fl(and)36 b(al)5 b(l)37 b Fs(m)28 b Fk(\025)h Fs(n)3095 234 y Fr(0)3139 219 y Fv(\()p Fs(!)t Fv(\))22 b(+)h Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))p Fl(,)36 b(we)0 343 y(have)f Fv(Leb)q(\()p Fs(E)510 306 y Fp(\003)504 367 y Fq(m)579 343 y Fv(\()p Fs(!)t Fv(\)\))26 b Fk(\024)j Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))p Fs(e)1136 306 y Fp(\000)p Fq(\020)e Fr(\()p Fq(\017)p Fr(\))p Fq(m)1410 343 y Fl(.)0 577 y(Pr)-5 b(o)g(of)34 b(of)i(L)-5 b(emma)36 b(3.8.)50 b Fv(Applying)33 b(Pliss')f(Lemma)g(as) h(in)h([A,)e(Prop)s(osition)g(2.6],)g(w)m(e)i(\014nd)1029 886 y Fs(I)29 b Fk(n)22 b Fs(E)1248 901 y Fq(m)1323 886 y Fv(\()p Fs(!)t Fv(\))27 b Fk(\032)1657 762 y Fq(m)1637 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1713 y Fo([)1171 1927 y Fq(k)r Fr(=)p Fq(p)1317 1937 y Fi(0)1373 1807 y Fs(H)1464 1766 y Fp(\003)1456 1832 y Fq(k)1509 1807 y Fv(\()p Fs(!)t Fv(\))27 b Fk(\033)1785 1674 y Fq(m)p Fp(\000)p Fq(C)19 b Fr(log)q(\(1)p Fq(=\017)p Fr(\))1974 1713 y Fo([)1936 1927 y Fq(k)r Fr(=)p Fq(p)2082 1937 y Fi(0)2289 1807 y Fs(H)2372 1822 y Fq(k)2421 1807 y Fv(\()p Fs(!)t Fv(\))e Fs(:)0 2150 y Fv(If)49 b Fs(y)57 b Fv(=)d Fk(j)p Fs(f)440 2114 y Fq(m)429 2175 y(!)515 2150 y Fv(\()p Fs(x)p Fv(\))p Fk(j)e Fs(>)861 2078 y Fk(p)p 944 2078 V 72 x Fs(\017)e Fv(then)f(the)g(in)m(terv)-6 b(al)49 b([)p Fk(\006)p Fs(y)36 b Fk(\000)2139 2078 y(p)p 2222 2078 V 72 x Fs(\017=)p Fv(2)p Fs(;)17 b Fk(\006)p Fs(y)35 b Fv(+)2678 2078 y Fk(p)p 2761 2078 V 72 x Fs(\017)q(=)p Fv(2])48 b(do)s(es)h(not)g(in)m(tersect)0 2270 y(\()p Fk(\000)116 2198 y(p)p 199 2198 V 72 x Fs(\017)q(=)p Fv(2)p Fs(;)385 2198 y Fk(p)p 467 2198 V 467 2270 a Fs(\017=)p Fv(2\).)81 b(The)46 b(heart)g(of)g(the)h(pro)s(of)f(lies)f(in)h(the)g (observ)-6 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Fs(\017\034)804 2626 y Fq(k)853 2662 y Fs(=C)7 b Fv(.)54 b(If)37 b Fs(k)f(>)d Fv(log)q(\(2)p Fs(=C)7 b(\017)p Fv(\))p Fs(=)17 b Fv(log)g Fs(\034)43 b Fk(\030)34 b Fs(C)24 b Fv(log\(1)p Fs(=\017)p Fv(\))36 b(then)i Fs(C)2985 2590 y Fk(p)p 3068 2590 V 72 x Fs(\017)3108 2590 y Fk(p)p 3191 2590 V 72 x Fs(\017)q(\034)3287 2626 y Fq(k)3369 2662 y Fs(>)33 b Fv(2)g(=)h Fk(j)p Fs(I)8 b Fk(j)0 2782 y Fv(so)36 b(that)f(our)h(in)m(terv)-6 b(al)35 b(of)h(length)1308 2710 y Fk(p)p 1391 2710 V 72 x Fs(\017)g Fv(cen)m(tered)h(at)f Fs(y)j Fv(will)34 b(ha)m(v)m(e)i(in)m(tersected)h(\()p Fk(\000)3120 2710 y(p)p 3204 2710 V 3204 2782 a Fs(\017;)3289 2710 y Fk(p)p 3371 2710 V 3371 2782 a Fs(\017)q Fv(\))e(for)h(the)0 2901 y(\014rst)e(time)e(b)m(y)h(the)g(time)f Fs(C)24 b Fv(log)q(\(1)p Fs(=\017)p Fv(\).)120 3040 y(T)-8 b(o)33 b(\014nish,)i(since)807 2965 y Fo(S)890 2990 y Fq(m)890 3070 y(k)r Fr(=)p Fq(p)1036 3080 y Fi(0)1097 3040 y Fs(H)1188 3004 y Fp(\003)1180 3068 y Fq(k)1233 3040 y Fv(\()p Fs(!)t Fv(\))28 b Fk(\033)1512 2965 y Fo(S)1595 2988 y Fq(m)p Fp(\000)p Fq(C)19 b Fr(log)q(\(1)p Fq(=\017)p Fr(\))1595 3070 y Fq(k)r Fr(=)p Fq(p)1741 3080 y Fi(0)2104 3040 y Fs(H)2187 3055 y Fq(k)2236 3040 y Fv(\()p Fs(!)t Fv(\))33 b(and)i Fs(I)30 b Fk(n)22 b Fs(E)2834 3004 y Fp(\003)2828 3065 y Fq(m)2904 3040 y Fv(\()p Fs(!)t Fv(\))28 b(=)3182 2965 y Fo(S)3265 2990 y Fq(m)3265 3070 y(k)r Fr(=)p Fq(p)3411 3080 y Fi(0)3472 3040 y Fs(H)3563 3004 y Fp(\003)3555 3068 y Fq(k)3609 3040 y Fv(\()p Fs(!)t Fv(\),)0 3163 y(w)m(e)34 b(ha)m(v)m(e)1323 3282 y Fs(E)1403 3241 y Fp(\003)1397 3307 y Fq(m)1472 3282 y Fv(\()p Fs(!)t Fv(\))27 b Fk(\032)h Fs(E)1822 3300 y Fq(m)p Fp(\000)p Fq(C)18 b Fr(log)q(\(1)p Fq(=\017)p Fr(\))2314 3282 y Fv(\()p Fs(!)t Fv(\))0 3468 y(giving)32 b(the)i(claim,)d(with)i Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))28 b(=)g(log)17 b(1)p Fs(=\017)p Fv(,)33 b(b)m(y)h(de\014nition)f(of)h(the)f Fs(B)2605 3483 y Fq(n)2659 3468 y Fv(,)g(see)h(Lemma)e(3.7.)98 b Fe(\003)0 3702 y Fx(3.C)38 b(The)f(random)h(partitions.)120 3822 y Fv(The)e(\014rst)g(step)g(is)g(to)f(obtain)h(for)f(\014xed)i Fs(!)e Fk(2)d Fv(\012,)k(and)g(eac)m(h)h Fs(m)32 b Fk(\025)h Fs(p)2719 3837 y Fr(0)2799 3822 y Fv(a)i(mo)s(d-0)g(partition)g(of)h Fs(I)0 3942 y Fv(in)m(to)d(in)m(terv)-6 b(als)1176 4116 y Fs(I)36 b Fv(=)1418 3992 y Fq(m)1398 4021 y Fo([)1361 4236 y Fq(k)r Fr(=)p Fq(p)1507 4246 y Fi(0)1677 4021 y Fo([)1562 4240 y Fq(J)7 b Fp(\032R)1743 4251 y Fh(k)1787 4240 y Fr(\()p Fq(!)s Fr(\))1919 4116 y Fs(J)65 b Fk([)2246 4021 y Fo([)2128 4240 y Fq(L)p Fp(\032S)2294 4250 y Fh(m)2360 4240 y Fr(\()p Fq(!)s Fr(\))2491 4116 y Fs(L)17 b(:)0 4417 y Fv(The)24 b(families)e(of)i(in)m(terv)-6 b(als)24 b Fk(R)1126 4432 y Fq(k)1203 4417 y Fv(=)k Fk(R)1392 4432 y Fq(k)1442 4417 y Fv(\()p Fs(!)t Fv(\))22 b(and)i Fk(S)1852 4432 y Fq(m)1928 4417 y Fv(\()p Fs(!)t Fv(\))f(are)g (constructed)i(inductiv)m(ely)-8 b(,)25 b(simplifying)0 4537 y(the)33 b(strategy)f(in)h([A,)f Fk(x)p Fv(3])g(\(in)g(particular) 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654 y Fv(\()p Fs(e)2352 613 y Fp(\000)p Fr(1)2481 654 y Fk(\000)23 b Fs(e)2627 613 y Fp(\000)p Fr(2)2734 654 y Fv(\))p Fs(e)2819 613 y Fp(\000)p Fq(c)2917 583 y Fg(0)2944 613 y Fr(\()p Fq(m)p Fp(\000)p Fq(j)t Fr(\))3198 654 y Fs(:)0 862 y Fv(If)34 b Fs(j)44 b(=)-61 b Fk(2)28 b Fs(G)346 877 y Fq(m)422 862 y Fv(\()p Fs(!)t(;)17 b(y)t Fv(\))31 b(then)j Fs(r)967 877 y Fq(j)1009 862 y Fv(\()p Fs(!)t(;)17 b(y)t Fv(\))25 b Fk(\024)k Fv(\()1433 822 y Fr(1)p 1432 839 40 4 v 1432 896 a(2)1506 862 y Fk(\000)23 b Fv(2)p Fs(\021)t Fv(\))17 b(log\(1)p Fs(=\017)p Fv(\))32 b(and)1508 1133 y Fk(j)p Fs(I)1580 1148 y Fq(r)1617 1158 y Fh(j)1659 1133 y Fk(j)27 b(\025)1831 1065 y Fs(e)1877 1029 y Fp(\000)p Fr(2)p 1831 1110 154 4 v 1883 1201 a Fv(2)1997 1133 y Fs(\017)2037 1092 y Fr(1)p Fp(\000)p Fr(2)p Fq(\021)2244 1133 y Fs(:)120 1378 y Fv(Finally)-8 b(,)26 b(the)h(distortion)g(con)m (trol)g(\(3\))f(with)h Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))28 b Fk(\030)g Fs(\017)2160 1342 y Fp(\000)p Fr(7)p Fq(=)p Fr(2)2375 1378 y Fv(is)f(obtained)h(b)m(y)f(a)g(one-dimensional)0 1498 y(v)m(ersion)38 b(of)g(the)h(pro)s(of)f(of)g(Prop)s(osition)f(4.2) g(in)h([A],)f(adapting)h(the)g(estimates)g(for)g(the)g(term)f Fs(A)3735 1513 y Fr(2)0 1617 y Fv(there.)45 b(\(W)-8 b(e)33 b(lea)m(v)m(e)g(the)h(details)f(to)g(the)g(reader.\))100 b Fe(\003)120 1838 y Fv(Let)43 b(us)h(de\014ne)h(the)f(basic)g(subin)m (terv)-6 b(als)44 b(\003)1814 1853 y Fp(\006)1925 1838 y Fv(on)f(whic)m(h)i(our)e(random)g(to)m(w)m(ers)h(will)e(b)s(e)i(con-) 0 1965 y(structed.)g(F)-8 b(or)32 b(this,)f(w)m(e)h(partition)f(\()p Fk(\000)p Fs(\016)n(;)17 b(\016)t Fv(\))31 b(\()p Fs(\016)k Fv(as)d(in)g(\(H2\))e(and)i(small)e(enough\))i(in)m(to)3323 1891 y Fo(S)3406 1995 y Fp(j)p Fq(k)r Fp(j\025)p Fq(K)3627 2005 y Fi(0)3699 1940 y Fv(^)3687 1965 y Fs(I)3731 1980 y Fq(k)0 2135 y Fv(with)244 2110 y(^)232 2135 y Fs(I)276 2150 y Fq(k)360 2135 y Fv(=)k(\()p Fs(e)558 2099 y Fp(\000)p Fq(k)r Fp(\000)p Fr(1)771 2135 y Fs(;)17 b(e)862 2099 y Fp(\000)p Fq(k)973 2135 y Fv(\),)1091 2110 y(^)1079 2135 y Fs(I)1123 2150 y Fp(\000)p Fq(k)1269 2135 y Fv(=)36 b Fk(\000)1471 2110 y Fv(^)1459 2135 y Fs(I)1503 2150 y Fq(k)1591 2135 y Fv(and)i(then)h(w)m(e)f(sub)s(divide)2638 2110 y(^)2626 2135 y Fs(I)2670 2150 y Fq(k)2754 2135 y Fv(=)2867 2060 y Fo(S)2950 2085 y Fq(k)2994 2055 y Fi(2)2950 2165 y Fq(`)p Fr(=1)3118 2110 y Fv(^)3106 2135 y Fs(I)3150 2150 y Fq(k)r(;`)3294 2135 y Fv(so)g(that)f(the)12 2244 y(^)0 2269 y Fs(I)44 2284 y Fq(k)r(;`)185 2269 y Fv(are)e(disjoin)m(t)g(and)h Fk(j)945 2244 y Fv(^)933 2269 y Fs(I)977 2284 y Fq(k)r(;`)1082 2269 y Fk(j)31 b Fv(=)g Fs(k)1304 2233 y Fp(\000)p Fr(2)1441 2269 y Fk(j)1512 2244 y Fv(^)1500 2269 y Fs(I)1544 2284 y Fq(k)1592 2269 y Fk(j)p Fv(.)49 b(\(Note)35 b(that)f Fs(\017)i Fv(do)s(es)f(not)g(in)m(terv)m(ene.\))50 b(W)-8 b(e)35 b(set)h(\003)3589 2284 y Fp(\006)3691 2269 y Fv(to)0 2389 y(b)s(e)e(the)f(righ)m(tmost)f(and)i(leftmost)e(in)m(terv)-6 b(als)33 b(of)h(this)f(partition)g(of)g(\()p Fk(\000)p Fs(\016)n(;)17 b(\016)t Fv(\),)33 b(i.e.,)1234 2596 y(\003)1303 2611 y Fr(+)1397 2596 y Fv(=)1514 2571 y(^)1502 2596 y Fs(I)1546 2618 y Fq(K)1613 2628 y Fi(0)1652 2618 y Fq(;K)1748 2594 y Fi(2)1743 2642 y(0)1808 2596 y Fs(;)116 b Fv(\003)2021 2611 y Fp(\000)2116 2596 y Fv(=)2233 2571 y(^)2221 2596 y Fs(I)2265 2611 y Fp(\000)p Fq(K)2394 2621 y Fi(0)2433 2611 y Fq(;)p Fr(1)2518 2596 y Fs(:)979 b Fv(\(3.17\))0 2822 y(W)-8 b(e)34 b(also)e(de\014ne)669 2797 y(~)659 2822 y(\003)728 2837 y Fr(+)828 2822 y Fv(to)g(b)s(e)i (the)g(in)m(terv)-6 b(al)33 b(of)g(length)h(3)p Fk(j)p Fv(\003)2182 2837 y Fr(+)2247 2822 y Fk(j)f Fv(cen)m(tered)i(at)e(\003) 2895 2837 y Fr(+)2961 2822 y Fv(,)g(similarly)e(for)3592 2797 y(~)3582 2822 y(\003)3651 2837 y Fp(\000)3719 2822 y Fv(.)120 2942 y(W)-8 b(e)40 b(close)g(this)g(section)h(with)e(a)h (lemma)e(that)h(will)g(b)s(e)h(instrumen)m(tal)g(to)f(pro)m(v)m(e)i (Lemma)d(4.1)0 3061 y(\(replacing)e(ideas)g(in)g(the)g(App)s(endix)g (of)g(a)f(preprin)m(t)i(v)m(ersion)f(of)g([A])f(whic)m(h)i(circulated)f (in)g(1997;)0 3181 y(note)d(that)g(w)m(e)h(do)f Fl(not)h Fv(use)g(the)f(top)s(ological)f(mixing)f(assumption)i(\(H4\)\):)0 3351 y Fx(Lemma)52 b(3.10)h(\(Large)e(size)g(of)h(image\).)57 b Fl(Assume)47 b(\(H1\)-\(H3\))e(and)i(let)g Fs(\014)53 b(<)48 b Fv(1)e Fl(b)-5 b(e)47 b(as)f(in)0 3471 y(L)-5 b(emma)43 b(3.9.)63 b(Then)42 b(ther)-5 b(e)42 b(is)f Fs(C)46 b(>)39 b Fv(1)j Fl(and)g(for)f(every)h(smal)5 b(l)43 b(enough)f Fs(\017)g Fl(and)g(lar)-5 b(ge)41 b(enough)h Fk(j)p Fs(k)s Fk(j)0 3590 y Fl(ther)-5 b(e)46 b(is)f(a)h(c)-5 b(onstant)45 b Fs(C)7 b Fv(\()p Fs(k)s Fv(\))47 b Fs(>)g Fv(1)f Fl(\(indep)-5 b(endent)44 b(of)i Fs(\017)p Fl(\))g(so)g(that)g (for)f(e)-5 b(ach)46 b Fs(!)k Fk(2)d Fv(\012)p Fl(,)h(and)e(every)0 3720 y(interval)377 3694 y Fv(^)365 3720 y Fs(I)409 3735 y Fq(k)r(;`)551 3720 y Fl(ther)-5 b(e)35 b(ar)-5 b(e)35 b(a)h(time)1363 3931 y Fs(t)p Fv(\()p Fs(k)s Fv(\))27 b(=)h Fs(t)1700 3850 y Fo(\000)1758 3906 y Fv(^)1746 3931 y Fs(I)1790 3946 y Fq(k)r(;`)1895 3931 y Fs(;)17 b(!)2006 3850 y Fo(\001)2079 3931 y Fk(\024)28 b Fs(C)7 b Fk(j)p Fs(k)s Fk(j)17 b Fs(;)0 4148 y Fl(and)35 b(a)h(subinterval)803 4123 y Fv(~)785 4148 y Fs(U)853 4163 y Fq(!)939 4148 y Fk(\032)1056 4123 y Fv(^)1044 4148 y Fs(I)1088 4163 y Fq(k)r(;`)1229 4148 y Fl(such)g(that)1303 4269 y Fo(\()1417 4361 y Fk(j)1460 4336 y Fo(e)1445 4361 y Fs(U)1513 4376 y Fq(!)1570 4361 y Fk(j)27 b Fs(>)i Fv(1)p Fs(=C)7 b Fv(\()p Fs(k)s Fv(\))17 b Fs(;)1417 4530 y(f)1477 4479 y Fq(t)p Fr(\()p Fq(k)r Fr(\))1466 4542 y Fq(!)1618 4450 y Fo(\000)1681 4505 y Fv(~)1663 4530 y Fs(U)1731 4545 y Fq(!)1789 4450 y Fo(\001)1862 4530 y Fv(=)1977 4505 y(~)1967 4530 y(\003)2036 4545 y Fr(+)2138 4530 y Fl(or)2277 4505 y Fv(~)2267 4530 y(\003)2336 4545 y Fp(\000)2420 4530 y Fs(:)3525 4440 y Fv(\(3.18\))0 4732 y Fl(F)-8 b(urthermor)j(e,)37 b Fs(\036)30 b Fv(=)g Fs(f)863 4695 y Fq(t)852 4756 y(!)910 4732 y Fk(j)951 4745 y Fr(~)938 4762 y Fq(U)992 4772 y Fh(!)1084 4732 y Fl(is)36 b(inje)-5 b(ctive)35 b(and)h(enjoys)g(b)-5 b(oth)37 b(uniform)f(c)-5 b(ontr)g(action)36 b(on)h(b)-5 b(ackwar)g(ds)0 4856 y(br)g(anches)35 b Fv(\(3.14\))f Fl(for)h Fs(C)43 b Fl(and)35 b Fs(\014)5 b Fl(,)36 b(and)g(distortion)e(b)-5 b(ounds)36 b Fv(\(3.15\))e Fl(for)h Fk(K)29 b Fv(=)f Fs(C)7 b Fv(\()p Fs(k)s Fv(\))p Fl(.)0 5077 y(Pr)-5 b(o)g(of)36 b(of)h(L)-5 b(emma)38 b(3.10.)50 b Fv(W)-8 b(e)35 b(shall)g(use)h(again)f(the)g(random)f(b)s (ound)i(p)s(erio)s(d)f(ideas)h(from)e([BeY].)0 5197 y(W)-8 b(e)38 b(\014rst)g(state)f(an)h(easy)g(consequence)h(of)f(Sublemma)e (3.2)h(\(3\).)57 b(F)-8 b(or)37 b(ev)m(ery)g(1)p Fs(=)p Fv(4)e Fs(>)g(\021)3354 5161 y Fp(0)3417 5197 y Fs(>)g(\021)k(>)d Fv(0)1847 5316 y Ft(21)p eop %%Page: 22 22 22 21 bop 0 100 a Fv(\(recall)31 b(that)h Fs(\021)j Fv(w)m(as)d (\014xed)h(in)e(the)i(pro)s(of)e(of)h(Lemma)f(3.4\))f(eac)m(h)j(small)d (enough)j Fs(\017)p Fv(,)f(all)f Fs(!)g Fk(2)d Fv(\012,)k(and)0 229 y(ev)m(ery)275 204 y(^)263 229 y Fs(I)307 244 y Fq(k)r(;`)413 229 y Fv(,)h(taking)f Fs(p)p Fv(\()p Fs(k)s(;)17 b(\017)p Fv(\))33 b(as)g(in)g(Sublemma)f(3.2:)1312 399 y Fo(\014)1312 459 y(\014)1345 483 y Fs(f)1405 442 y Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\)+1)1394 508 y Fq(!)1715 483 y Fv(\()1766 458 y(^)1754 483 y Fs(I)1798 498 y Fq(k)r(;`)1903 483 y Fv(\))1942 399 y Fo(\014)1942 459 y(\014)2003 483 y Fk(\025)c Fs(e)2154 442 y Fp(\000)p Fr(2)p Fq(\021)2300 412 y Fg(0)2327 442 y Fp(j)p Fq(k)r Fp(j)2440 483 y Fs(:)1057 b Fv(\(3.19\))0 720 y(Indeed,)34 b(just)g(observ)m(e)h(that)70 938 y Fo(\014)70 998 y(\014)103 1023 y Fs(f)163 982 y Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\)+1)152 1047 y Fq(!)472 1023 y Fv(\()523 998 y(^)511 1023 y Fs(I)555 1038 y Fq(k)r(;`)661 1023 y Fv(\))700 938 y Fo(\014)700 998 y(\014)761 1023 y Fk(\025)28 b Fv(inf)1004 938 y Fo(\014)1004 998 y(\014)1037 1023 y Fv(\()p Fs(f)1136 982 y Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\))1125 1047 y Fq(\033)s(!)1344 1023 y Fv(\))1383 982 y Fp(0)1411 938 y Fo(\014)1411 998 y(\014)1489 955 y Fs(e)1535 919 y Fp(\000j)p Fq(k)r Fp(j\000)p Fr(1)p 1489 1000 307 4 v 1604 1091 a Fs(C)1853 955 y(e)1899 919 y Fp(\000j)p Fq(k)r Fp(j\000)p Fr(1)p 1853 1000 V 1957 1091 a Fs(k)2012 1062 y Fr(2)2217 1023 y Fk(\025)2350 955 y Fs(e)2396 919 y Fr(\(2)p Fp(\000)p Fr(2)p Fq(\021)s Fr(\))p Fp(j)p Fq(k)r Fp(j)p 2350 1000 391 4 v 2506 1091 a Fs(C)2798 955 y(e)2844 919 y Fp(\000)p Fr(2\()p Fp(j)p Fq(k)r Fp(j)p Fr(+1\))p 2798 1000 408 4 v 2913 1091 a Fs(C)7 b(k)3046 1062 y Fr(2)3245 1023 y Fs(>)28 b(e)3396 982 y Fp(\000)p Fr(2)p Fq(\021)3542 952 y Fg(0)3569 982 y Fp(j)p Fq(k)r Fp(j)3682 1023 y Fs(:)3525 1176 y Fv(\(3.20\))120 1325 y(Next,)k(w)m(e)h(claim)f(that)h(there)h(is)f(an)g(in)m(teger)h Fs(i)28 b Fv(=)g Fs(i)2043 1340 y Fr(0)2116 1325 y Fk(\024)g Fs(C)7 b Fk(j)p Fs(k)s Fk(j)p Fv(,)33 b(so)g(that)g(for)g(some)g Fs(k)3263 1340 y Fr(1)3341 1325 y Fv(and)g Fs(`)3576 1340 y Fr(1)326 1572 y Fs(f)386 1530 y Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\)+1+)p Fq(i)780 1540 y Fi(0)375 1596 y Fq(!)824 1572 y Fv(\()875 1546 y(^)863 1572 y Fs(I)907 1587 y Fq(k)r(;`)1012 1572 y Fv(\))28 b Fk(\033)1196 1546 y Fv(^)1184 1572 y Fs(I)1228 1587 y Fq(k)1270 1597 y Fi(1)1309 1587 y Fq(;`)1366 1597 y Fi(1)1405 1587 y Fp(\000)p Fr(1)1534 1572 y Fk([)1634 1546 y Fv(^)1622 1572 y Fs(I)1666 1587 y Fq(k)1708 1597 y Fi(1)1747 1587 y Fq(;`)1804 1597 y Fi(1)1870 1572 y Fk([)1971 1546 y Fv(^)1959 1572 y Fs(I)2003 1587 y Fq(k)2045 1597 y Fi(1)2084 1587 y Fq(;`)2141 1597 y Fi(1)2180 1587 y Fr(+1)2302 1572 y Fs(;)149 b Fv(and)34 b Fk(j)p Fs(k)2753 1587 y Fr(1)2797 1572 y Fk(j)28 b(\024)g Fv(2)p Fs(\021)3061 1530 y Fp(0)3089 1572 y Fk(j)p Fs(k)s Fk(j)325 b Fv(\(3.21\))0 1808 y(\(with)33 b(the)g(ob)m(vious)h(in)m(terpretation)f(if)g Fs(`)1565 1823 y Fr(1)1637 1808 y Fv(=)28 b(1)33 b(or)g Fs(`)1989 1823 y Fr(1)2061 1808 y Fv(=)28 b Fs(k)2221 1772 y Fr(2)2218 1833 y(1)2266 1808 y Fv(\).)120 1928 y(T)-8 b(o)32 b(c)m(hec)m(k)j(\(3.21\))c(w)m(e)j(\014rst)g(note)f(that) g(there)h(is)f(a)g(\014rst)h(iterate)e Fs(j)2612 1943 y Fr(0)2685 1928 y Fk(\024)c Fs(C)7 b Fk(j)p Fs(k)s Fk(j)33 b Fv(so)g(that)1211 2175 y Fs(f)1271 2133 y Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\)+1+)p Fq(j)1670 2143 y Fi(0)1260 2199 y Fq(!)1714 2175 y Fv(\()1765 2149 y(^)1753 2175 y Fs(I)1797 2190 y Fq(k)r(;`)1902 2175 y Fv(\))22 b Fk(\\)h Fv(\()p Fk(\000)p Fs(\016)n(;)17 b(\016)t Fv(\))27 b Fk(6)p Fv(=)h Fk(;)17 b Fs(:)0 2434 y Fv(Indeed,)40 b(while)e Fs(f)682 2383 y Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\)+1+)p Fq(i)671 2446 y(!)1081 2434 y Fv(\()1132 2409 y(^)1120 2434 y Fs(I)1164 2449 y Fq(k)r(;`)1269 2434 y Fv(\))g(sta)m(ys)g (outside)g(of)g(\()p Fk(\000)p Fs(\016)n(;)17 b(\016)t Fv(\))38 b(w)m(e)g(ha)m(v)m(e,)h(setting)e Fs(i)f Fv(=)g Fs(q)t(M)11 b Fv(\()p Fs(\016)t Fv(\))24 b(+)i Fs(r)0 2554 y Fv(with)33 b(0)27 b Fk(\024)i Fs(r)h(<)e(M)11 b Fv(\()p Fs(\016)t Fv(\))32 b(and)i(applying)f(\(H2\)\(i\),)13 2782 y Fo(\014)13 2841 y(\014)46 2866 y Fs(f)106 2825 y Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\)+1+)p Fq(i)95 2891 y(!)505 2866 y Fv(\()556 2841 y(^)544 2866 y Fs(I)588 2881 y Fq(k)r(;`)693 2866 y Fv(\))732 2782 y Fo(\014)732 2841 y(\014)793 2866 y Fk(\025)1234 2773 y Fv(~)1230 2799 y Fs(\025)1288 2763 y Fq(q)s(M)p 910 2843 825 4 v 910 2935 a Fv(\(min)1115 2953 y Fp(j)p Fq(x)p Fp(j\025)p Fq(\016)1330 2935 y Fk(j)p Fs(f)1418 2906 y Fp(0)1445 2935 y Fv(\()p Fs(x)p Fv(\))p Fk(j)p Fv(\))1647 2906 y Fq(M)1746 2782 y Fo(\014)1746 2841 y(\014)1780 2866 y Fs(f)1840 2825 y Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\)+1)1829 2891 y Fq(!)2149 2866 y Fv(\()2200 2841 y(^)2188 2866 y Fs(I)2232 2881 y Fq(k)r(;`)2338 2866 y Fv(\))2377 2782 y Fo(\014)2377 2841 y(\014)2437 2866 y Fk(\025)2878 2773 y Fv(~)2874 2799 y Fs(\025)2932 2763 y Fq(q)s(M)p 2554 2843 V 2554 2935 a Fv(\(min)2759 2953 y Fp(j)p Fq(x)p Fp(j\025)p Fq(\016)2974 2935 y Fk(j)p Fs(f)3062 2906 y Fp(0)3090 2935 y Fv(\()p Fs(x)p Fv(\))p Fk(j)p Fv(\))3292 2906 y Fq(M)3407 2866 y Fs(e)3453 2825 y Fp(\000)p Fr(2)p Fq(\021)3599 2795 y Fg(0)3626 2825 y Fp(j)p Fq(k)r Fp(j)3739 2866 y Fs(:)120 3201 y Fv(No)m(w,)c(if)h Fs(f)519 3150 y Fq(p)p Fr(\()p Fq(k)r(;\017)p Fr(\)+)p Fq(j)817 3160 y Fi(0)508 3213 y Fq(!)860 3201 y Fv(\()911 3176 y(^)899 3201 y Fs(I)943 3216 y Fq(k)r(;`)1049 3201 y Fv(\))d Fk(\032)i Fv(\()p Fk(\000)p Fs(\016)n(;)17 b(\016)t Fv(\))e Fk([)g Fv(\003)1676 3216 y Fr(++)1817 3201 y Fk([)g Fv(\003)1967 3216 y Fp(\000\000)2097 3201 y Fv(,)30 b(where)g(\003)2508 3216 y Fr(++)2665 3201 y Fv(is)g(the)g(in)m(terv)-6 b(al)29 b(to)g(the)h(righ)m(t)0 3321 y(of)37 b(\003)186 3336 y Fr(+)288 3321 y Fv(in)g(an)f(augmen)m(ted)h(partition,)f(and)g(\003) 1771 3336 y Fp(\000\000)1938 3321 y Fv(is)g(the)h(corresp)s(onding)g (in)m(terv)-6 b(al)36 b(to)g(the)h(left)f(of)0 3440 y(\003)69 3455 y Fp(\000)136 3440 y Fv(,)d(w)m(e)h(set)g Fs(i)534 3404 y Fp(0)589 3440 y Fv(=)29 b Fs(j)736 3455 y Fr(0)780 3440 y Fv(,)k(and)h(b)m(y)f(\(H2\)\(ii\))737 3607 y Fo(\014)737 3667 y(\014)771 3691 y Fs(f)831 3650 y Fq(p)p Fr(\()p Fq(k)r(;\017)p 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Fl(\):)120 3741 y Fv(Fix)37 b Fs(!)j Fk(2)d Fv(\012.)59 b(F)-8 b(or)38 b(eac)m(h)h Fs(m)e Fk(\025)g Fs(p)1368 3756 y Fr(0)1413 3741 y Fv(,)i(and)g Fs(J)46 b Fk(2)37 b(R)1967 3756 y Fq(m)2043 3741 y Fv(\()p Fs(!)t Fv(\),)h(w)m(e)h(no)m(w)g(de\014ne)h(the)e(auxiliary)f(return)0 3873 y(time)247 3848 y Fo(b)228 3873 y Fs(R)304 3888 y Fq(!)361 3873 y Fv(\()p Fs(x)p Fv(\))29 b Fk(2)g Ff(Z)689 3888 y Fr(+)776 3873 y Fk([)23 b(f1g)33 b Fv(of)i(all)e Fs(x)c Fk(2)h Fs(J)44 b Fv(inductiv)m(ely)-8 b(.)46 b(Let)34 b Fs(t)p Fv(\()p Fs(J)10 b Fv(\),)34 b(and)2814 3848 y Fo(e)2792 3873 y Fs(J)44 b Fv(b)s(e)35 b(as)f(in)g(Lemma)f(4.1.)0 4005 y(Then,)h(if)f Fs(f)439 3969 y Fq(m)428 4030 y(!)514 4005 y Fv(\()p Fs(x)p Fv(\))27 b Fk(2)792 3980 y Fo(e)771 4005 y Fs(J)9 b Fv(,)33 b(and)h Fs(f)1150 3969 y Fq(m)p Fr(+)p Fq(t)1139 4030 y(!)1316 4005 y Fv(\()p Fs(x)p Fv(\))27 b Fk(2)i Fv(\003)1642 4020 y Fr(+)1741 4005 y Fv(or)k(\003)1932 4020 y Fp(\000)2033 4005 y Fv(w)m(e)g(set)1492 4216 y Fo(b)1474 4241 y Fs(R)1550 4256 y Fq(!)1607 4241 y Fv(\()p Fs(x)p Fv(\))27 b(=)h Fs(m)23 b Fv(+)f Fs(t)p Fv(\()p Fs(J)10 b Fv(\))17 b Fs(:)0 4478 y Fv(If)34 b Fs(f)160 4442 y Fq(m)149 4502 y(!)235 4478 y Fv(\()p Fs(x)p Fv(\))27 b Fk(2)512 4453 y Fo(e)491 4478 y Fs(J)10 b Fv(,)33 b(but)g Fs(f)859 4442 y Fq(m)p Fr(+)p Fq(t)848 4502 y(!)1026 4478 y Fv(\()p Fs(x)p Fv(\))38 b Fs(=)-61 b Fk(2)28 b Fv(\003)1351 4493 y Fp(\006)1452 4478 y Fv(\(for)33 b(all)f Fs(r)s Fv(\))h(then)1042 4689 y Fo(b)1023 4714 y Fs(R)1099 4729 y Fq(!)1156 4714 y Fv(\()p Fs(x)p Fv(\))27 b(=)h Fs(m)23 b Fv(+)g Fs(t)p Fv(\()p Fs(J)10 b Fv(\))21 b(+)1952 4689 y Fo(b)1933 4714 y Fs(R)2009 4733 y Fq(\033)2058 4713 y Fh(m)p Fi(+)p Fh(t)2203 4733 y Fq(!)2260 4633 y Fo(\000)2306 4714 y Fs(f)2366 4673 y Fq(m)p Fr(+)p Fq(t)2355 4739 y(!)2532 4714 y Fv(\()p Fs(x)p Fv(\))2667 4633 y Fo(\001)2729 4714 y Fs(:)0 4960 y Fv(Finally)-8 b(,)31 b(if)j Fs(f)510 4924 y Fq(m)499 4985 y(!)585 4960 y Fv(\()p Fs(x)p Fv(\))k Fs(=)-61 b Fk(2)862 4935 y Fo(e)841 4960 y Fs(J)10 b Fv(,)33 b(w)m(e)g(set)1277 5172 y Fo(b)1258 5197 y Fs(R)1334 5212 y Fq(!)1391 5197 y Fv(\()p Fs(x)p Fv(\))28 b(=)g Fs(m)22 b Fv(+)1887 5172 y Fo(b)1868 5197 y Fs(R)1944 5212 y Fq(\033)1993 5192 y Fh(m)2058 5212 y Fq(!)2116 5116 y Fo(\000)2162 5197 y Fs(f)2222 5156 y Fq(m)2211 5221 y(!)2297 5197 y Fv(\()p Fs(x)p Fv(\))2432 5116 y Fo(\001)2494 5197 y Fs(:)1847 5316 y Ft(25)p eop %%Page: 26 26 26 25 bop 120 100 a Fv(W)-8 b(e)33 b(in)m(tro)s(duce)i(a)e(sequence)j (of)d Fl(stopping)i(times)2022 74 y Fo(b)2006 100 y Fs(T)2064 115 y Fq(i)2125 100 y Fv(=)2248 74 y Fo(b)2231 100 y Fs(T)2289 115 y Fq(!)s(;i)2427 100 y Fv(:)28 b(\003)2552 115 y Fr(+)2641 100 y Fk([)22 b Fv(\003)2798 115 y Fp(\000)2894 100 y Fk(!)28 b(f)p Fv(0)p Fs(;)17 b(:)g(:)g(:)30 b(;)17 b(n)p Fk(g)k([)i(f1g)0 219 y Fv(with)572 339 y(0)28 b Fk(\021)772 314 y Fo(b)755 339 y Fs(T)813 354 y Fq(!)s(;)p Fr(0)962 339 y Fk(\024)1084 314 y Fo(b)1067 339 y Fs(T)1125 354 y Fq(!)s(;)p Fr(1)1246 339 y Fv(\()p Fs(x)p 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b(precisely)-8 b(,)45 b(w)m(e)e(shall)f(exhibit)g(a)g(random)g(v)-6 b(ariable)42 b Fs(n)3352 2191 y Fr(1)3397 2176 y Fv(\()p Fs(!)t Fv(\))f(on)i(a)0 2296 y(full)37 b(measure)f(set)h(\012)800 2311 y Fr(1)881 2296 y Fv(\(with)f(con)m(trolled)h(distribution,)g(see) g(\(4.8\))o(\),)g(and)g Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))33 b Fk(\025)h Fv(1,)j(so)f(that)g(for)0 2415 y Fs(`)27 b(>)h(n)234 2430 y Fr(1)279 2415 y Fv(\()p Fs(!)t Fv(\))154 2678 y(Leb)316 2568 y Fo(\020)375 2598 y(\010)433 2678 y Fs(x)g Fk(2)g Fv(\003)681 2693 y Fr(+)770 2678 y Fk([)22 b Fv(\003)927 2693 y Fp(\000)1022 2678 y Fk(j)1078 2589 y(p)p 1161 2589 42 4 v 89 x Fs(`)27 b Fk(\025)h Fs(k)1387 2693 y Fr(max)1539 2678 y Fv(\()p Fs(x)p Fv(\))1028 2898 y(and)34 b Fk(9)p Fs(i)28 b Fk(\024)g Fs(k)1496 2913 y Fr(max)1648 2898 y Fv(\()p Fs(x)p Fv(\))33 b(with)2059 2873 y Fo(b)2043 2898 y Fs(T)2101 2913 y Fq(!)s(;i)2210 2898 y Fv(\()p Fs(x)p Fv(\))27 b Fs(>)i(`)2520 2817 y Fo(\011)2577 2787 y(\021)2664 2898 y Fk(\024)f Fs(C)7 b 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Fv(\(4.5\))120 3817 y(Let)30 b(us)h(no)m(w)g(de\014ne)h(the)f(stopping)g(times,)e(using)i(again)f (the)h(notation)e(from)h(Lemma)f(4.1.)42 b(W)-8 b(e)0 3944 y(sa)m(y)41 b(that)421 3918 y Fo(b)404 3944 y Fs(T)462 3959 y Fq(!)s(;)p Fr(1)625 3944 y Fv(is)g(de\014ned)i(at)e Fs(x)g Fk(2)g Fv(\003)1488 3959 y Fr(+)1582 3944 y Fk([)28 b Fv(\003)1745 3959 y Fp(\000)1854 3944 y Fv(if)41 b(there)h(is)f Fs(m)2411 3959 y Fr(1)2498 3944 y Fk(\025)g Fs(p)2666 3959 y Fr(0)2752 3944 y Fv(and)h Fs(J)3009 3959 y Fr(1)3095 3944 y Fk(2)f(R)3286 3959 y Fq(m)3357 3969 y Fi(1)3401 3944 y Fv(\()p Fs(!)t Fv(\))g(with)0 4079 y Fs(x)32 b Fk(2)g Fs(J)242 4094 y Fr(1)322 4079 y Fv(\(hence,)37 b(the)f(comp)s(onen)m(t)g(of)g Fs(f)1530 4043 y Fq(m)p Fr(+)p Fq(t)1696 4079 y Fv(\()p Fs(J)1790 4094 y Fr(1)1835 4079 y Fv(\))f(con)m(taining)h Fs(f)2456 4043 y Fq(m)p Fr(+)p Fq(t)2622 4079 y Fv(\()p Fs(x)p Fv(\))f(co)m(v)m(ers)3098 4054 y Fo(e)3092 4079 y Fv(\003)3161 4094 y Fr(+)3262 4079 y Fv(or)3394 4054 y Fo(e)3387 4079 y Fv(\003)3456 4094 y Fp(\000)3523 4079 y Fv(\).)51 b(W)-8 b(e)0 4199 y(then)34 b(set)1105 4339 y Fo(b)1088 4364 y Fs(T)1146 4379 y Fq(!)s(;)p Fr(1)1268 4364 y Fv(\()p Fs(x)p Fv(\))27 b(=)1535 4224 y Fo(\032)1643 4295 y Fs(m)435 b Fv(if)33 b Fs(f)2316 4260 y Fq(m)2305 4320 y(!)2391 4295 y Fs(x)39 b(=)-61 b Fk(2)2592 4271 y Fo(e)2570 4296 y Fs(J)2635 4295 y(;)1643 4445 y(m)22 b Fv(+)h Fs(t)p Fv(\()p Fs(J)1982 4460 y Fr(1)2027 4445 y Fv(\))99 b(otherwise.)0 4644 y(Clearly)-8 b(,)403 4619 y Fo(b)384 4644 y Fs(R)460 4659 y Fq(!)517 4644 y Fv(\()p Fs(x)p Fv(\))52 b Fk(\025)850 4619 y Fo(b)833 4644 y Fs(T)891 4659 y Fq(!)s(;)p Fr(1)1012 4644 y Fv(\()p Fs(x)p Fv(\),)f(and)d(equalit)m(y)f(is)h(only)f(p)s (ossible)h(in)g(the)g(second)h(case:)74 b(There,)0 4811 y(at)49 b(time)398 4786 y Fo(b)381 4811 y Fs(T)439 4826 y Fr(1)484 4811 y Fv(\()p Fs(x)p Fv(\),)j(part)e(of)f(the)h(comp)s (onen)m(t)f(of)h Fs(f)1971 4755 y Fo(b)1964 4760 y Fq(T)2011 4770 y Fi(1)2049 4760 y Fr(\()p Fq(x)p Fr(\))1953 4823 y Fq(!)2162 4811 y Fv(\()p Fs(J)2256 4826 y Fr(1)2300 4811 y Fv(\))g(con)m(taining)f Fs(f)2956 4755 y Fo(b)2949 4760 y Fq(T)2996 4770 y Fi(1)3034 4760 y Fr(\()p Fq(x)p Fr(\))2938 4823 y Fq(!)3147 4811 y Fv(\()p Fs(x)p Fv(\))g(returns)h(to) 0 4931 y(\003)69 4946 y Fr(+)160 4931 y Fk([)25 b Fv(\003)320 4946 y Fp(\000)387 4931 y Fv(.)54 b(W)-8 b(e)38 b(shall)e(estimate)g (the)h(\\o)m(v)m(er\015o)m(wing)f(parts")h(using)g(the)g(distortion)g (con)m(trol)f(from)0 5061 y(Lemmas)41 b(3.9)h(and)h(4.1.)71 b(F)-8 b(or)43 b(this,)h(let)e(\002)1656 5076 y Fq(!)s(;)p Fr(1)1821 5061 y Fv(=)i Fk(f)p Fs(x)f Fk(2)h Fv(\003)2271 5076 y Fr(+)2365 5061 y Fk([)29 b Fv(\003)2529 5076 y Fp(\000)2640 5061 y Fk(j)2727 5036 y Fo(b)2711 5061 y Fs(T)2769 5076 y Fq(!)s(;)p Fr(1)2890 5061 y Fv(\()p Fs(x)p Fv(\))j(is)i(de\014ned)18 b Fk(g)p Fv(.)72 b(F)-8 b(or)0 5197 y Fs(x)39 b Fk(2)h Fv(\002)279 5212 y Fq(!)s(;)p Fr(1)428 5197 y Fk(n)26 b(f)p Fs(R)q Fv(\()p Fs(x)p Fv(\))39 b(=)938 5172 y Fo(b)922 5197 y Fs(T)980 5212 y Fr(1)1025 5197 y Fv(\()p Fs(x)p Fv(\))p Fk(g)p Fv(,)i(w)m(e)f(sa)m(y)h(that)1851 5172 y Fo(b)1835 5197 y Fs(T)1893 5212 y Fr(2)1978 5197 y Fv(is)f(de\014ned)i(at)e Fs(x)g Fv(if)h(there)g(are)f Fs(m)3282 5212 y Fr(2)3367 5197 y Fs(>)g(p)3534 5212 y Fr(0)3619 5197 y Fv(and)1847 5316 y Ft(26)p eop %%Page: 27 27 27 26 bop 0 128 a Fs(J)55 143 y Fr(2)151 128 y Fk(2)51 b(R)352 143 y Fq(m)423 153 y Fi(2)467 128 y Fv(\()p Fs(\033)574 88 y Fo(b)567 93 y Fq(T)614 103 y Fi(1)652 92 y Fr(\()p Fq(x)p Fr(\))765 128 y Fs(!)t Fv(\))46 b(with)h Fs(f)1224 72 y Fo(b)1217 77 y Fq(T)1264 87 y Fi(1)1302 77 y Fr(\()p Fq(x)p Fr(\))1206 140 y Fq(!)1415 128 y Fv(\()p Fs(x)p Fv(\))j Fk(2)h Fs(J)1772 143 y Fr(2)1817 128 y Fv(,)g(setting)2253 103 y Fo(b)2237 128 y Fs(T)2295 143 y Fr(2)2339 128 y Fv(\()p Fs(x)p Fv(\))c(to)g(b)s(e)g(either)3119 103 y Fo(b)3103 128 y Fs(T)3161 143 y Fr(1)3206 128 y Fv(\()p Fs(x)p Fv(\))30 b(+)i Fs(m)3567 143 y Fr(2)3612 128 y Fv(,)51 b(or)17 235 y Fo(b)0 260 y Fs(T)58 275 y Fr(1)103 260 y Fv(\()p Fs(x)p Fv(\))28 b(+)h Fs(m)459 275 y Fr(2)533 260 y Fv(+)g Fs(t)p Fv(\()p Fs(J)769 275 y Fr(2)813 260 y Fv(\).)73 b(F)-8 b(or)42 b(general)h Fs(k)j Fk(\025)f Fv(2,)f(w)m(e)f(let)g(\002)2223 275 y Fq(!)s(;k)2392 260 y Fv(=)h Fk(f)p Fs(x)g Fk(j)2752 235 y Fo(b)2735 260 y Fs(T)2793 275 y Fq(!)s(;k)2919 260 y Fv(\()p Fs(x)p Fv(\))32 b(is)h(de\014ned)r Fk(g)p Fv(,)45 b(and)0 396 y(w)m(e)j(de\014ne)481 371 y Fo(b)464 396 y Fs(T)522 411 y Fq(!)s(;k)r Fr(+1)796 396 y Fv(on)h(\002)1027 411 y Fq(k)r Fr(+1)p Fq(;!)1305 396 y Fk(\032)k Fv(\002)1512 411 y Fq(k)r(;!)1670 396 y Fk(n)32 b(f)1820 371 y Fo(b)1802 396 y Fs(R)1878 411 y Fq(!)1935 396 y Fv(\()p Fs(x)p Fv(\))52 b(=)2268 371 y Fo(b)2251 396 y Fs(T)2309 411 y Fq(!)s(;k)2435 396 y Fv(\()p Fs(x)p Fv(\))p Fk(g)47 b Fv(if)h(there)h(is)f Fs(m)3245 411 y Fq(k)3347 396 y Fk(\025)k Fs(p)3526 411 y Fr(0)3619 396 y Fv(and)0 563 y Fs(J)55 578 y Fq(k)154 563 y Fk(2)e(R)354 578 y Fq(m)425 589 y Fh(k)474 563 y Fv(\()p Fs(\033)581 522 y Fo(b)574 527 y Fq(T)621 538 y Fh(k)663 527 y Fr(\()p Fq(x)p Fr(\))776 563 y Fs(!)t Fv(\))c(with)g Fs(f)1234 506 y Fo(b)1227 511 y Fq(T)1274 522 y Fh(k)1317 511 y Fr(\()p Fq(x)p Fr(\))1216 575 y Fq(!)1429 563 y Fv(\()p Fs(x)p Fv(\))j Fk(2)h Fs(J)1784 578 y Fq(k)1834 563 y Fv(.)83 b(The)47 b(relation)f(\(4.1\))f(\(and)i(th)m(us)g(\(4.2\))o(\)) f(is)g(an)0 683 y(immediate)31 b(consequence)k(of)f(the)f (de\014nition.)0 851 y Fl(Estimate)i Fv(\(4.3\))f Fl(for)837 826 y Fo(b)818 851 y Fs(R)894 866 y Fq(!)952 851 y Fl(:)120 976 y Fv(The)40 b(estimate)e(\(4.3\))g(can)i(b)s(e)g(restated)g(as)g (Leb)q(\(\002)2107 991 y Fq(!)s(;k)2232 976 y Fv(\))e Fk(\024)h Fs(\022)2475 940 y Fq(k)2563 976 y Fv(for)h(some)f(0)f Fs(<)h(\022)h(<)f Fv(1)g(and)h(all)0 1096 y Fs(k)50 b Fk(2)c Ff(Z)283 1111 y Fr(+)347 1096 y Fv(,)h Fs(n)f Fk(2)h Ff(Z)710 1111 y Fr(+)774 1096 y Fv(,)g Fs(!)j Fk(2)d Fv(\012)1146 1111 y Fr(0)1191 1096 y Fv(.)77 b(This)45 b(exp)s(onen)m(tial)g(b)s(ound)g(will)f(b)s(e)g(an)h(easy)g (consequence)h(of)0 1223 y(Lemma)28 b(4.1.)43 b(Indeed,)31 b(for)g(all)e Fs(!)i Fk(2)d Fv(\012)1443 1238 y Fr(0)1488 1223 y Fv(,)i Fs(n)1606 1187 y Fp(0)1634 1223 y Fv(,)g(and)h Fs(p)1933 1238 y Fr(0)2005 1223 y Fk(\024)d Fs(m)p Fv(,)j(if)f Fs(J)40 b Fv(is)30 b(an)g(in)m(terv)-6 b(al)30 b(of)g Fk(R)3222 1238 y Fq(m)3298 1223 y Fv(\()p Fs(\033)3398 1187 y Fq(n)3447 1157 y Fg(0)3479 1223 y Fs(!)t Fv(\),)f(the)0 1343 y(uniform)k(distortion)g(b)s(ounds)h(from)f(Lemma)e(4.1)i(imply)e (\(using)j(the)f(notation)g(there\))g(that)320 1657 y(Leb)482 1576 y Fo(\000)528 1657 y Fs(L)596 1616 y Fp(0)651 1657 y Fv(:=)805 1632 y Fo(e)784 1657 y Fs(J)f Fk(\\)22 b Fv(\()p Fs(f)1058 1616 y Fq(t)1093 1657 y Fv(\))1132 1614 y Fp(\000)p Fr(1)1132 1701 y Fq(\033)1181 1681 y Fh(n)1225 1666 y Fg(0)1251 1681 y Fi(+)p Fh(m)1368 1701 y Fq(!)1426 1657 y Fv(\(\003)1534 1672 y Fr(+)1622 1657 y Fk([)h Fv(\003)1780 1672 y Fp(\000)1847 1657 y Fv(\))1886 1576 y Fo(\001)1959 1657 y Fs(>)2150 1590 y Fv(1)p 2076 1634 197 4 v 2076 1725 a Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))2330 1590 y Fk(j)p Fv(\003)2427 1605 y Fr(+)2515 1590 y Fk([)23 b Fv(\003)2673 1605 y Fp(\000)2740 1590 y Fk(j)p 2330 1634 438 4 v 2524 1725 a Fv(2)2813 1657 y(Leb)q(\()3035 1632 y Fo(e)3014 1657 y Fs(J)9 b Fv(\))1959 1932 y Fk(\025)2172 1865 y Fv(1)p 2076 1909 241 4 v 2076 2001 a Fs(C)e Fv(\()p Fs(\017)p Fv(\))2272 1972 y Fr(2)2375 1865 y Fk(j)p Fv(\003)2472 1880 y Fr(+)2560 1865 y Fk([)22 b Fv(\003)2717 1880 y Fp(\000)2785 1865 y Fk(j)p 2375 1909 438 4 v 2569 2001 a Fv(2)2841 1932 y Fs(;)1012 2208 y Fv(Leb)1174 2127 y Fo(\000)1220 2208 y Fs(J)32 b Fk(\\)22 b Fv(\()p Fs(f)1494 2166 y Fq(m)1569 2208 y Fv(\))1608 2165 y Fp(\000)p Fr(1)1608 2251 y Fq(\033)1657 2232 y Fh(n)1701 2217 y Fg(0)1733 2251 y Fq(!)1790 2208 y Fs(L)1858 2166 y Fp(0)1886 2127 y Fo(\001)1959 2208 y Fs(>)2150 2140 y Fv(1)p 2076 2185 197 4 v 2076 2276 a Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))2425 2140 y(1)p 2330 2185 241 4 v 2330 2276 a Fs(C)g Fv(\()p Fs(\017)p Fv(\))2526 2247 y Fr(2)2628 2140 y Fk(j)p Fv(\003)2725 2155 y Fr(+)2813 2140 y Fk([)23 b Fv(\003)2971 2155 y Fp(\000)3038 2140 y Fk(j)p 2628 2185 438 4 v 2822 2276 a Fv(4)3111 2208 y(Leb)q(\()p Fs(J)10 b Fv(\))17 b Fs(:)0 2523 y Fv(\(In)34 b(the)g(ab)s(o)m(v)m(e)g(b)s(ounds,)h Fs(J)43 b Fv(ma)m(y)33 b(b)s(e)h(replaced)h(b)m(y)f(a)g(subin)m(terv)-6 b(al)34 b Fs(L)29 b Fk(\032)g Fs(J)43 b Fv(with)34 b Fk(j)p Fs(L)p Fk(j)28 b(\025)h(j)p Fs(J)10 b Fk(j)p Fs(=C)d Fv(,)33 b(up)0 2643 y(to)g(adapting)g(the)g(constan)m(ts)i(corresp)s (ondingly)-8 b(.\))120 2781 y(Therefore,)34 b(setting)f Fs(n)986 2744 y Fp(0)1014 2781 y Fv(\()p Fs(x)p Fv(\))27 b(=)1298 2755 y Fo(b)1281 2781 y Fs(T)1339 2796 y Fq(!)s(;k)r Fp(\000)p Fr(1)1567 2781 y Fv(\()p Fs(x)p Fv(\))32 b(for)i Fs(x)27 b Fk(2)h Fv(\002)2142 2796 y Fq(!)s(;k)r Fp(\000)p Fr(1)2370 2781 y Fv(,)33 b(w)m(e)h(ha)m(v)m(e)159 3086 y(Leb)321 3006 y Fo(\000)367 3086 y Fs(f)434 3028 y Fo(b)427 3033 y Fq(T)474 3044 y Fh(k)q Fg(\000)p Fi(1)416 3098 y Fq(!)610 3086 y Fv(\(\002)726 3101 y Fq(!)s(;k)r Fp(\000)p Fr(1)953 3086 y Fv(\))22 b Fk(\\)1103 3006 y Fo(\010)1161 3086 y Fs(y)31 b Fk(2)d Fv(\002)1412 3114 y Fq(\033)1461 3094 y Fh(n)1505 3079 y Fg(0)1537 3114 y Fq(!)s(;)p Fr(1)1686 3086 y Fk(j)1760 3061 y Fo(b)1741 3086 y Fs(R)1817 3114 y Fq(\033)1866 3094 y Fh(n)1910 3079 y Fg(0)1941 3114 y Fq(!)1999 3086 y Fv(\()p Fs(y)t Fv(\))e(=)2278 3061 y Fo(b)2262 3086 y Fs(T)2320 3114 y Fq(\033)2369 3094 y Fh(n)2413 3079 y Fg(0)2444 3114 y Fq(!)s(;)p Fr(1)2565 3086 y Fv(\()p Fs(y)t Fv(\))2696 3006 y Fo(\011)o(\001)p 159 3144 2640 4 v 1040 3286 a Fv(Leb)1202 3205 y Fo(\000)1247 3286 y Fs(f)1314 3227 y Fo(b)1307 3232 y Fq(T)1354 3243 y Fh(k)q Fg(\000)p Fi(1)1296 3297 y Fq(!)1490 3286 y Fv(\(\002)1606 3301 y Fq(!)s(;k)r Fp(\000)p Fr(1)1834 3286 y Fv(\))1873 3205 y Fo(\001)2839 3167 y Fs(>)2956 3099 y Fk(j)p Fv(\003)3053 3114 y Fr(+)3141 3099 y Fk([)c Fv(\003)3298 3114 y Fp(\000)3366 3099 y Fk(j)p 2956 3144 438 4 v 3029 3235 a Fv(4)p Fs(C)3157 3206 y Fr(3)3202 3235 y Fv(\()p Fs(\017)p Fv(\))3433 3167 y Fs(>)28 b Fv(0)17 b Fs(:)0 3565 y Fv(Since)34 b(\002)337 3580 y Fq(!)s(;k)491 3565 y Fk(\032)28 b Fv(\002)673 3580 y Fq(!)s(;k)r Fp(\000)p Fr(1)923 3565 y Fk(\\)22 b(f)1080 3540 y Fo(b)1061 3565 y Fs(R)1137 3593 y Fq(\033)1186 3573 y Fh(n)1230 3558 y Fg(0)1261 3593 y Fq(!)1341 3565 y Fk(\016)g Fs(f)1473 3529 y Fq(n)1522 3499 y Fg(0)1462 3590 y Fq(!)1581 3565 y Fs(>)1703 3540 y Fo(b)1687 3565 y Fs(T)1745 3593 y Fq(\033)1794 3573 y Fh(n)1838 3558 y Fg(0)1869 3593 y Fq(!)s(;)p Fr(1)2012 3565 y Fk(\016)g Fs(f)2144 3529 y Fq(n)2193 3499 y Fg(0)2133 3590 y Fq(!)2225 3565 y Fk(g)p Fv(,)33 b(setting)1368 3896 y Fs(\022)e Fv(=)d(1)22 b Fk(\000)1734 3829 y(j)p Fv(\003)1831 3844 y Fr(+)1920 3829 y Fk([)g Fv(\003)2077 3844 y Fp(\000)2145 3829 y Fk(j)p 1734 3873 V 1808 3964 a Fv(4)p Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))2054 3936 y Fr(6)2212 3896 y Fs(<)28 b Fv(1)17 b Fs(;)0 4221 y Fv(one)43 b(more)g(\(inductiv)m(e\))g (application)f(of)i(the)f(distortion)g(b)s(ounds)h(yields)f(Leb)q (\(\002)3189 4236 y Fq(!)s(;k)3314 4221 y Fv(\))h Fk(\024)h Fs(\022)3569 4185 y Fq(k)3617 4221 y Fv(,)g(as)0 4341 y(claimed.)e(\(Note)33 b(that)f Fs(\022)k Fv(is)d(uniform)g(in)g Fs(!)k Fv(but)d(tends)g(to)f(1)g(as)g Fk(j)p Fv(\003)2544 4356 y Fr(+)2632 4341 y Fk([)23 b Fv(\003)2790 4356 y Fp(\000)2857 4341 y Fk(j)28 b(!)f Fv(0)33 b(or)g Fs(\017)28 b Fk(!)f Fv(0.\))0 4515 y Fl(Estimate)35 b Fv(\(4.4\))f Fl(for)837 4490 y Fo(b)818 4515 y Fs(R)894 4530 y Fq(!)952 4515 y Fl(:)120 4640 y Fv(W)-8 b(e)32 b(no)m(w)g(mo)m(v)m(e)e(to)i(the) g(estimate)e(\(4.4\))o(.)44 b(F)-8 b(or)31 b(\014xed)h Fs(`;)17 b(i)27 b Fk(\025)h Fv(1,)j(\014xed)i(0)27 b(=)h Fs(\034)2960 4655 y Fr(0)3032 4640 y Fk(\024)g Fs(p)3187 4655 y Fr(0)3260 4640 y Fk(\024)g Fs(\034)3409 4655 y Fr(1)3481 4640 y Fs(<)g(\034)3630 4655 y Fr(2)3702 4640 y Fs(<)0 4760 y Fk(\001)17 b(\001)g(\001)26 b Fs(<)i(\034)293 4775 y Fq(i)353 4760 y Fk(\024)h Fs(`)p Fv(,)j(and)i Fs(\034)k Fk(\024)28 b Fs(`)p Fv(,)33 b(de\014ne)i Fs(k)s Fv(\()p Fs(\034)11 b Fv(\))27 b(=)h(max)n Fk(f)p Fv(0)f Fk(\024)i Fs(k)h Fk(\024)e Fs(i)g Fk(j)g Fs(\034)2420 4775 y Fq(k)2496 4760 y Fk(\024)g Fs(\034)11 b Fk(g)33 b Fv(and)117 5029 y Fs(A)192 5044 y Fq(\034)241 5029 y Fv(\()p Fs(\034)324 5044 y Fr(1)368 5029 y Fs(;)17 b(:)g(:)g(:)31 b(;)17 b(\034)651 5044 y Fq(i)683 5029 y Fv(\))27 b(=)i Fk(f)p Fs(x)e Fk(2)h Fv(\003)1152 5044 y Fr(+)1241 5029 y Fk([)22 b Fv(\003)1398 5044 y Fp(\000)1493 5029 y Fk(j)28 b Fs(k)s Fv(\()p Fs(\034)11 b Fv(\))21 b(+)i(1)k Fk(\024)h Fs(k)2092 5044 y Fr(max)2244 5029 y Fv(\()p Fs(x)p Fv(\))17 b Fs(;)2456 5004 y Fo(b)2440 5029 y Fs(T)2498 5047 y Fq(!)s(;k)r Fr(\()p Fq(\034)8 b Fr(\)+1)2831 5029 y Fv(\()p Fs(x)p Fv(\))27 b Fs(>)h(\034)g(;)2456 5202 y Fv(and)2666 5177 y Fo(b)2650 5202 y Fs(T)2708 5217 y Fq(!)s(;j)2826 5202 y Fv(\()p Fs(x)p Fv(\))f(=)i Fs(\034)3138 5217 y Fq(j)3196 5202 y Fs(;)17 b Fk(8)p Fs(\034)3340 5217 y Fq(j)3409 5202 y Fk(\024)28 b Fs(\034)11 b Fk(g)17 b Fs(:)1847 5316 y Ft(27)p eop %%Page: 28 28 28 27 bop 0 100 a Fv(Applying)32 b(the)h(absolute)g(b)s(ound)h(in)f (Corollary)e(4.2)h(w)m(e)h(\014nd)h(that,)e(whenev)m(er)i Fs(\034)3087 115 y Fr(1)3152 100 y Fk(\000)22 b Fv(1)27 b Fs(>)h(n)3493 115 y Fr(0)3538 100 y Fv(\()p Fs(!)t Fv(\))20 b(+)0 219 y(2)p Fs(C)k Fv(log\(1)p Fs(=\017)p Fv(\),)490 480 y(Leb)q(\()p Fs(A)766 495 y Fq(\034)803 505 y Fi(1)841 495 y Fp(\000)p Fr(1)948 480 y Fv(\()p Fs(\034)1031 495 y Fr(1)1075 480 y Fs(;)17 b(:)g(:)g(:)31 b(;)17 b(\034)1358 495 y Fq(i)1390 480 y Fv(\)\))27 b(=)h(Leb)q(\()p Fk(f)p Fs(x)f Fk(2)h Fv(\003)2098 495 y Fr(+)2187 480 y Fk([)22 b Fv(\003)2344 495 y Fp(\000)2439 480 y Fk(j)2511 455 y Fo(b)2495 480 y Fs(T)2553 495 y Fq(!)s(;)p Fr(1)2674 480 y Fv(\()p Fs(x)p Fv(\))27 b Fs(>)h(\034)2985 495 y Fr(1)3052 480 y Fk(\000)22 b Fv(1)p Fk(g)p Fv(\))1495 648 y Fk(\024)28 b Fs(C)7 b(e)1724 607 y Fp(\000)p Fq(\020)e Fr(\()p Fq(\034)1895 617 y Fi(1)1934 607 y Fp(\000)p Fr(1\))p Fp(\000)p Fr(2)p Fq(C)19 b Fr(log)q(\(1)p Fq(=\017)p Fr(\))2546 648 y Fs(:)0 919 y Fv(F)-8 b(or)42 b Fs(j)49 b Fk(\025)44 b Fv(2,)g(w)m(e)f(let)g Fs(L)f Fv(b)s(e)h(a)g(comp)s(onen) m(t)f(of)h Fs(A)1895 934 y Fq(\034)1932 944 y Fh(j)s Fg(\000)p Fi(1)2061 919 y Fv(\()p Fs(\034)2144 934 y Fr(1)2188 919 y Fs(;)17 b(:)g(:)g(:)31 b(;)17 b(\034)2471 934 y Fq(i)2503 919 y Fv(\))42 b(with)2837 894 y Fo(b)2821 919 y Fs(T)2879 934 y Fq(!)s(;j)t Fp(\000)p Fr(1)3099 919 y Fk(j)p Fs(L)h Fv(=)h Fs(\034)3403 934 y Fq(j)t Fp(\000)p Fr(1)3546 919 y Fv(,)h(and)0 1061 y(decomp)s(ose)e Fs(L)29 b Fk(\000)g(f)783 1036 y Fo(b)764 1061 y Fs(R)840 1076 y Fq(!)941 1061 y Fv(=)45 b Fs(\034)1107 1076 y Fq(j)t Fp(\000)p Fr(1)1250 1061 y Fk(g)e Fv(in)m(to)g(connected)i(comp) s(onen)m(ts)2588 986 y Fo(S)2671 1091 y Fq(r)2732 1061 y Fs(L)2800 1076 y Fq(r)2887 1061 y Fv(\(with)d(p)s(ossible)i(times)17 1173 y Fo(b)0 1198 y Fs(T)58 1213 y Fq(j)t Fp(\000)p Fr(1)247 1198 y Fv(=)i Fs(m)p Fv(,)g(and)f Fs(m)29 b Fv(+)h Fs(t)p Fv(\).)75 b(W)-8 b(e)44 b(apply)g(again)f(the)h(absolute) g(b)s(ounds)h(from)e(Corollary)f(4.2)h(to)0 1318 y Fs(`)27 b Fv(=)h Fs(\034)218 1333 y Fq(j)290 1318 y Fv(and)j Fs(f)541 1282 y Fq(\034)578 1292 y Fh(j)615 1282 y Fp(\000)p Fr(1)722 1318 y Fs(L)790 1333 y Fq(r)864 1318 y Fv(and)g(get,)g(using)g (once)g(more)e(the)i(distortion)f(con)m(trol)g(in)h(Lemma)e(4.1)h(when) 0 1437 y(pulling)j(bac)m(k)h(that)e(whenev)m(er)j Fs(\034)1262 1452 y Fq(j)1326 1437 y Fk(\000)22 b Fs(\034)1469 1452 y Fq(j)t Fp(\000)p Fr(1)1640 1437 y Fs(>)29 b(n)1806 1452 y Fr(0)1850 1437 y Fv(\()p Fs(\033)1950 1401 y Fq(\034)1987 1411 y Fh(j)s Fg(\000)p Fi(1)2116 1437 y Fs(!)t Fv(\))22 b(+)g(2)p Fs(C)i Fv(log)q(\(1)p Fs(=\017)p Fv(\))228 1747 y(Leb)q(\()p Fs(L)497 1762 y Fq(r)563 1747 y Fk(\\)e Fs(A)726 1762 y Fq(\034)763 1772 y Fh(j)800 1762 y Fp(\000)p Fr(1)907 1747 y Fv(\()p Fs(\034)990 1762 y Fr(1)1034 1747 y Fs(;)17 b(:)g(:)g(:)30 b(;)17 b(\034)1316 1762 y Fq(i)1348 1747 y Fv(\)\))28 b Fk(\024)g Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))1919 1680 y(Leb)q(\()p Fs(L)2188 1695 y Fq(r)2231 1680 y Fv(\))p 1767 1724 656 4 v 1767 1820 a(Leb)q(\()p Fs(f)2028 1766 y Fq(\034)2065 1776 y Fh(j)s Fg(\000)p Fi(1)2017 1832 y Fq(!)2194 1820 y Fv(\()p Fs(L)2301 1835 y Fq(r)2345 1820 y Fv(\)\))2434 1747 y Fs(e)2480 1706 y Fp(\000)p Fq(\020)e Fr(\()p Fq(\034)2651 1716 y Fh(j)2689 1706 y Fp(\000)p Fr(1)p Fp(\000)p Fq(\034)2890 1716 y Fh(j)s Fg(\000)p Fi(1)3015 1706 y Fr(\))p Fp(\000)p Fr(2)p Fq(C)18 b Fr(log)q(\(1)p Fq(=\017)p Fr(\))3524 1747 y Fs(:)0 2059 y Fv(If)39 b Fs(\034)149 2074 y Fq(j)216 2059 y Fk(\000)26 b Fs(\034)363 2074 y Fq(j)t Fp(\000)p Fr(1)542 2059 y Fk(\024)37 b Fs(n)716 2074 y Fr(0)761 2059 y Fv(\()p Fs(\033)861 2022 y Fq(\034)898 2032 y Fh(j)s Fg(\000)p Fi(1)1027 2059 y Fs(!)t Fv(\))25 b(+)h(2)p Fs(C)d Fv(log)q(\(1)p Fs(=\017)p Fv(\),)39 b(w)m(e)g(only)e(ha)m(v)m (e,)j(b)m(y)f(the)f(distortion)g(con)m(trol)g(from)0 2178 y(Lemma)32 b(4.1,)g(that)765 2479 y(Leb\()p Fs(L)1033 2494 y Fq(r)1099 2479 y Fk(\\)23 b Fs(A)1263 2494 y Fq(\034)1300 2504 y Fh(j)1336 2494 y Fp(\000)p Fr(1)1443 2479 y Fv(\()p Fs(\034)1526 2494 y Fr(1)1570 2479 y Fs(;)17 b(:)g(:)g(:)31 b(;)17 b(\034)1853 2494 y Fq(i)1885 2479 y Fv(\)\))27 b Fk(\024)h Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))2456 2411 y(Leb\()p Fs(L)2724 2426 y Fq(r)2768 2411 y Fv(\))p 2303 2456 V 2303 2552 a(Leb)r(\()p Fs(f)2565 2498 y Fq(\034)2602 2508 y Fh(j)s Fg(\000)p Fi(1)2554 2563 y Fq(!)2731 2552 y Fv(\()p Fs(L)2838 2567 y Fq(r)2881 2552 y Fv(\)\))2987 2479 y Fs(:)0 2785 y Fv(Th)m(us,)60 b(b)m(y)54 b(de\014nition)g(of)h (the)f Fs(L)1323 2800 y Fq(r)1421 2785 y Fv(and)g Fs(A)1710 2800 y Fq(\034)1760 2785 y Fv(,)k(and)d(using)f(the)g(\\large)g(image") e(prop)s(erties)j(in)0 2904 y(Lemma)32 b(4.1,)g(there)i(is)f Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))33 b(suc)m(h)i(that)e(for)g(all)g Fs(j)g Fk(\025)28 b Fv(2,)178 3088 y Fo(8)178 3178 y(<)178 3357 y(:)366 3111 y Fr(Leb\()p Fq(A)585 3121 y Fh(\034)619 3137 y(j)657 3121 y Fg(\000)p Fi(1)749 3111 y Fr(\()p Fq(\034)817 3121 y Fi(1)856 3111 y Fq(;:::)12 b(;\034)1025 3121 y Fh(i)1056 3111 y Fr(\)\))p 312 3154 861 4 v 312 3212 a(Leb)q(\()p Fq(A)532 3222 y Fh(\034)566 3238 y(j)s Fg(\000)p Fi(1)691 3222 y Fg(\000)p Fi(1)784 3212 y Fr(\()p Fq(\034)852 3222 y Fi(1)890 3212 y Fq(;)33 b(:::)o(;\034)1079 3222 y Fh(i)1110 3212 y Fr(\)\))1212 3177 y Fk(\024)28 b Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))p Fs(e)1559 3141 y Fp(\000)p Fq(\020)e Fr(\()p Fq(\034)1730 3151 y Fh(j)1768 3141 y Fp(\000)p Fq(\034)1867 3151 y Fh(j)s Fg(\000)p Fi(1)1992 3141 y Fp(\000)p Fr(1\))2263 3177 y Fv(if)33 b Fs(\034)2398 3192 y Fq(j)2462 3177 y Fk(\000)23 b Fs(\034)2606 3192 y Fq(j)t Fp(\000)p Fr(1)2771 3177 y Fk(\000)g Fv(1)k Fk(\025)i Fs(n)3114 3192 y Fr(0)3158 3177 y Fv(\()p Fs(\033)3258 3141 y Fq(\034)3295 3151 y Fh(j)s Fg(\000)p Fi(1)3424 3177 y Fs(!)t Fv(\))17 b Fs(;)366 3340 y Fr(Leb\()p Fq(A)585 3350 y Fh(\034)619 3366 y(j)657 3350 y Fg(\000)p Fi(1)749 3340 y Fr(\()p Fq(\034)817 3350 y Fi(1)856 3340 y Fq(;:::)12 b(;\034)1025 3350 y Fh(i)1056 3340 y Fr(\)\))p 312 3383 V 312 3440 a(Leb)q(\()p Fq(A)532 3450 y Fh(\034)566 3466 y(j)s Fg(\000)p Fi(1)691 3450 y Fg(\000)p Fi(1)784 3440 y Fr(\()p Fq(\034)852 3450 y Fi(1)890 3440 y Fq(;)33 b(:::)o(;\034)1079 3450 y Fh(i)1110 3440 y Fr(\)\))1212 3406 y Fk(\024)28 b Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))750 b(if)33 b Fs(\034)2398 3421 y Fq(j)2462 3406 y Fk(\000)23 b Fs(\034)2606 3421 y Fq(j)t Fp(\000)p Fr(1)2771 3406 y Fk(\000)g Fv(1)k Fs(<)i(n)3114 3421 y Fr(0)3158 3406 y Fv(\()p Fs(\033)3258 3370 y Fq(\034)3295 3380 y Fh(j)s Fg(\000)p Fi(1)3424 3406 y Fs(!)t Fv(\))17 b Fs(:)120 3711 y Fv(Therefore)34 b(for)g(an)m(y)f(0)27 b Fs(<)i(\034)1139 3726 y Fr(1)1210 3711 y Fs(<)g Fk(\001)17 b(\001)g(\001)25 b Fs(<)k(\034)1609 3726 y Fq(i)1669 3711 y Fk(\024)f Fs(`)255 4013 y Fv(Leb)q(\()p Fs(A)531 4028 y Fq(`)568 4013 y Fv(\()p Fs(\034)651 4028 y Fr(1)695 4013 y Fs(;)17 b(:)g(:)g(:)e(;)i(\034)962 4028 y Fq(i)994 4013 y Fv(\)\))27 b Fk(\024)h Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))1400 3972 y Fq(i)1433 4013 y Fs(e)1479 3972 y Fp(\000)p Fq(\020)e(`)1643 4013 y Fk(\001)22 b Fv(exp)1862 3873 y Fo(\024)1914 4013 y Fs(\020)7 b Fv(\()p Fs(\017)p Fv(\))2464 3919 y Fo(X)2100 4138 y Fq(\034)2137 4148 y Fh(j)2174 4138 y Fp(\000)p Fq(\034)2273 4148 y Fh(j)s Fg(\000)p Fi(1)2398 4138 y Fp(\000)p Fr(1)p Fp(\024)p Fq(n)2611 4148 y Fi(0)2650 4138 y Fr(\()p Fq(\033)2730 4106 y Fh(\034)2764 4122 y(i)p Fg(\000)p Fi(1)2888 4138 y Fq(!)s Fr(\))2972 4013 y Fv(\()p Fs(\034)3055 4028 y Fq(j)3119 4013 y Fk(\000)22 b Fs(\034)3262 4028 y Fq(j)t Fp(\000)p Fr(1)3406 4013 y Fv(\))3445 3873 y Fo(\025)3497 4013 y Fs(;)0 4382 y Fv(and)34 b(\(w)m(e)f(shall)g(so)s(on)g(c)m(ho)s(ose)h Fs(k)d Fv(=)d Fs(k)s Fv(\()p Fs(`)p Fv(\)\))138 4728 y(Leb)300 4648 y Fo(\000)346 4728 y Fk(f)p Fs(k)i Fk(\025)f Fs(k)636 4743 y Fr(max)804 4728 y Fs(;)16 b Fk(9)p Fs(i)29 b Fk(\024)f Fs(k)1123 4743 y Fr(max)1291 4728 y Fs(;)1352 4703 y Fo(b)1336 4728 y Fs(T)1394 4743 y Fq(!)s(;i)1531 4728 y Fs(>)g(`)p Fk(g)1728 4648 y Fo(\001)1801 4728 y Fk(\024)1956 4604 y Fq(k)1906 4634 y Fo(X)1913 4846 y Fq(i)p Fr(=0)2263 4634 y Fo(X)2066 4848 y Fr(0)p Fp(\024)p Fq(\034)2205 4858 y Fi(1)2244 4848 y Fq(<)p Fp(\001\001\001)o Fq(<\034)2476 4858 y Fh(i)2508 4848 y Fp(\024)p Fq(`)2620 4728 y Fv(Leb)q(\()p Fs(A)2896 4743 y Fq(`)2934 4728 y Fv(\()p Fs(\034)3017 4743 y Fr(1)3061 4728 y Fs(;)17 b(:)g(:)g(:)d(;)j(\034)3327 4743 y Fq(i)3359 4728 y Fv(\)\))859 5094 y Fk(\024)1014 4969 y Fq(k)965 4999 y Fo(X)972 5212 y Fq(i)p Fr(=0)1322 4999 y Fo(X)1125 5214 y Fr(0)p Fp(\024)p Fq(\034)1264 5224 y Fi(1)1303 5214 y Fq(<)p Fp(\001\001\001)o Fq(<\034)1535 5224 y Fh(i)1566 5214 y Fp(\024)p Fq(`)1679 5094 y Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))1875 5053 y Fq(i)1908 5094 y Fs(e)1954 5053 y Fp(\000)p Fq(\020)e(`)2118 5094 y Fk(\001)22 b Fv(exp)2337 4953 y Fo(\024)2389 5094 y Fs(\020)2515 4969 y Fq(i)2457 4999 y Fo(X)2460 5212 y Fq(j)t Fr(=1)2617 5094 y Fs(n)2677 5109 y Fr(0)2722 5094 y Fv(\()p Fs(\033)2822 5053 y Fq(\034)2859 5063 y Fh(i)p Fg(\000)p Fi(1)2982 5094 y Fs(!)t Fv(\))3087 4953 y Fo(\025)3156 5094 y Fs(:)3575 4916 y Fv(\(4.6\))1847 5316 y Ft(28)p eop %%Page: 29 29 29 28 bop 0 100 a Fv(W)-8 b(e)39 b(no)m(w)h(estimate)d(the)j(last)e (factor)h(in)g(\(4.6\))o(,)h(i.e.,)f(the)g(e\013ect)h(of)f(the)g (random)g(w)m(aiting)f(times:)0 219 y(This)h(is)f(where)i(w)m(e)f (shall)f(lose)h(the)g(exp)s(onen)m(tial)f(deca)m(y)-8 b(.)61 b(Fix)37 b(0)g Fs(<)g(\032)f(<)h Fv(1.)60 b(Since)40 b Fs(P)14 b Fv(\()p Fk(f)p Fs(n)3479 234 y Fr(0)3522 219 y Fv(\()p Fs(!)t Fv(\))36 b Fs(>)0 339 y(n)p Fk(g)p Fv(\))27 b Fk(\024)h Fs(C)7 b(e)405 303 y Fp(\000)p Fq(\020)e(n)563 339 y Fv(,)33 b(for)h(eac)m(h)g(\014xed)f(1)28 b Fk(\024)g Fs(i)g Fk(\024)g Fs(k)36 b Fv(and)e Fs(\034)1917 354 y Fr(1)1961 339 y Fs(;)17 b Fk(\001)g(\001)g(\001)e Fs(;)i(\034)2228 354 y Fq(i)2260 339 y Fv(,)554 634 y Fs(P)632 464 y Fo( )711 494 y(\032)860 510 y Fq(i)802 539 y Fo(X)805 752 y Fq(j)t Fr(=1)963 634 y Fs(n)1023 649 y Fr(0)1067 634 y Fv(\()p Fs(\033)1167 593 y Fq(\034)1204 603 y Fh(j)s Fg(\000)p Fi(1)1333 634 y Fs(!)t Fv(\))27 b Fs(>)h(\032)17 b(`)1681 494 y Fo(\033)1755 464 y(!)1861 634 y Fk(\024)2024 510 y Fq(i)1966 539 y Fo(X)1969 752 y Fq(j)t Fr(=1)2127 634 y Fs(P)2205 494 y Fo(\022)2278 524 y(n)2344 634 y Fs(n)2404 649 y Fr(0)2449 634 y Fv(\()p Fs(\033)2549 593 y Fq(\034)2586 603 y Fh(j)s Fg(\000)p Fi(1)2715 634 y Fs(!)t Fv(\))27 b Fs(>)2964 567 y(\032)17 b(`)p 2964 611 110 4 v 3002 702 a(i)3086 524 y Fo(o)3152 494 y(\023)1861 924 y Fk(\024)28 b Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))17 b Fs(k)j(e)2297 883 y Fp(\000)p Fq(\020)5 b Fr(\()p Fq(\017)p Fr(\))2521 850 y Fh(\032)13 b(`)p 2521 867 79 4 v 2541 907 a(k)2633 924 y Fs(:)3575 731 y Fv(\(4.7\))0 1146 y(Consider)37 b(the)f(partition)g(of)g(\003)g(in)m(to)g(maximal)d(atoms)i(\000)2207 1161 y Fq(!)2297 1146 y Fv(=)e(\000)2469 1161 y Fq(!)2527 1146 y Fv(\()p Fs(k)s Fv(\))j(on)g(whic)m(h)h(the)3316 1121 y Fo(b)3299 1146 y Fs(T)3357 1161 y Fq(!)s(;j)3476 1146 y Fv(\()p Fs(x)p Fv(\))e(are)0 1265 y(constan)m(t)g(for)f(0)29 b Fk(\024)g Fs(j)35 b Fk(\024)29 b Fs(k)s Fv(.)46 b(W)-8 b(e)35 b(will)e(sa)m(y)g(that)h(suc)m(h)h(an)g(atom)d(\000)i(is)g(\()p Fs(`;)17 b(k)s Fv(\)-)p Fl(go)-5 b(o)g(d)32 b Fv(if)i(for)g(all)f Fs(x)c Fk(2)h Fv(\000)3722 1280 y Fq(!)0 1385 y Fv(and)k Fs(i)28 b Fk(\024)g Fs(k)s Fv(,)1455 1470 y Fq(i)1397 1500 y Fo(X)1400 1712 y Fq(j)t Fr(=1)1558 1595 y Fs(n)1618 1610 y Fr(0)1662 1595 y Fv(\()p Fs(\033)1769 1549 y Fo(b)1762 1554 y Fq(T)1809 1564 y Fh(!)r(;j)s Fg(\000)p Fi(1)2008 1595 y Fs(!)t Fv(\))f Fk(\024)h Fs(\032`)17 b(:)0 1873 y Fv(The)29 b(other)h(atoms)d(are)i(called)g(\()p Fs(`;)17 b(k)s Fv(\)-)p Fl(b)-5 b(ad.)40 b Fv(De\014ning)30 b Fs(M)2143 1888 y Fq(`;k)2277 1873 y Fk(\032)e Fv(\012)13 b Fk(\002)g Fs(I)38 b Fv(to)28 b(b)s(e)h(the)h(set)f(of)g(\()p Fs(!)t(;)17 b(x)p Fv(\))27 b(suc)m(h)0 2015 y(that)e Fs(x)h Fv(b)s(elongs)g(to)f(an)h(\()p Fs(`;)17 b(k)s Fv(\)-bad)25 b(\000)1386 2030 y Fq(!)1444 2015 y Fv(,)i(\(4.7\))d (implies)h(that)h(\()p Fs(P)20 b Fk(\002)7 b Fv(Leb)r(\)\()p Fs(M)2814 2030 y Fq(`;k)2919 2015 y Fv(\))27 b Fk(\024)i Fs(C)7 b(k)s(e)3270 1979 y Fp(\000)3345 1946 y Fh(\020)s(\032`)p 3345 1963 102 4 v 3376 2003 a(k)3463 2015 y Fv(.)42 b(Using)0 2135 y(a)f(F)-8 b(ubini)41 b(argumen)m(t)f(as)h(in)g(Corollaries)f (3.5{3.6,)h(w)m(e)g(get)g(that)g(the)g(set)g Fs(M)3001 2098 y Fp(0)2990 2163 y Fq(`;k)3137 2135 y Fv(of)g Fs(!)j Fv(suc)m(h)f(that)0 2186 y Fo(Z)116 2322 y Fs(\037)178 2337 y Fq(M)254 2348 y Fh(`;k)356 2322 y Fv(\()p Fs(!)t(;)17 b(x)p Fv(\))g Fs(d)p Fv(Leb)n(\()p Fs(x)p Fv(\))27 b Fs(>)i(k)s(e)1199 2281 y Fp(\000)1273 2248 y Fh(\020)s(\032)p 1273 2265 73 4 v 1292 2305 a Fi(3)1374 2254 y Fh(`)p 1369 2265 39 4 v 1369 2305 a(k)1456 2322 y Fv(has)j Fs(P)14 b Fv(-measure)32 b(smaller)e(than)i Fs(e)2749 2286 y Fp(\000)2823 2253 y Fi(2)p Fh(\020)s(\032)p 2823 2270 106 4 v 2859 2310 a Fi(3)2957 2259 y Fh(`)p 2953 2270 39 4 v 2953 2310 a(k)3009 2322 y Fv(.)43 b(Therefore,)33 b(there)0 2492 y(is)h(a)g(set)h(of)f(full)h(measure)f(\012)1093 2507 y Fr(1)1167 2492 y Fk(\032)c Fv(\012)1346 2507 y Fr(0)1425 2492 y Fv(suc)m(h)36 b(that)e(for)g(eac)m(h)h Fs(!)e Fk(2)d Fv(\012)2509 2507 y Fr(1)2554 2492 y Fv(,)k(there)h (exists)f Fs(n)3208 2507 y Fr(1)3253 2492 y Fv(\()p Fs(!)t Fv(\))28 b Fk(\025)i Fs(n)3592 2507 y Fr(0)3636 2492 y Fv(\()p Fs(!)t Fv(\))0 2612 y(with)j(the)h(prop)s(ert)m(y)f(that)g Fs(!)42 b(=)-61 b Fk(2)28 b Fs(M)1317 2576 y Fp(0)1306 2640 y Fq(`;k)1445 2612 y Fv(for)33 b(all)g Fs(`)27 b Fk(\025)h Fs(n)1970 2627 y Fr(1)2015 2612 y Fv(\()p Fs(!)t Fv(\).)43 b(No)m(w,)32 b(for)i Fs(!)d Fk(2)d Fv(\012)2896 2627 y Fr(1)2974 2612 y Fv(and)34 b Fs(`)27 b Fk(\025)h Fs(n)3402 2627 y Fr(1)3447 2612 y Fv(\()p Fs(!)t Fv(\))146 2927 y(Leb)q(\()p Fk(f)p Fs(k)i Fk(\025)e Fs(k)636 2942 y Fr(max)804 2927 y Fs(;)17 b Fk(9)p Fs(i)28 b Fk(\024)g Fs(k)1123 2942 y Fr(max)1292 2927 y Fs(;)1353 2901 y Fo(b)1337 2927 y Fs(T)1395 2942 y Fq(!)s(;i)1531 2927 y Fs(>)h(`)p Fk(g)p Fv(\))d Fk(\024)1949 2802 y Fq(k)1900 2832 y Fo(X)1907 3044 y Fq(i)p Fr(=0)2243 2832 y Fo(X)2072 3034 y Fi(0)p Fg(\024)p Fh(\034)2194 3049 y Fi(1)2233 3034 y Fh(<)p Fg(\001\001\001)o Fh(<\034)2443 3050 y(i)2475 3034 y Fg(\024)p Fh(`)2102 3099 y Fi(\()p Fh(`;k)q Fi(\)-go)r(o)r(d)21 b(\000)2477 3109 y Fh(!)2570 2927 y Fv(Leb)q(\()p Fs(A)2846 2942 y Fq(`)2884 2927 y Fv(\()p Fs(\034)2967 2942 y Fr(1)3011 2927 y Fs(;)c(:)g(:)g(:)31 b(;)17 b(\034)3294 2942 y Fq(i)3326 2927 y Fv(\))22 b Fk(\\)g Fv(\000)3537 2942 y Fq(!)3595 2927 y Fv(\))2592 3266 y(+)2843 3171 y Fo(X)2692 3390 y Fr(\()p Fq(`;k)r Fr(\)-bad)27 b(\000)3086 3400 y Fh(!)3154 3266 y Fv(Leb\(\000)3416 3281 y Fq(!)3474 3266 y Fv(\))17 b Fs(:)120 3600 y Fv(Therefore,)38 b(taking)d Fs(k)h Fv(=)1111 3515 y Fk(p)p 1194 3515 42 4 v 85 x Fs(`)p Fv(,)g(applying)g(\(4.6\))o(,)h(and)g(using)f(the)h(Stirling)f (form)m(ula)f(w)m(e)i(get)f(for)0 3719 y(1)p Fs(=)p Fv(2)27 b Fs(<)h(v)k(<)c Fv(1)33 b(and)h Fs(`)27 b Fk(\025)h Fs(n)978 3734 y Fr(1)1023 3719 y Fv(\()p Fs(!)t Fv(\))476 3932 y(Leb)638 3851 y Fo(\000\010)742 3932 y Fs(k)i Fv(=)930 3842 y Fk(p)p 1013 3842 V 90 x Fs(`)d Fk(\025)h Fs(k)1239 3947 y Fr(max)1407 3932 y Fs(;)p Fk(9)p Fs(i)h Fk(\024)f Fs(k)1710 3947 y Fr(max)1878 3932 y Fs(;)1939 3907 y Fo(b)1923 3932 y Fs(T)1981 3947 y Fq(!)s(;i)2118 3932 y Fs(>)g(`)2265 3851 y Fo(\011\001)1463 4121 y Fk(\024)1568 4031 y(p)p 1651 4031 V 90 x Fs(`e)1739 4080 y Fq(`)1772 4050 y Fh(v)1820 4121 y Fv([)p Fs(C)7 b Fv(\()p Fs(\017)p Fv(\)])2072 4020 y Fp(p)p 2137 4020 34 4 v 60 x Fq(`)2175 4121 y Fs(e)2221 4080 y Fp(\000)p Fq(`)p Fr(\(1)p Fp(\000)p Fq(\032)p Fr(\))p Fq(\020)e Fr(\()p Fq(\017)p Fr(\))2686 4121 y Fv(+)2786 4031 y Fk(p)p 2869 4031 42 4 v 90 x Fs(`)o(e)2956 4080 y Fp(\000)3031 4044 y Fh(\020)s Fi(\()p Fh(\017)p Fi(\))p Fh(\032)p 3031 4064 157 4 v 3092 4104 a Fi(3)3200 4020 y Fp(p)p 3265 4020 34 4 v 60 x Fq(`)1463 4304 y Fk(\024)28 b Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))p Fs(e)1810 4262 y Fp(\000)p Fr(\()1903 4203 y Fp(p)p 1969 4203 V 1969 4262 a Fq(`)q(=C)2101 4272 y Fi(3)2140 4262 y Fr(\()p Fq(\017)p Fr(\)\))2288 4304 y Fs(:)0 4525 y Fv(Com)m(bining)37 b(this)h(with)f(\(4.3\))g(ends)i(the)f(pro)s(of)g (of)g(the)g(b)s(ound)h(\(4.5\))d(for)i(the)g(return)h(times)3637 4500 y Fo(b)3619 4525 y Fs(R)3695 4540 y Fq(!)3752 4525 y Fv(.)0 4645 y(Moreo)m(v)m(er,)33 b(w)m(e)h(ma)m(y)e(estimate)g Fs(P)14 b Fv(\()p Fk(f)p Fs(n)1454 4660 y Fr(1)1498 4645 y Fv(\()p Fs(!)t Fv(\))27 b Fs(>)h(`)p Fk(g)p Fv(\):)299 4874 y Fs(P)377 4794 y Fo(\000)423 4874 y Fk(f)p Fs(!)j Fk(j)c Fs(n)681 4889 y Fr(1)725 4874 y Fv(\()p Fs(!)t Fv(\))g Fs(>)h(`)p Fk(g)1093 4794 y Fo(\001)1166 4874 y Fk(\024)g Fs(P)1349 4794 y Fo(\000\010)1453 4874 y Fk(9)p Fs(j)34 b(>)28 b(`)f Fk(j)g Fs(!)k Fk(2)e Fs(M)2108 4833 y Fp(0)2097 4916 y Fq(j;)2154 4856 y Fp(p)p 2218 4856 V 2218 4916 a Fq(`)2256 4794 y Fo(\011\001)2382 4874 y Fv(+)23 b Fs(P)2560 4764 y Fo(\020)2619 4794 y(\010)2677 4874 y Fs(n)2737 4889 y Fr(0)2782 4874 y Fv(\()p Fs(!)t Fv(\))k Fs(>)h(`)3100 4794 y Fo(\011)3157 4764 y(\021)1166 5093 y Fk(\024)1271 4999 y Fo(X)1277 5213 y Fq(j)t(>`)1432 5093 y Fs(e)1478 5052 y Fp(\000)1552 5016 y Fi(2)p Fh(\020)s Fi(\()p Fh(\017)p Fi(\))p Fh(\032)p 1552 5037 191 4 v 1631 5076 a Fi(3)1755 5000 y Fp(p)p 1820 5000 37 4 v 52 x Fq(j)1901 5093 y Fv(+)39 b Fs(C)7 b(e)2141 5052 y Fp(\000)p Fq(\020)e Fr(\()p Fq(\017)p Fr(\))p Fq(`)2406 5093 y Fk(\024)29 b Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))17 b Fs(e)2771 5052 y Fp(\000)p Fr(\()2864 4992 y Fp(p)p 2930 4992 34 4 v 2930 5052 a Fq(`=C)3061 5062 y Fi(4)3100 5052 y Fr(\()p Fq(\017)p Fr(\)\))3248 5093 y Fs(:)3575 5023 y Fv(\(4.8\))1847 5316 y Ft(29)p eop %%Page: 30 30 30 29 bop 0 100 a Fv(Note)33 b(that)g Fs(C)528 115 y Fr(4)573 100 y Fv(\()p Fs(\017)p Fv(\))691 63 y Fp(\000)p Fr(1)825 100 y Fs(>)28 b(C)1001 115 y Fr(3)1046 100 y Fv(\()p Fs(\017)p Fv(\))1164 63 y Fp(\000)p Fr(1)1271 100 y Fv(.)0 247 y Fl(Go)-5 b(o)g(d)35 b(r)-5 b(eturns)36 b(to)g Fv(\003)788 262 y Fr(+)890 247 y Fl(\(estimating)e Fs(R)1491 211 y Fp(\003)1490 272 y Fq(!)1547 247 y Fl(\):)120 367 y Fv(F)-8 b(or)30 b Fs(x)e Fk(2)g Fv(\003)544 382 y Fr(+)627 367 y Fk([)18 b Fv(\003)780 382 y Fp(\000)878 367 y Fv(w)m(e)31 b(no)m(w)g(consider)h(the)f(\\concrete")h(return)f (times)f Fs(R)2916 331 y Fp(\003)2915 392 y Fq(!)2972 367 y Fv(\()p Fs(x)p Fv(\))h(to)f(\003)e(=)g(\003)3528 382 y Fr(+)3594 367 y Fv(.)44 b(As)0 487 y(observ)m(ed)38 b(in)f(the)g(b)s(eginning)g(of)g(the)g(pro)s(of,)g(the)g(abstract)g (times)e(satisfy)i Fs(R)2980 502 y Fq(!)3037 487 y Fv(\()p Fs(x)p Fv(\))c Fk(\024)h Fs(R)3393 450 y Fp(\003)3392 511 y Fq(!)3449 487 y Fv(\()p Fs(x)p Fv(\).)54 b(T)-8 b(o)0 606 y(pro)m(v)m(e)36 b(the)g(desired)h(asymptotics)d(for)i Fs(R)1572 570 y Fp(\003)1571 631 y Fq(!)1628 606 y Fv(\()p Fs(x)p Fv(\),)g(follo)m(wing)f Fk(x)h Fv(7.6)f(in)g([Y)-8 b(o1],)35 b(w)m(e)h(in)m(tro)s(duce)g(second)0 726 y(stopping)e(times)e Fs(S)732 741 y Fq(!)s(;i)875 726 y Fv(on)h(\003)1082 741 y Fr(+)1170 726 y Fk([)23 b Fv(\003)1328 741 y Fp(\000)1429 726 y Fv(b)m(y)33 b(setting)g Fs(S)1955 741 y Fq(!)s(;)p Fr(0)2104 726 y Fk(\021)28 b Fv(0,)33 b(and)894 966 y Fs(S)955 981 y Fq(!)s(;k)1081 966 y Fv(\()p Fs(x)p Fv(\))27 b(=)h Fs(S)1409 981 y Fq(!)s(;k)r Fp(\000)p Fr(1)1636 966 y Fv(\()p Fs(x)p Fv(\))22 b(+)1911 941 y Fo(b)1892 966 y Fs(R)1968 1001 y Fq(\033)2017 966 y Fh(S)2060 984 y(!)r(;k)q Fg(\000)p Fi(1)2265 1001 y Fr(\()p Fq(!)s Fr(\))2384 966 y Fv(\()p Fs(f)2483 925 y Fq(S)2532 936 y Fh(k)q Fg(\000)p Fi(1)2472 991 y Fq(!)2668 966 y Fv(\()p Fs(x)p Fv(\)\))17 b Fs(:)0 1234 y Fv(If)31 b(\004)h(is)f(the)g (partition)g(\003)945 1249 y Fr(+)1029 1234 y Fk([)18 b Fv(\003)1182 1249 y Fp(\000)1250 1234 y Fv(,)31 b(and)g(if)h(w)m(e)f (de\014ne)i(\004)2086 1249 y Fq(k)2135 1234 y Fv(\()p Fs(!)t Fv(\))27 b(=)h(\004)2494 1159 y Fo(W)2594 1234 y Fs(f)2654 1198 y Fp(\000)p Fq(S)2765 1208 y Fi(1)2643 1259 y Fq(!)2808 1234 y Fv(\(\004\))2969 1159 y Fo(W)3068 1234 y Fk(\001)17 b(\001)g(\001)3201 1159 y Fo(W)3301 1234 y Fs(f)3361 1181 y Fp(\000)p Fq(S)3472 1192 y Fh(k)q Fg(\000)p Fi(1)3350 1246 y Fq(!)3608 1234 y Fv(\(\004\),)0 1354 y(then)31 b Fs(f)284 1318 y Fq(S)333 1329 y Fh(k)411 1354 y Fv(maps)e(eac)m(h)i(elemen)m(t)f Fs(\030)k Fv(of)c(\004)1508 1369 y Fq(k)1558 1354 y Fv(\()p Fs(!)t Fv(\))f(on)m(to)h(\003)2021 1369 y Fr(+)2117 1354 y Fv(or)g(\003)2305 1369 y Fp(\000)2372 1354 y Fv(,)g(and)h Fs(f)2681 1318 y Fq(S)2730 1329 y Fh(k)2670 1378 y Fq(!)2808 1354 y Fv(restricted)g(to)e(eac)m(h)i(suc)m (h)0 1473 y Fs(\030)j Fv(has)d(b)s(ounded)h(distortion)e(and)g(uniform) g(con)m(traction)h(in)f(the)h(sense)g(of)g(Lemma)d(3.9.)43 b(With)30 b(the)0 1593 y(help)k(of)f(ideas)h(already)f(discussed,)i (these)f(t)m(w)m(o)f(facts)h(yield)f(the)g(follo)m(wing)g(t)m(w)m(o)g (claims:)120 1757 y(\(i\))27 b(The)j(map)e Fs(f)739 1694 y Fo(b)731 1699 y Fq(R)791 1723 y Fh(\033)834 1708 y(j)870 1723 y(!)720 1792 y Fq(\033)769 1772 y Fh(j)805 1792 y Fq(!)954 1757 y Fv(b)s(eha)m(v)m(es)i(lik)m(e)e(an)h(irreducible)h(t) m(w)m(o-state)f(random)f(Mark)m(o)m(v)g(c)m(hain.)43 b(Con-)0 1877 y(sider)23 b(for)g(a)g(momen)m(t)e(the)i(unp)s(erturb)s (ed)i(map)c Fs(f)11 b Fv(,)24 b(writing)e Fs(R)2276 1841 y Fp(\003)2344 1877 y Fv(and)h Fs(S)2588 1892 y Fq(k)2660 1877 y Fv(for)g(its)g(return)g(and)g(stopping)0 1997 y(time.)43 b(Since)34 b(the)f(in)m(terv)-6 b(als)33 b(\003)1165 2012 y Fp(\006)1266 1997 y Fv(are)g(indep)s(enden)m(t)i(of)e Fs(\017)h Fv(there)g(are)f Fs(\017)p Fv(-indep)s(enden)m(t)j(return)e (times)0 2116 y Fs(T)58 2131 y Fr(+)158 2116 y Fv(and)f Fs(T)409 2131 y Fp(\000)510 2116 y Fv(with)86 2429 y(min)269 2258 y Fo( )348 2429 y Fv(Leb)510 2318 y Fo(\020)569 2348 y(\010)627 2429 y Fs(x)28 b Fk(2)g Fv(\003)875 2444 y Fr(+)969 2429 y Fk(j)f Fs(R)1101 2388 y Fp(\003)1147 2429 y Fv(\()p Fs(x)p Fv(\))g(=)h Fs(S)1475 2444 y Fq(T)1522 2454 y Fi(+)1583 2429 y Fv(\()p Fs(x)p Fv(\))1718 2348 y Fo(\011)1776 2318 y(\021)1852 2429 y Fs(;)17 b Fv(Leb)2058 2318 y Fo(\020)2118 2348 y(\010)2176 2429 y Fs(x)27 b Fk(2)h Fv(\003)2423 2444 y Fp(\000)2518 2429 y Fk(j)g Fs(R)2651 2388 y Fp(\003)2696 2429 y Fv(\()p Fs(x)p Fv(\))f(=)h Fs(S)3024 2444 y Fq(T)3071 2454 y Fg(\000)3135 2429 y Fv(\()p Fs(x)p Fv(\))3270 2348 y Fo(\011)3328 2318 y(\021)3387 2258 y(!)3494 2429 y Fs(>)g Fv(0)17 b Fs(:)0 2738 y Fv(Th)m(us,)34 b(if)f Fs(\017)h Fv(is)f(small)f(enough,)860 3004 y(inf)777 3068 y Fp(\006)p Fq(;!)s Fp(2)p Fr(\012)1027 3078 y Fi(2)1081 3004 y Fv(Leb)1243 2923 y Fo(\010)1301 3004 y Fs(x)c Fk(2)g Fv(\003)1549 3019 y Fp(\006)1644 3004 y Fk(j)f Fs(R)1776 2963 y Fp(\003)1775 3028 y Fq(!)1833 3004 y Fv(\()p Fs(x)p Fv(\))g(=)h Fs(S)2161 3019 y Fq(!)s(;T)2285 3029 y Fg(\006)2348 3004 y Fv(\()p Fs(x)p Fv(\))2483 2923 y Fo(\011)2569 3004 y Fk(\025)2700 2936 y Fv(1)p 2686 2981 79 4 v 2686 3072 a Fs(C)2804 3004 y(>)g Fv(0)17 b Fs(:)0 3276 y Fv(Hence,)34 b(there)g(is)f Fs(K)764 3291 y Fr(0)836 3276 y Fk(\025)28 b Fv(1)33 b(so)g(that)g(for)h(all)e Fs(!)37 b Fv(and)d Fs(k)s Fv(,)937 3549 y(Leb)1099 3438 y Fo(\020)1158 3468 y(\010)1217 3549 y Fs(x)27 b Fk(2)h Fv(\003)g Fk(j)f Fs(R)1624 3508 y Fp(\003)1623 3573 y Fq(!)1708 3549 y Fs(>)i(S)1875 3564 y Fq(!)s(;k)r(K)2063 3574 y Fi(0)2106 3468 y Fo(\011)2164 3438 y(\021)2251 3549 y Fk(\024)2356 3438 y Fo(\020)2416 3549 y Fv(1)22 b Fk(\000)2614 3481 y Fv(1)p 2599 3526 V 2599 3617 a Fs(C)2690 3438 y Fo(\021)2749 3459 y Fq(k)2815 3549 y Fs(:)732 b Fv(\(4.9\))120 3807 y(Note)46 b(also)h(for)g(further)h(use)g (that)f(if)g(\(H4\))f(holds,)k(then)e(there)g(is)f Fs(N)2882 3822 y Fr(1)2926 3807 y Fv(\()p Fs(f)5 b(;)17 b Fv(\003\))46 b(so)h(that)g(\()p Fs(q)35 b Fk(\000)0 3926 y Fv(1)p Fs(=C)r(;)17 b(q)26 b Fv(+)d(1)p Fs(=C)7 b Fv(\))29 b Fk(\032)h Fs(f)801 3890 y Fq(n)855 3926 y Fv(\(\003\))k(for)h(all)e Fs(n)c Fv(=)i Fs(N)1606 3941 y Fr(1)1650 3926 y Fv(\()p Fs(f)5 b(;)17 b Fv(\003\),)33 b(and)i(th)m(us)h(for)e Fs(n)c Fk(\025)g Fs(N)2803 3941 y Fr(1)2847 3926 y Fv(\()p Fs(f)5 b(;)17 b Fv(\003\),)34 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Fq(!)2747 4842 y Fk(\002)h Fv(\001)2930 4857 y Fq(!)2988 4842 y Fv(\))33 b(de\014ned)i(b)m(y:)702 5136 y Ff(P)p Fv(\()p Fs(A)p Fv(\))29 b(=)1048 5000 y Fo(Z)1104 5227 y Fr(\012)1182 5000 y Fo(Z)1237 5227 y Fr(\001)1302 5237 y Fh(!)1354 5227 y Fp(\002)p Fr(\001)1481 5237 y Fh(!)1554 5136 y Fs(\037)1616 5151 y Fq(A)1681 5136 y Fv(\()p Fs(!)t(;)17 b(x;)g(x)1990 5095 y Fp(0)2016 5136 y Fv(\))g Fk(K)2149 5095 y Fr(2)2148 5161 y Fq(!)2222 5136 y Fs(d)p Fv(Leb)2435 5095 y Fr(2)2480 5136 y Fv(\()p Fs(x;)g(x)2678 5095 y Fp(0)2705 5136 y Fv(\))g Fs(dP)d Fv(\()p Fs(!)t Fv(\))j Fs(;)1847 5316 y Ft(41)p eop %%Page: 42 42 42 41 bop 0 100 a Fv(using)34 b(\(5.6\))e(and)h(\(5.7\))o(,)g(w)m(e)h (\014nd)g(for)f(large)g(enough)h Fs(m)29 359 y Ff(P)p Fv(\()p Fs(M)235 318 y Fp(0)224 384 y Fq(m)301 359 y Fv(\))27 b(=)472 224 y Fo(Z)589 359 y Fs(\037)651 374 y Fq(M)734 354 y Fg(0)727 394 y Fh(m)798 359 y Fv(\()p Fs(!)t(;)17 b(x;)g(x)1107 318 y Fp(0)1133 359 y Fv(\))g Fk(K)1266 318 y Fr(2)1265 384 y Fq(!)1338 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Fk(62)3210 1582 y Fo(f)3198 1607 y Fs(M)3295 1622 y Fq(n)3349 1607 y Fk(g)p Fv(,)c(a)e(large)0 1727 y(deviations)28 b(argumen)m(t)g(as)h(in)f(Lemma)f(7.1)h(together)g (with)g(the)h(\014rst)g(claim)e(of)i(the)g(corollary)e(giv)m(es)0 1846 y(the)34 b(second)g(claim.)98 b Fe(\003)0 2072 y Fx(7.B)39 b(Estimates)e(on)h(stopping)g(times)f(and)h(join)m(t)f (return)g(times.)120 2194 y Fv(F)-8 b(rom)40 b(no)m(w)j(on,)h(the)f (notations)e Fs(\025)p Fv(,)k Fs(\025)1600 2158 y Fp(0)1628 2194 y Fv(,)1704 2168 y(~)1700 2194 y Fs(\025)d Fv(will)f(b)s(e)i(used) g(to)f(denote)h(probabilit)m(y)e(measures,)0 2314 y(absolutely)33 b(con)m(tin)m(uous)h(with)f(resp)s(ect)h(to)f(Leb)h(on)f(\001)g(or)g (Leb)23 b Fk(\002)f Fv(Leb)34 b(on)f(\001)22 b Fk(\002)h Fv(\001.)44 b(There)34 b(should)0 2433 y(b)s(e)44 b(no)g(confusion)h (with)e(the)h(constan)m(ts)g(from)f(\(H1\){\(H2\))e(whic)m(h)k(will)d (not)i(app)s(ear)g(an)m(ymore.)0 2553 y(Before)29 b(pro)m(ving)f(the)h (main)e(estimate)h(of)h(this)f(section)h(\(Prop)s(osition)e(7.6\),)i(w) m(e)g(state)f(t)m(w)m(o)g(lemmas)0 2672 y(whic)m(h)34 b(are)f(randomised)h(v)m(ersions)f(of)h(Lemmas)e(1)h(and)g(2)g(in)h([Y) -8 b(o2].)0 2845 y Fx(Lemma)46 b(7.3)h(\(Lo)m(w)m(er)e(b)s(ound)g(for)i Fs(P)14 b Fv(\()p Fk(f)p Fs(T)1772 2860 y Fq(!)1868 2845 y Fv(=)40 b Fs(\034)2029 2860 y Fq(i)2062 2845 y Fk(g)p Fv(\))p Fx(\).)56 b Fl(L)-5 b(et)42 b Fs(\025)p Fl(,)i Fs(\025)2653 2809 y Fp(0)2723 2845 y Fl(b)-5 b(e)41 b(absolutely)j(c)-5 b(ontinuous)0 2977 y(pr)g(ob)g(ability)47 b(me)-5 b(asur)g(es)47 b(on)h Fk(f)p Fv(\001)1218 2992 y Fq(!)1275 2977 y Fk(g)p Fl(,)i(with)e(densities)e Fs(')p Fl(,)k Fs(')2268 2941 y Fp(0)2344 2977 y Fl(in)d Fk(F)2560 2934 y Fr(+)2550 3007 y Fq(\014)2626 2977 y Fl(.)81 b(If)48 b Fv(\000)h Fk(2)3088 2952 y Fo(e)3082 2977 y Fv(\004)3148 2941 y Fq(!)3148 3003 y(i)3254 2977 y Fl(is)e(such)g(that)0 3126 y Fv(\()p Fs(T)97 3141 y Fq(!)155 3126 y Fv(\))194 3144 y Fp(j)p Fr(\000)299 3126 y Fs(>)28 b(\034)448 3141 y Fq(i)p Fp(\000)p Fr(1)583 3126 y Fl(,)35 b(then,)g(letting)h Fs(V)1287 3074 y Fq(\034)1324 3084 y Fh(i)1355 3074 y Fp(\000)p Fq(\034)1454 3084 y Fh(i)p Fg(\000)p Fi(1)1265 3161 y Fq(\033)1314 3129 y Fh(\034)1348 3146 y(i)p Fg(\000)p Fi(1)1472 3161 y Fq(!)1614 3126 y Fl(b)-5 b(e)35 b(asso)-5 b(ciate)g(d)34 b(to)i(the)f Fs(\034)2531 3141 y Fq(j)2573 3126 y Fv(\(\000\))p Fl(,)867 3357 y Fv(\()p Fs(\025)22 b Fk(\002)h Fs(\025)1144 3316 y Fp(0)1172 3357 y Fv(\)\()p Fk(f)p Fs(T)1358 3372 y Fq(!)1443 3357 y Fs(>)28 b(\034)1592 3372 y Fq(i)1624 3357 y Fk(gj)p Fv(\000\))g Fk(\024)g Fv(1)22 b Fk(\000)g Fs(V)2188 3304 y Fq(\034)2225 3314 y Fh(i)2256 3304 y Fp(\000)p Fq(\034)2355 3314 y Fh(i)p Fg(\000)p Fi(1)2165 3392 y Fq(\033)2214 3360 y Fh(\034)2248 3376 y(i)p Fg(\000)p Fi(1)2372 3392 y Fq(!)2479 3357 y Fs(=C)2600 3372 y Fq(\025;\025)2718 3352 y Fg(0)2750 3357 y Fv(\()p Fs(\017)p Fv(\))17 b Fs(;)0 3566 y Fl(wher)-5 b(e)41 b Fs(C)358 3581 y Fq(\025;\025)476 3561 y Fg(0)509 3566 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y(i)3749 3990 y Fl(,)0 4113 y(we)36 b(have)410 4297 y Fv(\()p Fs(\025)22 b Fk(\002)g Fs(\025)686 4256 y Fp(0)714 4297 y Fv(\))753 4312 y Fq(!)811 4297 y Fv(\()p Fk(f)p Fs(\034)944 4312 y Fq(i)p Fr(+1)1099 4297 y Fk(\000)h Fs(\034)1243 4312 y Fq(i)1303 4297 y Fs(>)28 b(`)1450 4312 y Fr(0)1516 4297 y Fv(+)23 b Fs(n)p Fk(g)35 b(j)h Fv(\000\))1303 4456 y Fk(\024)28 b Fs(C)1479 4471 y Fq(\025;\025)1597 4451 y Fg(0)1629 4456 y Fv(\()p Fs(\017)p Fv(\)Leb)q(\()p Fk(f)p Fs(R)2074 4480 y Fq(\033)2123 4457 y Fh(\034)2157 4473 y(i)2188 4457 y Fi(+)p Fh(`)2269 4472 y Fi(0)2312 4480 y Fq(!)2398 4456 y Fs(>)g(n)p Fk(g)p Fv(\))21 b Fk(\001)h Fv(Leb)q(\(\001)3007 4480 y Fq(\033)3056 4457 y Fh(\034)3090 4473 y(i)3121 4457 y Fi(+)p Fh(`)3202 4472 y Fi(0)3246 4480 y Fq(!)3304 4456 y Fv(\))p Fs(;)0 4666 y Fl(wher)-5 b(e)43 b Fs(C)360 4681 y Fq(\025;\025)478 4661 y Fg(0)510 4666 y Fv(\()p Fs(\017)p Fv(\))g Fl(dep)-5 b(ends)42 b(on)i(the)f(Lipschitz)f(c)-5 b(onstants)42 b(of)h Fs(')p Fl(,)h Fs(')2578 4629 y Fp(0)2607 4666 y Fl(.)67 b(This)43 b(dep)-5 b(endenc)g(e)42 b(may)i(b)-5 b(e)0 4785 y(r)g(emove)g(d)35 b(if)g(we)h(c)-5 b(onsider)34 b Fs(i)28 b Fk(\025)g Fs(i)1227 4800 y Fr(0)1272 4785 y Fv(\()p Fs(\025;)17 b(\025)1472 4749 y Fp(0)1499 4785 y Fv(\))p Fl(.)120 4958 y Fv(The)35 b(pro)s(ofs)g(of)g(Lemmas)f(7.3)g (and)h(7.4)f(are)h(based)h(on)f(the)g(follo)m(wing)g(sublemma,)e(whic)m (h)j(is)f(a)0 5077 y(randomised)g(v)m(ersion)f(of)h(Sublemmas)f(1)g (and)i(2)e(in)h([Y)-8 b(o2])33 b(\(recall)h(that)g(the)h(b)s(ounded)h (distortion)0 5197 y(inequalit)m(y)c(\(A.IV\))g(is)h(uniform)g(in)g Fs(!)t Fv(\).)1847 5316 y Ft(42)p eop %%Page: 43 43 43 42 bop 0 100 a Fx(Sublemma)37 b(7.5)i(\(Consequences)d(of)i(b)s (ounded)f(distorsion\).)0 219 y Fl(\(1\))e(Ther)-5 b(e)35 b(is)g Fs(M)658 234 y Fr(0)738 219 y Fl(such)h(that)g(for)f(al)5 b(l)36 b Fs(n)28 b Fk(2)g Ff(Z)1721 234 y Fr(+)1784 219 y Fl(,)36 b(and)f Fs(!)c Fk(2)e Fv(\012)p Fl(,)1267 436 y Fs(d)p Fv(\()p Fs(F)1436 400 y Fq(n)1422 461 y(!)1490 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Fk(g)22 b(\\)h Fv(\000)2565 5057 y Fq(i)2598 5042 y Fv(\))p 1761 5086 V 1765 5165 a(~)1761 5191 y Fs(\025)p Fv(\()p Fk(f)p Fs(T)42 b(>)28 b(\034)2157 5206 y Fq(i)p Fp(\000)p Fr(1)2292 5191 y Fk(g)21 b(\\)i Fv(\000)2514 5206 y Fq(i)p Fp(\000)p Fr(1)2649 5191 y Fv(\))2700 5109 y Fs(:)1847 5316 y Ft(44)p eop %%Page: 45 45 45 44 bop 0 100 a Fv(Hence)34 b(\(making)d(use)j(of)g(the)f (consequences)j(of)e Fs(i)28 b Fk(\025)g Fs(i)2060 115 y Fr(0)2105 100 y Fv(\()p Fs(\025;)17 b(\025)2305 63 y Fp(0)2332 100 y Fv(\))33 b(in)g(Lemma)f(7.3\),)183 198 y Fo(X)111 441 y Fr(\000)p Fp(2)213 436 y Fo(e)214 441 y Fr(\004)267 411 y Fh(!)267 464 y Fi([)p Fh(n)331 449 y(v)373 464 y Fi(])48 541 y Fr(\000)h([)p Fq(n)202 511 y Fh(v)245 541 y Fr(]-go)r(o)r(d)466 266 y Fv(~)462 293 y Fs(\025)p Fv(\()p Fk(f)p Fs(T)41 b(>)28 b(\034)868 252 y Fq(!)857 320 y Fr([)p Fq(n)929 300 y Fh(v)971 320 y Fr(])999 293 y Fk(g)22 b(\\)g Fv(\000\))489 768 y Fk(\024)622 674 y Fo(X)594 887 y Fr(\000)643 897 y Fi(2)683 887 y Fp(\032)p Fr(\000)611 958 y(go)r(o)r(d)815 742 y Fv(~)811 768 y Fs(\025)p Fv(\(\000)970 783 y Fr(2)1015 768 y Fv(\))1150 674 y Fo(X)1103 887 y Fr(\000)1152 897 y Fi(3)1191 887 y Fp(\032)p Fr(\000)1302 897 y Fi(2)1139 958 y Fr(go)r(o)r(d)1374 675 y Fv(~)1370 701 y Fs(\025)p Fv(\()p Fk(f)p Fs(T)42 b(>)28 b(\034)1766 716 y Fr(2)1810 701 y Fk(g)22 b(\\)h Fv(\000)2033 716 y Fr(3)2077 701 y Fv(\))p 1370 745 747 4 v 1374 824 a(~)1370 850 y Fs(\025)p Fv(\()p Fk(f)p Fs(T)42 b(>)28 b(\034)1766 865 y Fr(2)1810 850 y Fk(g)22 b(\\)h Fv(\000)2033 865 y Fr(2)2077 850 y Fv(\))2150 768 y Fk(\002)g Fs(:)17 b(:)g(:)882 1217 y Fk(\002)1133 1122 y Fo(X)1015 1336 y Fr(\000)p Fp(\032)p Fr(\000)1175 1349 y Fi([)p Fh(n)1239 1334 y(v)1283 1349 y Fi(])p Fg(\000)p Fi(1)1122 1422 y Fr(go)r(o)r(d)1550 1116 y Fv(~)1546 1142 y Fs(\025)p Fv(\()p Fk(f)p Fs(T)41 b(>)29 b(\034)1942 1160 y Fr([)p Fq(n)2014 1140 y Fh(v)2056 1160 y Fr(])p Fp(\000)p Fr(1)2185 1142 y Fk(g)22 b(\\)h Fv(\000\))p 1424 1194 1145 4 v 1428 1272 a(~)1424 1298 y Fs(\025)p Fv(\()p Fk(f)p Fs(T)41 b(>)29 b(\034)1820 1316 y Fr([)p Fq(n)1892 1296 y Fh(v)1934 1316 y Fr(])p Fp(\000)p Fr(1)2063 1298 y Fk(g)22 b(\\)h Fv(\000)2286 1316 y Fr([)p Fq(n)2358 1296 y Fh(v)2401 1316 y Fr(])p Fp(\000)p Fr(1)2530 1298 y Fv(\))2603 1217 y Fk(\002)2703 1083 y Fr([)p Fq(n)2775 1053 y Fh(v)2817 1083 y Fr(])2707 1122 y Fo(Y)2707 1334 y Fq(i)p Fr(=2)2840 1217 y Fv(\(1)e Fk(\000)i Fs(V)3130 1164 y Fq(\034)3167 1174 y Fh(i)3198 1164 y Fp(\000)p Fq(\034)3297 1174 y Fh(i)p Fg(\000)p Fi(1)3108 1251 y Fq(\033)3157 1220 y Fh(\034)3191 1236 y(i)p Fg(\000)p Fi(1)3315 1251 y Fq(!)3422 1217 y Fs(=C)3543 1232 y Fq(\025;\025)3661 1212 y Fg(0)3693 1217 y Fv(\))489 1579 y Fk(\024)28 b Fs(e)640 1538 y Fp(\000)p Fr([)p Fq(n)774 1507 y Fh(v)818 1538 y Fr(])p Fq(\032=C)1006 1579 y Fs(;)0 1777 y Fv(where)34 b(w)m(e)g(used)g Fs(!)d Fk(62)862 1752 y Fo(f)850 1777 y Fs(M)947 1795 y Fr([)p Fq(n)1019 1775 y Fh(v)1061 1795 y Fr(])1122 1777 y Fv(and)i(also)g(the)h(fact)f(that)148 1949 y Fo(X)120 2162 y Fr(\000)169 2172 y Fi(2)208 2162 y Fp(\032)p Fr(\000)149 2233 y(go)r(o)r(d)340 2017 y Fv(~)336 2043 y Fs(\025)p Fv(\(\000)495 2058 y Fr(2)540 2043 y Fv(\))676 1949 y Fo(X)628 2162 y Fr(\000)677 2172 y Fi(3)717 2162 y Fp(\032)p Fr(\000)828 2172 y Fi(2)677 2233 y Fr(go)r(o)r(d)900 1950 y Fv(~)895 1976 y Fs(\025)q Fv(\()p Fk(f)p Fs(T)41 b(>)28 b(\034)1291 1991 y Fr(2)1335 1976 y Fk(g)22 b(\\)h Fv(\000)1558 1991 y Fr(3)1603 1976 y Fv(\))p 895 2020 747 4 v 899 2099 a(~)895 2125 y Fs(\025)q Fv(\()p Fk(f)p Fs(T)41 b(>)28 b(\034)1291 2140 y Fr(2)1335 2125 y Fk(g)22 b(\\)h Fv(\000)1558 2140 y Fr(2)1603 2125 y Fv(\))1670 2043 y Fs(:)17 b(:)g(:)1954 1949 y Fo(X)1836 2162 y Fr(\000)p Fp(\032)p Fr(\000)1996 2175 y Fi([)p Fh(n)2060 2160 y(v)2104 2175 y Fi(])p Fg(\000)p Fi(1)1956 2248 y Fr(go)r(o)r(d)2387 1943 y Fv(~)2383 1969 y Fs(\025)p Fv(\()p Fk(f)p Fs(T)41 b(>)28 b(\034)2778 1987 y Fr([)p Fq(n)2850 1967 y Fh(v)2892 1987 y Fr(])p 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Fq(i)p Fp(\000)p Fr(1)923 2748 y Fk(\024)28 b Fs(n)g(<)g(\034)1265 2763 y Fq(i)1298 2748 y Fk(g)p Fv(\))353 3003 y(=)588 2908 y Fo(X)470 3112 y Fi(\()p Fh(k)535 3127 y Fi(1)586 3112 y Fh(:::)11 b(k)704 3128 y(i)p Fg(\000)p Fi(1)823 3112 y(\))515 3148 y Fo(P)632 3210 y Fh(k)670 3226 y(j)707 3210 y Fg(\024)p Fh(n)883 2977 y Fv(~)879 3003 y Fs(\025)937 2892 y Fo(\020)997 3003 y Fk(f)p Fs(\034)1091 3018 y Fq(i)1145 3003 y Fk(\000)23 b Fs(\034)1289 3018 y Fq(i)p Fp(\000)p Fr(1)1451 3003 y Fs(>)28 b(n)22 b Fk(\000)1745 2878 y Fq(i)p Fp(\000)p Fr(1)1738 2908 y Fo(X)1741 3121 y Fq(j)t Fr(=1)1899 3003 y Fs(k)1951 3018 y Fq(j)2009 3003 y Fv(;)33 b Fs(\034)2114 3018 y Fq(j)2178 3003 y Fk(\000)22 b Fs(\034)2321 3018 y Fq(j)t Fp(\000)p Fr(1)2492 3003 y Fv(=)29 b Fs(k)2650 3018 y Fq(j)2725 3003 y Fs(;)17 b(j)32 b Fv(=)c(1)p Fs(;)17 b(:)g(:)g(:)31 b(;)17 b(i)22 b Fk(\000)g Fv(1)p Fk(g)3492 2892 y Fo(\021)0 3396 y Fv(Fixing)35 b Fs(k)366 3411 y Fr(1)410 3396 y Fs(;)17 b(:)g(:)g(:)31 b(;)17 b(k)701 3411 y Fq(i)p Fp(\000)p Fr(1)835 3396 y Fv(,)37 b(conditioning,)f(using)g(Lemma)e(7.4)h(and)h (the)g(asymptotics)f(\(A.V\))f(on)i(the)0 3516 y(return)e(times,)e(w)m (e)i(get)f(if)g Fs(n)27 b(>)1196 3441 y Fo(P)1317 3516 y Fs(k)1369 3531 y Fq(j)1433 3516 y Fv(+)c Fs(n)1593 3531 y Fr(4)1637 3516 y Fv(\()p Fs(\033)1737 3479 y Fq(\034)1774 3489 y Fh(i)p Fg(\000)p Fi(1)1893 3479 y Fr(+)p Fq(`)1987 3489 y Fi(0)2031 3516 y Fs(!)t Fv(\))f(+)g Fs(`)2299 3531 y Fr(0)2343 3516 y Fv(:)210 3762 y(~)206 3788 y Fs(\025)p Fv(\()p Fk(f)p Fs(\034)397 3803 y Fq(i)452 3788 y Fk(\000)g Fs(\034)595 3803 y Fq(i)p Fp(\000)p Fr(1)758 3788 y Fs(>)28 b(n)22 b Fk(\000)1051 3664 y Fq(i)p Fp(\000)p Fr(1)1045 3694 y Fo(X)1048 3906 y Fq(j)t Fr(=1)1205 3788 y Fs(k)1257 3803 y Fq(j)1299 3788 y Fv(;)17 b Fs(\034)1388 3803 y Fq(j)1451 3788 y Fk(\000)22 b Fs(\034)1594 3803 y Fq(j)t Fp(\000)p Fr(1)1766 3788 y Fv(=)28 b Fs(k)1923 3803 y Fq(j)1998 3788 y Fs(;)17 b(j)33 b Fv(=)28 b(1)p Fs(;)17 b(:)g(:)g(:)30 b(;)17 b(i)22 b 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Fv(and)h(pro)s(ceeding)g(as)f(in)g(the) h(pro)s(of)f(of)h(Prop)s(osition)e(4.3,)i(w)m(e)f(get)g(for)h(0)i Fs(<)38 b Fv(^)-60 b Fs(\032)28 b(<)g Fv(1)0 355 y(a)36 b(subset)h(\012)466 370 y Fr(6)543 355 y Fk(\032)32 b Fv(\012)724 370 y Fr(5)793 355 y Fk(\\)24 b Fv(\012)955 370 y Fr(4)1036 355 y Fv(of)36 b(full)g(measure)g(with)g(the)g(follo)m (wing)g(prop)s(ert)m(y:)49 b(F)-8 b(or)36 b Fs(!)f Fk(2)e Fv(\012)3449 370 y Fr(6)3494 355 y Fv(,)j(there)0 475 y(exists)28 b Fs(n)330 490 y Fr(6)374 475 y Fv(\()p Fs(!)t Fv(\))f Fk(\025)h Fv(max)o(\()p Fs(n)935 490 y Fr(5)979 475 y Fv(\()p Fs(!)t Fv(\))p Fs(;)17 b(n)1228 490 y Fr(4)1271 475 y Fv(\()p Fs(!)t Fv(\)\))26 b(\(with)i(the)g(b)s(ounds)h(stated)f (in)g(Prop)s(osition)f(7.6\))g(suc)m(h)i(that)0 602 y Fk(8)p Fs(n)f Fk(\025)g Fs(n)308 617 y Fr(6)352 602 y Fv(\()p Fs(!)t Fv(\),)k(the)732 576 y(~)728 602 y Fs(\025)p Fv(-measure)i(of)f(the)h(cylinders)f(in)h(\(I\))e(whic)m(h)i(violate)e (the)i(condition)250 768 y Fq(i)192 798 y Fo(X)195 1010 y Fq(j)t Fr(=0)336 892 y Fv(\()p Fs(n)435 907 y Fr(4)479 892 y Fv(\()p Fs(\033)579 851 y Fq(\034)616 861 y Fh(j)652 851 y Fr(+)p Fq(`)746 861 y Fi(0)791 892 y Fs(!)t Fv(\)\))935 851 y Fr(1)p Fq(=)p Fr(4)1087 892 y Fk(\024)j Fv(^)-59 b Fs(\032)o(n)1303 851 y Fr(1)p Fq(=)p Fr(4)1462 892 y Fv(and)1676 768 y Fq(i)p Fp(\000)p Fr(1)1677 798 y Fo(Y)1672 1010 y Fq(j)t Fr(=0)1826 892 y Fs(n)1886 907 y Fr(3)1931 892 y Fv(\()p Fs(\033)2031 851 y Fq(\034)2068 861 y Fh(j)2104 851 y Fr(+)p Fq(`)2198 861 y Fi(0)2243 892 y Fs(!)t Fv(\))27 b Fk(\024)h Fs(e)2526 851 y Fq(n)2575 821 y Fh(v)2632 851 y Fr(log)q(\()7 b(^)-47 b Fq(\032n)p Fr(\))2909 892 y Fs(;)116 b Fk(8)p Fs(i)28 b Fk(\024)h Fs(n)3336 851 y Fq(v)3575 892 y Fv(\(7.6\))0 1221 y(is)k(less)h(than)f Fs(e)562 1185 y Fp(\000)p Fr(\()p Fq(n)704 1155 y Fi(1)p Fh(=)p Fi(4)p Fg(\000)p Fh(v)907 1185 y Fq(=C)5 b Fr(\()p Fq(\017)p Fr(\)\))1141 1221 y Fv(.)120 1341 y(Next,)50 b(summing)d(\(7.5\))g(o)m(v)m(er)h(the)h Fs(k)1571 1356 y Fq(j)1661 1341 y 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2260 y Fq(=C)5 b Fr(\()p Fq(\017)p Fr(\)\))3667 2301 y Fs(:)3575 2481 y Fv(\(7.8\))0 2620 y(Assume)27 b(\014rst)h(that)f Fs( )k Fk(2)d(F)1045 2578 y Fr(+)1035 2651 y Fq(\014)1121 2620 y Fk(\\)10 b(L)1266 2584 y Fp(K)1327 2594 y Fh(!)1266 2645 y Fp(1)1382 2620 y Fv(.)42 b(Prop)s(osition)27 b(7.9)f(applied)i(to)f Fs(\025)2654 2635 y Fq(!)2739 2620 y Fv(=)2844 2540 y Fo(\000)16 b(R)2990 2620 y Fs( )3055 2635 y Fq(!)3129 2620 y Fs(d)p Fv(Leb)3342 2540 y Fo(\001)3388 2560 y Fp(\000)p Fr(1)3495 2620 y Fs( )3560 2635 y Fq(!)3618 2620 y Fv(Leb)0 2749 y(and)34 b Fs(\026)254 2764 y Fq(!)345 2749 y Fv(giv)m(es)f(that)g(for)g Fs(n)28 b Fk(\025)g Fs(n)1210 2764 y Fr(7)1254 2749 y Fv(\()p Fs(!)t Fv(\),)66 2894 y Fo(\014)66 2954 y(\014)66 3014 y(\014)116 2873 y(Z)232 3009 y Fs(')297 3024 y Fq(\033)346 3004 y Fh(n)395 3024 y Fq(!)475 3009 y Fk(\016)22 b Fs(F)625 2968 y Fq(n)611 3033 y(!)701 3009 y Fk(\001)g Fs( )816 3024 y Fq(!)890 3009 y Fs(d)p Fv(Leb)h Fk(\000)1259 2873 y Fo(Z)1375 3009 y Fs(')1440 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Fk(\025)d Fs(n)1358 4011 y Fr(7)1403 3996 y Fv(\()p Fs(!)t Fv(\))f Fk(j)g(K)1711 4016 y Fq(\033)1760 3996 y Fh(k)1803 4016 y Fq(!)1891 3996 y Fk(\024)i Fs(k)s Fk(g)p Fv(.)49 b(By)34 b(\(5.7\))g(and)i(the)f(b)s(ounds)h(on)g Fs(n)3498 4011 y Fr(7)3577 3996 y Fv(from)0 4152 y(Prop)s(osition)c(7.9,)h(w)m(e)g (get)g Fs(P)14 b Fv(\()p Fk(f)p Fs(n)1264 4167 y Fr(8)1308 4152 y Fv(\()p Fs(!)t Fv(\))27 b Fs(>)h(k)s Fk(g)p Fv(\))f Fk(\024)h Fs(e)1906 4116 y Fp(\000)p Fr(\()p Fq(k)2073 4063 y Fi(1)p 2056 4072 V 2056 4109 a(16)2141 4116 y Fq(=)2192 4111 y Fo(e)2182 4116 y Fq(C)2239 4126 y Fi(2)2278 4116 y Fr(\))2314 4152 y Fv(.)44 b(W)-8 b(e)33 b(\014nd)i(for)e Fs(n)28 b(>)g(n)3162 4167 y Fr(8)3206 4152 y Fv(\()p Fs(!)t Fv(\),)122 4276 y Fo(\014)122 4336 y(\014)122 4396 y(\014)122 4456 y(\014)155 4285 y(Z)271 4421 y Fs(')336 4436 y Fq(\033)385 4416 y Fh(n)434 4436 y Fq(!)514 4421 y Fk(\016)22 b Fs(F)664 4380 y Fq(n)650 4446 y(!)740 4421 y Fk(\001)g Fs( )855 4436 y Fq(!)929 4421 y Fs(d)p Fv(Leb)56 b Fk(\000)1331 4285 y Fo(Z)1447 4421 y Fs(')1512 4436 y Fq(\033)1561 4416 y Fh(n)1610 4436 y Fq(!)1667 4421 y Fs(d\026)1779 4436 y Fq(\033)1828 4416 y Fh(n)1877 4436 y Fq(!)1951 4285 y Fo(Z)2067 4421 y Fs( )2132 4436 y Fq(!)2206 4421 y Fs(d)p Fv(Leb)2420 4276 y Fo(\014)2420 4336 y(\014)2420 4396 y(\014)2420 4456 y(\014)2481 4421 y Fk(\024)61 b Fs(C)2690 4436 y Fq('; )2827 4421 y Fv(\()p Fs(\017)p Fv(\))p Fk(K)3021 4436 y Fq(!)3078 4421 y Fs(n)33 b(e)3217 4380 y Fp(\000)p Fr(\()p Fq(n)3389 4327 y Fi(1)p 3372 4336 V 3372 4372 a(16)3457 4380 y Fq(=)3508 4375 y Fo(e)3498 4380 y Fq(C)3555 4390 y Fi(1)3594 4380 y Fr(\))3630 4421 y Fs(:)0 4690 y Fv(If)h Fs(n)27 b Fk(\024)h Fs(n)352 4705 y Fr(8)397 4690 y Fv(\()p Fs(!)t Fv(\),)536 4835 y Fo(\014)536 4895 y(\014)536 4955 y(\014)586 4814 y(Z)702 4950 y Fs(')767 4965 y Fq(\033)816 4945 y Fh(n)865 4965 y Fq(!)945 4950 y Fk(\016)22 b Fs(F)1095 4908 y Fq(n)1081 4974 y(!)1171 4950 y Fk(\001)g Fs( )1286 4965 y Fq(!)1360 4950 y Fs(d)p Fv(Leb)56 b Fk(\000)1762 4814 y Fo(Z)1878 4950 y Fs(')1943 4965 y Fq(\033)1992 4945 y Fh(n)2041 4965 y Fq(!)2098 4950 y Fs(d\026)2210 4965 y Fq(\033)2259 4945 y Fh(n)2308 4965 y Fq(!)2382 4814 y Fo(Z)2498 4950 y Fs( )2563 4965 y Fq(!)2621 4950 y Fs(d)p Fv(Leb)2835 4835 y Fo(\014)2835 4895 y(\014)2835 4955 y(\014)2033 5202 y Fk(\024)28 b Fs(C)2209 5217 y Fq('; )2346 5202 y Fv(\()p Fs(\017)p Fv(\))22 b Fk(\001)g Fs(C)7 b Fv(\()p Fs(!)t Fv(\))p Fs(e)2804 5161 y Fp(\000)p Fr(\()p Fq(n)2975 5107 y Fi(1)p 2958 5116 V 2958 5153 a(16)3043 5161 y Fq(=)3094 5156 y Fo(e)3084 5161 y Fq(C)3141 5171 y Fi(1)3180 5161 y Fr(\))3216 5202 y Fs(;)1847 5316 y Ft(48)p eop %%Page: 49 49 49 48 bop 0 100 a Fv(setting)982 256 y Fs(C)7 b Fv(\()p Fs(!)t Fv(\))27 b(:=)h Fs(e)1410 214 y Fr(\()p Fq(n)1490 224 y Fi(7)1529 214 y Fr(\()p Fq(!)s Fr(\))1673 161 y Fi(1)p 1656 170 68 4 v 1656 207 a(16)1741 214 y Fq(=)1792 209 y Fo(e)1782 214 y Fq(C)1839 224 y Fi(1)1878 214 y Fr(\))1936 256 y Fk(\001)22 b(K)2062 271 y Fq(!)2141 256 y Fk(\001)87 b Fv(max)2191 324 y Fq(n)p Fp(\024)p Fq(n)2351 334 y Fi(8)2391 324 y Fr(\()p Fq(!)s Fr(\))2523 256 y Fk(K)2599 271 y Fq(\033)2648 251 y Fh(n)2696 271 y Fq(!)2770 256 y Fs(:)0 510 y Fv(This)44 b(giv)m(es)f(\(7.8\))g(if)g Fs( )k Fv(b)s(elongs)d(to)g Fk(F)1537 468 y Fr(+)1527 540 y Fq(\014)1631 510 y Fk(\\)30 b(L)1796 474 y Fp(K)1857 484 y Fh(!)1796 535 y Fp(1)1912 510 y Fv(.)75 b(F)-8 b(or)43 b(non)h(negativ)m(e)g(real-v)-6 b(alued)44 b Fs( )k Fk(2)e(F)3636 466 y Fp(K)3697 476 y Fh(!)3626 540 y Fq(\014)3752 510 y Fv(,)0 657 y(remark)29 b(that)571 630 y(~)551 657 y Fs( )616 672 y Fq(!)701 657 y Fv(=)f Fs( )871 672 y Fq(!)945 657 y Fv(+)16 b(\()p Fs(C)1148 672 y Fq( )1224 657 y Fv(+)g(1\))p Fk(K)1482 672 y Fq(!)1570 657 y Fv(b)s(elongs)30 b(to)g Fk(F)2125 614 y Fr(+)2115 687 y Fq(\014)2207 657 y Fk(\\)16 b(L)2358 621 y Fp(K)2419 631 y Fh(!)2358 681 y Fp(1)2504 657 y Fv(and)31 b(apply)f(the)g(ab)s(o) m(v)m(e)h(estimate)0 803 y(to)148 777 y(~)128 803 y Fs( )s Fv(.)62 b(General)40 b(real-v)-6 b(alued)40 b(functions)h(are)e(decomp) s(osed)h(in)m(to)f(p)s(ositiv)m(e)g(and)h(negativ)m(e)f(parts.)0 923 y(Complex-v)-6 b(alued)33 b(functions)h(are)f(decomp)s(osed)h(in)m (to)f(real)g(and)h(imaginary)d(parts.)120 1042 y(Next,)39 b(w)m(e)g(pro)m(v)m(e)g(that)f Fs(C)7 b Fv(\()p Fs(!)t Fv(\))38 b(has)h(the)g(announced)i(b)s(eha)m(viour.)60 b(Fix)38 b(0)e Fs(<)h(u)g(<)g Fv(1)i(suc)m(h)h(that)22 1149 y Fo(e)12 1154 y Fq(C)69 1164 y Fi(1)108 1154 y Fr(\()p Fq(\017)p Fr(\)\(1)p Fp(\000)p Fq(u)p Fr(\))p 12 1179 403 4 v 127 1255 a Fo(e)118 1260 y Fq(C)175 1270 y Fi(2)214 1260 y Fr(\()p Fq(\017)p Fr(\))454 1202 y Fs(>)29 b Fv(1,)j(and)i(use)g(\(5.7\))e(and)i(Prop)s(osition)e(7.9)h (again)55 1462 y Fs(P)14 b Fv(\()p Fk(f)p Fs(C)7 b Fv(\()p Fs(!)t Fv(\))26 b Fs(>)i(m)p Fk(g)p Fv(\))470 1663 y Fk(\024)g Fs(P)14 b Fv(\()p Fk(f)82 b Fv(sup)742 1751 y Fq(n)p Fp(\024)p Fq(n)902 1761 y Fi(8)941 1751 y Fr(\()p Fq(!)s Fr(\))1073 1663 y Fk(K)1149 1678 y Fq(\033)1198 1658 y Fh(n)1246 1678 y Fq(!)1332 1663 y Fs(>)28 b(m)1536 1595 y Fh(u)p 1536 1606 43 4 v 1540 1646 a Fi(2)1595 1663 y Fk(g)p Fv(\))22 b(+)h Fs(P)14 b Fv(\()p Fk(f)p Fs(e)2019 1622 y Fq(n)2068 1632 y Fi(7)2107 1622 y Fr(\()p Fq(!)s Fr(\))2251 1569 y Fi(1)p 2234 1578 68 4 v 2234 1615 a(16)2318 1622 y Fq(=)2369 1617 y Fo(e)2359 1622 y Fq(C)2416 1632 y Fi(1)2488 1663 y Fs(>)28 b(m)2680 1622 y Fr(1)p Fp(\000)p Fq(u)2835 1663 y Fk(g)p Fv(\))21 b(+)i Fs(P)14 b Fv(\()p Fk(fK)3288 1678 y Fq(!)3372 1663 y Fs(>)29 b(m)3577 1595 y Fh(u)p 3577 1606 43 4 v 3581 1646 a Fi(2)3636 1663 y Fk(g)p Fv(\))470 1983 y Fk(\024)f Fs(P)14 b Fv(\()p Fk(f)p Fs(n)802 1998 y Fr(8)846 1983 y Fv(\()p Fs(!)t Fv(\))27 b Fs(>)h(m)p Fk(g)p Fv(\))22 b(+)1460 1859 y Fq(m)1423 1889 y Fo(X)1420 2100 y Fq(n)p Fr(=1)1587 1983 y Fs(P)14 b Fv(\()p Fk(fK)1830 1998 y Fq(\033)1879 1978 y Fh(n)1927 1998 y Fq(!)2012 1983 y Fs(>)28 b(m)2216 1915 y Fh(u)p 2216 1926 V 2220 1966 a Fi(2)2276 1983 y Fk(g)p Fv(\))1063 2247 y(+)22 b Fs(P)14 b Fv(\()p Fk(f)p Fs(n)1389 2262 y Fr(7)1433 2247 y Fv(\()p Fs(!)t Fv(\))27 b Fs(>)h Fv([\(1)21 b Fk(\000)i Fs(u)p Fv(\))2062 2221 y Fo(e)2043 2247 y Fs(C)2114 2262 y Fr(1)2175 2247 y Fv(log)17 b Fs(m)p Fv(])2435 2205 y Fr(16)2520 2247 y Fk(g)p Fv(\))22 b(+)g Fs(P)14 b Fv(\()p Fk(fK)2973 2262 y Fq(!)3058 2247 y Fs(>)28 b(m)3262 2179 y Fh(u)p 3263 2190 V 3267 2229 a Fi(2)3322 2247 y Fk(g)p Fv(\))470 2448 y Fk(\024)g Fs(e)621 2407 y Fp(\000)p Fr(\()p Fq(m)815 2353 y Fi(1)p 798 2362 68 4 v 798 2399 a(16)883 2407 y Fq(=)934 2402 y Fo(e)924 2407 y Fq(C)981 2417 y Fi(2)1019 2407 y Fr(\))1078 2448 y Fv(+)22 b Fs(me)1310 2407 y Fp(\000)p Fr(\()p Fq(m)1487 2353 y Fh(u)p 1487 2362 43 4 v 1491 2399 a Fi(8)1546 2407 y Fq(=C)5 b Fr(\()p Fq(\017)p Fr(\)\))1803 2448 y Fv(+)23 b Fs(e)1949 2407 y Fp(\000)p Fr([log)14 b Fq(m)2231 2402 y Fo(e)2221 2407 y Fq(C)2278 2417 y Fi(1)2317 2407 y Fr(\(1)p Fp(\000)p Fq(u)p Fr(\))p Fq(=)2579 2402 y Fo(e)2569 2407 y Fq(C)2626 2417 y Fi(2)2665 2407 y Fr(])2715 2448 y Fv(+)23 b Fs(e)2861 2407 y Fp(\000)p Fq(m)3006 2353 y Fh(u)p 3006 2362 V 3010 2399 a Fi(8)3065 2407 y Fq(=C)5 b Fr(\()p Fq(\017)p Fr(\))3268 2448 y Fs(:)0 2680 y Fv(This)33 b(pro)m(v)m(es)h(the)g(claim)e(on)h(the)h (random)e(v)-6 b(ariable)33 b Fs(C)7 b Fv(\()p Fs(!)t Fv(\),)32 b(taking)h Fs(v)e Fv(=)2816 2655 y Fo(e)2796 2680 y Fs(C)2867 2695 y Fr(1)2912 2680 y Fv(\(1)22 b Fk(\000)h Fs(u)p Fv(\))p Fs(=)3288 2655 y Fo(e)3269 2680 y Fs(C)3340 2695 y Fr(2)3412 2680 y Fs(>)28 b Fv(1.)120 2799 y(T)-8 b(o)35 b(conclude,)j(it)e(remains)f(to)h(sho)m(w)h(that)f Fs(C)1852 2814 y Fq(')14 b( )1978 2799 y Fv(\()p Fs(\017)p Fv(\))33 b Fk(\024)g Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))p Fk(k)p Fs(')p Fk(k)2600 2814 y Fp(L)2655 2824 y Fg(1)2733 2799 y Fk(k)p Fs( )t Fk(k)2902 2814 y Fp(F)2970 2799 y Fv(.)53 b(W)-8 b(e)36 b(adapt)h(to)e(our)0 2919 y(random)23 b(setting)h(an)f(argumen)m(t)h(of)g(Collet)e([Co2])h(based)i(on)e(the)h (uniform)g(b)s(oundedness)i(principle.)0 3046 y(Fix)32 b Fs( )g Fk(2)c(F)451 3001 y Fp(K)512 3011 y Fh(!)441 3076 y Fq(\014)600 3046 y Fv(and)34 b(de\014ne)247 3378 y Fs(p)297 3337 y Fq( )297 3403 y(n;!)428 3378 y Fv(\()p Fs(')p Fv(\))27 b(=)715 3311 y Fs(e)761 3275 y Fr(\()p Fq(n)871 3221 y Fi(1)p 854 3230 68 4 v 854 3267 a(16)939 3275 y Fq(=)990 3270 y Fo(e)980 3275 y Fq(C)1037 3285 y Fi(1)1075 3275 y Fr(\))p 715 3355 397 4 v 803 3447 a Fs(C)7 b Fv(\()p Fs(!)t Fv(\))1123 3234 y Fo(\014)1123 3293 y(\014)1123 3353 y(\014)1123 3413 y(\014)1173 3243 y(Z)1289 3378 y Fs(')1354 3393 y Fq(\033)1403 3373 y Fh(n)1452 3393 y Fq(!)1532 3378 y Fk(\016)22 b Fs(F)1682 3337 y Fq(n)1668 3403 y(!)1758 3378 y Fk(\001)g Fs( )1873 3393 y Fq(!)1947 3378 y Fs(d)p Fv(Leb)56 b Fk(\000)2349 3243 y Fo(Z)2465 3378 y Fs(')2530 3393 y Fq(\033)2579 3373 y Fh(n)2628 3393 y Fq(!)2702 3378 y Fs(d\026)2814 3393 y Fq(\033)2863 3373 y Fh(n)2912 3393 y Fq(!)2986 3243 y Fo(Z)3102 3378 y Fs( )3167 3393 y Fq(!)3241 3378 y Fs(d)p Fv(Leb)3455 3234 y Fo(\014)3455 3293 y(\014)3455 3353 y(\014)3455 3413 y(\014)3505 3378 y Fs(:)0 3657 y Fv(It)33 b(follo)m(ws)g(from)f(\(7.8\))g(that)h(sup)1274 3682 y Fq(n;!)s Fp(2)p Fr(\012)1511 3692 y Fi(8)1570 3657 y Fs(p)1620 3621 y Fq( )1620 3682 y(n;!)1751 3657 y Fv(\()p Fs(')p Fv(\))27 b Fs(<)h Fk(1)33 b Fv(for)h(all)e Fs(')c Fk(2)g(L)2706 3621 y Fp(K)2767 3631 y Fh(!)2706 3682 y Fp(1)2822 3657 y Fv(.)44 b(The)34 b(uniform)e(b)s(ound-)0 3780 y(edness)j(principle)f(giv)m(es)f(a)g(constan)m(t)h Fs(D)1532 3795 y Fq( )1593 3780 y Fv(\()p Fs(\017)p Fv(\))f(suc)m(h)h (that)1493 3998 y(sup)1238 4086 y Fq(n;!)s Fp(2)p Fr(\012)1475 4096 y Fi(8)1513 4086 y Fq(;)p Fp(k)p Fq(')p Fp(k)1671 4096 y Fg(L)1719 4106 y(1)1796 4086 y Fp(\024)p Fr(1)1915 3998 y Fs(p)1965 3957 y Fq( )1965 4023 y(n;!)2096 3998 y Fv(\()p Fs(')p Fv(\))27 b Fk(\024)h Fs(D)2453 4013 y Fq( )2514 3998 y Fs(:)1033 b Fv(\(7.9\))0 4300 y(F)-8 b(or)33 b Fs(n)27 b Fk(2)h Ff(Z)429 4315 y Fr(+)493 4300 y Fv(,)33 b Fs(!)e Fk(2)d Fv(\012)813 4315 y Fr(8)891 4300 y Fv(and)34 b Fs(')28 b Fk(2)g(L)1341 4264 y Fp(K)1402 4274 y Fh(!)1341 4325 y Fp(1)1490 4300 y Fv(with)33 b Fk(k)p Fs(')p Fk(k)1882 4315 y Fp(L)1937 4325 y Fg(1)2043 4300 y Fk(\024)28 b Fv(1,)k(set)210 4614 y Fs(q)254 4629 y Fq(n;!)s(;')461 4614 y Fv(\()p Fs( )t Fv(\))27 b(=)752 4546 y Fs(e)798 4510 y Fr(\()p Fq(n)908 4457 y Fi(1)p 891 4466 68 4 v 891 4503 a(16)975 4510 y Fq(=)1026 4505 y Fo(e)1016 4510 y Fq(C)1073 4520 y Fi(1)1112 4510 y Fr(\))p 752 4591 397 4 v 839 4682 a Fs(C)7 b Fv(\()p Fs(!)t Fv(\))1160 4469 y Fo(\014)1160 4529 y(\014)1160 4589 y(\014)1160 4649 y(\014)1210 4478 y(Z)1326 4614 y Fs(')1391 4629 y Fq(\033)1440 4609 y Fh(n)1489 4629 y Fq(!)1569 4614 y Fk(\016)22 b Fs(F)1719 4573 y Fq(n)1705 4638 y(!)1795 4614 y Fk(\001)g Fs( )1910 4629 y Fq(!)1984 4614 y Fs(d)p Fv(Leb)56 b Fk(\000)2386 4478 y Fo(Z)2502 4614 y Fs(')2567 4629 y Fq(\033)2616 4609 y Fh(n)2665 4629 y Fq(!)2739 4614 y Fs(d\026)2851 4629 y Fq(\033)2900 4609 y Fh(n)2949 4629 y Fq(!)3023 4478 y Fo(Z)3139 4614 y Fs( )3204 4629 y Fq(!)3278 4614 y Fs(d)p Fv(Leb)3492 4469 y Fo(\014)3492 4529 y(\014)3492 4589 y(\014)3492 4649 y(\014)3542 4614 y Fs(:)0 4900 y Fv(It)33 b(follo)m(ws)g(from)g (\(7.9\))f(that)g(for)i(an)m(y)f Fs( )e Fk(2)d(F)1737 4856 y Fp(K)1798 4866 y Fh(!)1727 4930 y Fq(\014)1853 4900 y Fv(,)1389 5118 y(sup)1134 5207 y Fq(n;!)s Fp(2)p Fr(\012)1371 5217 y Fi(8)1409 5207 y Fq(;)p Fp(k)p Fq(')p Fp(k)1567 5217 y Fg(L)1615 5227 y(1)1692 5207 y Fp(\024)p Fr(1)1811 5118 y Fs(q)1855 5133 y Fq(n;!)s(;')2062 5118 y Fv(\()p Fs( )t Fv(\))f Fk(\024)h Fs(D)2423 5133 y Fq( )2484 5118 y Fv(\()p Fs(\017)p Fv(\))17 b Fs(:)1847 5316 y Ft(49)p eop %%Page: 50 50 50 49 bop 0 100 a Fv(Using)29 b(once)i(more)d(the)i(uniform)g(b)s (oundedness)i(principle,)e(w)m(e)h(conclude)f(that)g(there)g(exists)f Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))0 219 y(so)33 b(that)1430 344 y(sup)1013 432 y Fq(n;!)s Fp(2)p Fr(\012)1250 442 y Fi(8)1288 432 y Fq(;)p Fp(k)p Fq(')p Fp(k)1446 442 y Fg(L)1494 452 y(1)1572 432 y Fp(\024)p Fr(1)p Fq(;)p Fp(k)p Fq( )s Fp(k)1836 442 y Fg(F)1894 432 y Fp(\024)p Fr(1)2012 344 y Fs(q)2056 359 y Fq(n;!)s(;')2263 344 y Fv(\()p Fs( )t Fv(\))27 b Fk(\024)h Fs(C)7 b Fv(\()p Fs(\017)p Fv(\))p Fs(:)0 616 y Fv(This)33 b(ends)i(the)f(pro)s(of)f(of) g(Corollary)f(7.10.)98 b Fe(\003)566 928 y Fw(8.)55 b(Random)37 b(coupling)g(ar)n(gument,)f(\\p)-7 b(ast")37 b(correla)-7 b(tions)120 1109 y Fv(The)31 b(estimates)f(for)h(the)g(\\past")f (correlations)g(are)h(obtained)g(b)m(y)g(recycling)g(the)g(argumen)m (ts)f(of)0 1228 y(Section)k(7:)0 1413 y Fx(Lemma)46 b(8.1)h(\(Lo)m(w)m (er)e(b)s(ound)g(for)i Fs(P)14 b Fv(\()p Fk(f)p Fs(T)1772 1428 y Fq(!)1868 1413 y Fv(=)40 b Fs(\034)2029 1428 y Fq(i)2062 1413 y Fk(g)p Fv(\))p Fx(\).)56 b Fl(L)-5 b(et)42 b Fs(\025)p Fl(,)i Fs(\025)2653 1377 y Fp(0)2723 1413 y Fl(b)-5 b(e)41 b(absolutely)j(c)-5 b(ontinuous)0 1547 y(pr)g(ob)g(ability)36 b(me)-5 b(asur)g(es)36 b(on)g Fk(f)p Fv(\001)1184 1562 y Fq(!)1242 1547 y Fk(g)p Fl(,)g(with)g (densities)f Fs(')p Fl(,)h Fs(')2184 1511 y Fp(0)2249 1547 y Fl(in)g Fk(F)2454 1505 y Fr(+)2444 1577 y Fq(\014)2520 1547 y Fl(.)47 b(F)-8 b(or)37 b(e)-5 b(ach)36 b Fs(i)p Fl(,)h(if)e Fv(\000)30 b Fk(2)3402 1522 y Fo(e)3396 1547 y Fv(\004)3462 1511 y Fq(\033)3511 1481 y Fg(\000)p Fh(n)3614 1511 y Fq(!)3462 1573 y(i)3708 1547 y Fl(is)0 1696 y(such)36 b(that)f Fv(\()p Fs(T)527 1715 y Fq(\033)576 1695 y Fg(\000)p Fh(n)679 1715 y Fq(!)736 1696 y Fv(\))775 1714 y Fp(j)p Fr(\000)881 1696 y Fs(>)28 b(\034)1030 1711 y Fq(i)p Fp(\000)p Fr(1)1165 1696 y Fl(,)35 b(then,)g(asso)-5 b(ciating)33 b Fs(V)2070 1644 y Fq(\034)2107 1654 y Fh(i)2138 1644 y Fp(\000)p Fq(\034)2237 1654 y Fh(i)p Fg(\000)p Fi(1)2047 1743 y Fq(\033)2096 1711 y Fh(\034)2130 1727 y(i)p Fg(\000)p Fi(1)2250 1711 y Fg(\000)p Fh(n)2352 1743 y Fq(!)2446 1696 y Fl(to)i Fv(\000)h Fl(as)g(usual,)742 1974 y Fv(\()p Fs(\025)22 b Fk(\002)h Fs(\025)1019 1933 y Fp(0)1047 1974 y Fv(\)\()p Fk(f)p Fs(T)1233 1993 y Fq(\033)1282 1973 y Fg(\000)p Fh(n)1384 1993 y Fq(!)1469 1974 y Fs(>)28 b(\034)1618 1989 y Fq(i)1651 1974 y Fk(gj)p Fv(\000)p Fk(g)p Fv(\))f Fk(\024)h Fv(1)22 b Fk(\000)h Fs(V)2264 1921 y Fq(\034)2301 1931 y Fh(i)2332 1921 y Fp(\000)p Fq(\034)2431 1931 y Fh(i)p Fg(\000)p Fi(1)2242 2020 y Fq(\033)2291 1989 y Fh(\034)2325 2005 y(i)p Fg(\000)p Fi(1)2444 1989 y Fg(\000)p Fh(n)2547 2020 y Fq(!)2604 1974 y Fs(=C)2725 1989 y Fq(\025;\025)2843 1969 y Fg(0)2875 1974 y Fv(\()p Fs(\017)p Fv(\))17 b Fs(;)0 2230 y Fl(wher)-5 b(e)41 b Fs(C)358 2245 y Fq(\025;\025)476 2225 y Fg(0)508 2230 y Fv(\()p Fs(\017)p Fv(\))c Fs(>)h Fv(1)j Fl(dep)-5 b(ends)40 b(only)h(on)g(the)g(Lipschitz)f(c)-5 b(onstant)40 b(of)h Fs(')p Fl(,)h Fs(')2941 2194 y Fp(0)2969 2230 y Fl(.)61 b(This)41 b(dep)-5 b(endenc)g(e)0 2350 y(may)36 b(b)-5 b(e)36 b(r)-5 b(emove)g(d)35 b(if)g(we)g(c)-5 b(onsider)35 b Fs(i)28 b Fk(\025)g Fs(i)1566 2365 y Fr(0)1611 2350 y Fv(\()p Fs(\025;)17 b(\025)1811 2314 y Fp(0)1838 2350 y Fv(\))p Fl(.)0 2534 y Fx(Lemma)35 b(8.2)g(\(Relating)e(stopping) h(times)h(and)f(return)g(times\).)56 b Fl(L)-5 b(et)33 b Fs(\025)p Fl(,)h Fs(\025)3179 2498 y Fp(0)3240 2534 y Fl(b)-5 b(e)32 b(absolutely)0 2655 y(c)-5 b(ontinuous)43 b(pr)-5 b(ob)g(ability)42 b(me)-5 b(asur)g(es)42 b(on)h Fk(f)p Fv(\001)1712 2670 y Fq(!)1770 2655 y Fk(g)p Fl(,)g(with)g (densities)e Fs(')p Fl(,)j Fs(')2740 2619 y Fp(0)2811 2655 y Fl(in)f Fk(F)3023 2612 y Fr(+)3013 2685 y Fq(\014)3088 2655 y Fl(.)67 b(F)-8 b(or)43 b(e)-5 b(ach)42 b Fv(\000)f Fk(2)6 2783 y Fo(e)0 2808 y Fv(\004)66 2772 y Fq(\033)115 2742 y Fg(\000)p Fh(n)218 2772 y Fq(!)66 2834 y(i)276 2808 y Fl(,)35 b(we)g(have)h(for)f(al)5 b(l)36 b Fs(`)306 3053 y Fv(\()p Fs(\025)22 b Fk(\002)g Fs(\025)582 3012 y Fp(0)610 3053 y Fv(\))649 3072 y Fq(\033)698 3052 y Fg(\000)p Fh(n)801 3072 y Fq(!)858 3053 y Fv(\()p Fk(f)p Fs(\034)991 3068 y Fq(i)p Fr(+1)1124 3053 y Fk(\000)q Fs(\034)1246 3068 y Fq(i)1306 3053 y Fs(>)28 b(`)1453 3068 y Fr(0)1520 3053 y Fv(+)22 b Fs(`)p Fk(g)27 b(j)h Fv(\000\))1230 3213 y Fk(\024)g Fs(C)1406 3228 y Fq(\025;\025)1524 3208 y Fg(0)1556 3213 y Fv(\()p Fs(\017)p Fv(\)Leb)q(\()p Fk(f)p Fs(R)2001 3237 y Fq(\033)2050 3213 y Fh(\034)2084 3229 y(i)2114 3213 y Fi(+)p Fh(`)2195 3228 y Fi(0)2234 3213 y Fg(\000)p Fh(n)2337 3237 y Fq(!)2422 3213 y Fs(>)g(`)p Fk(g)p Fv(\))22 b Fk(\001)f Fv(Leb)q(\(\001)3013 3237 y Fq(\033)3062 3213 y Fh(\034)3096 3229 y(i)3127 3213 y Fi(+)p Fh(`)3208 3228 y Fi(0)3247 3213 y Fg(\000)p Fh(n)3350 3237 y Fq(!)3408 3213 y Fv(\))p Fs(:)0 3457 y Fl(wher)-5 b(e)43 b Fs(C)360 3472 y Fq(\025;\025)478 3452 y Fg(0)510 3457 y Fv(\()p Fs(\017)p Fv(\))g Fl(dep)-5 b(ends)42 b(on)i(the)f(Lipschitz)f(c)-5 b(onstants)42 b(of)h Fs(')p Fl(,)h Fs(')2578 3421 y Fp(0)2607 3457 y Fl(.)67 b(This)43 b(dep)-5 b(endenc)g(e)42 b(may)i(b)-5 b(e)0 3577 y(r)g(emove)g(d)35 b(if)g(we)h(c)-5 b(onsider)34 b Fs(i)28 b Fk(\025)g Fs(i)1227 3592 y Fr(0)1272 3577 y Fv(\()p Fs(\025;)17 b(\025)1472 3541 y Fp(0)1499 3577 y Fv(\))p Fl(.)0 3762 y Fx(Prop)s(osition)43 b(8.3)h(\(Join)m(t)e (return)g(time)h(asymptotics\).)56 b Fl(F)-8 b(or)40 b(every)f(p)-5 b(air)39 b Fs(\025)p Fl(,)i Fs(\025)3367 3725 y Fp(0)3434 3762 y Fl(of)e(abso-)0 3882 y(lutely)30 b(c)-5 b(ontinuous)29 b(pr)-5 b(ob)g(ability)28 b(me)-5 b(asur)g(es)28 b(on)g Fk(f)p Fv(\001)1915 3897 y Fq(!)1973 3882 y Fk(g)g Fl(having)g(densities)f Fs(')p Fl(,)j Fs(')2954 3846 y Fp(0)3010 3882 y Fl(in)e Fk(F)3207 3840 y Fr(+)3197 3912 y Fq(\014)3280 3882 y Fk(\\)7 b(L)3422 3846 y Fp(K)3483 3856 y Fh(!)3422 3907 y Fp(1)3567 3882 y Fl(ther)-5 b(e)0 4010 y(is)35 b Fs(C)178 4025 y Fq(\025;\025)296 4006 y Fg(0)328 4010 y Fv(\()p Fs(\017)p Fv(\))h Fl(so)f(that)h(for)f(e)-5 b(ach)35 b Fs(!)c Fk(2)d Fv(\012)1455 4025 y Fr(6)1536 4010 y Fl(and)35 b(al)5 b(l)37 b Fs(n)27 b(>)h(n)2124 4025 y Fr(6)2169 4010 y Fv(\()p Fs(!)t Fv(\))802 4300 y(\()p Fs(\025)22 b Fk(\002)h Fs(\025)1079 4259 y Fp(0)1107 4300 y Fv(\))1146 4319 y Fq(\033)1195 4299 y Fg(\000)p Fh(n)1297 4319 y Fq(!)1355 4300 y Fv(\()p Fk(f)p Fs(T)1502 4319 y Fq(\033)1551 4299 y Fg(\000)p Fh(n)1653 4319 y Fq(!)1739 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Fq(!)3518 5197 y Fk(g)g Fl(with)1847 5316 y Ft(50)p eop %%Page: 51 51 51 50 bop 0 100 a Fl(densities)34 b Fs(')p Fl(,)h Fs(')606 63 y Fp(0)670 100 y Fl(in)g Fk(F)874 57 y Fr(+)864 130 y Fq(\014)961 100 y Fk(\\)22 b(L)1118 63 y Fp(K)1179 73 y Fh(!)1118 124 y Fp(1)1235 100 y Fl(,)35 b(ther)-5 b(e)35 b(exists)g Fs(C)1893 115 y Fq(\025;\025)2011 95 y Fg(0)2043 100 y Fv(\()p Fs(\017)p Fv(\))p Fl(,)g(dep)-5 b(ending)34 b(on)i Fs(\025)f Fl(and)h Fs(\025)3178 63 y Fp(0)3241 100 y Fl(only)g(thr)-5 b(ough)0 228 y(the)35 b(Lipschitz)g(c)-5 b(onstants)35 b(of)g Fs(')p Fl(,)g Fs(')1338 192 y Fp(0)1366 228 y Fl(,)h(such)f(that,)g(for)h(e)-5 b(ach)35 b Fs(!)c Fk(2)d Fv(\012)2536 243 y Fr(9)2616 228 y Fl(and)36 b(al)5 b(l)36 b Fs(n)28 b Fk(\025)g Fs(n)3205 243 y Fr(9)3250 228 y Fv(\()p Fs(!)t Fv(\))p Fl(,)702 414 y Fo(\014)702 474 y(\014)736 499 y Fv(\()p Fs(F)853 458 y Fq(n)839 527 y(\033)888 507 y Fg(\000)p Fh(n)990 527 y Fq(!)1048 499 y Fv(\))1087 514 y Fp(\003)1132 499 y Fv(\()p Fs(\025)p Fv(\))22 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Fr(9)2837 1273 y Fv(\()p Fs(!)t Fv(\),)g(de\014ned)i(on)e(\012) 3604 1288 y Fr(9)3648 1273 y Fv(,)h(is)0 1392 y(suc)m(h)32 b(that,)f(on)f(the)h(one)h(hand,)f(for)g Fs(i)d Fk(\024)g Fs(n)1600 1356 y Fq(t)1666 1392 y Fv(\(where)j(0)d Fs(<)g(t)f(<)i Fv(1)p Fs(=)p Fv(8)h(will)f(b)s(e)i(\014xed)g(later)g(on\))f(and)h(all) 0 1512 y Fs(n)c Fk(\025)i Fs(n)253 1527 y Fr(9)297 1512 y Fv(\()p Fs(!)t Fv(\))1148 1603 y Fq(i)p Fp(\000)p Fr(1)1142 1633 y Fo(X)1145 1845 y Fq(j)t Fr(=0)1285 1727 y Fv(\()p Fs(n)1384 1742 y Fr(6)1429 1727 y Fv(\()p Fs(\033)1529 1683 y Fp(\000)p Fq(n)p Fr(+)p Fq(T)1748 1707 y Fh(j)o(;\033)1842 1692 y Fg(\000)p Fh(n)1944 1707 y(!)2000 1727 y Fs(!)t Fv(\)\))2144 1686 y Fr(1)p Fq(=)p Fr(8)2296 1727 y Fk(\024)38 b Fv(^)-60 b Fs(\032n)2513 1686 y Fr(1)p Fq(=)p Fr(8)0 2043 y Fv(for)36 b(the)g(\\go)s(o)s(d")f(atoms)g(of)h(the)g(partition) 1689 2018 y(^)1680 2043 y(\004)1746 2007 y Fq(\033)1795 1977 y Fg(\000)p Fh(n)1898 2007 y Fq(!)1746 2069 y(i)1956 2043 y 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Fo(\022)705 1462 y Fk(K)781 1477 y Fq(!)838 1462 y Fs(;)98 b Fv(max)882 1531 y Fq(n)p Fp(\024)p Fq(n)1042 1541 y Fi(10)1116 1531 y Fr(\()p Fq(!)s Fr(\))1247 1462 y Fs(e)1293 1421 y Fr(\()p Fq(n)1373 1431 y Fi(9)1413 1421 y Fr(\()p Fq(!)s Fr(\))1557 1368 y Fi(1)p 1540 1377 V 1540 1413 a(16)1625 1421 y Fq(=)p 1666 1361 63 4 v(C)1728 1431 y Fi(1)1767 1421 y Fr(\))1803 1462 y Fk(\002)1181 1598 y Fo(\014)1181 1658 y(\014)1181 1718 y(\014)1181 1778 y(\014)1214 1607 y(Z)1331 1743 y Fs(')1396 1758 y Fq(!)1476 1743 y Fk(\016)22 b Fs(F)1626 1702 y Fq(n)1612 1771 y(\033)1661 1751 y Fg(\000)p Fh(n)1763 1771 y Fq(!)1843 1743 y Fk(\001)g Fs( )1958 1761 y Fq(\033)2007 1742 y Fg(\000)p Fh(n)2109 1761 y Fq(!)2183 1743 y Fs(d)p Fv(Leb)h Fk(\000)2519 1607 y Fo(Z)2635 1743 y Fs(')2700 1758 y Fq(!)2758 1743 y Fs(d\026)2870 1758 y Fq(!)2944 1607 y Fo(Z)3060 1743 y Fs( )3125 1761 y Fq(\033)3174 1742 y Fg(\000)p Fh(n)3277 1761 y Fq(!)3351 1743 y Fs(d)p Fv(Leb)3565 1598 y Fo(\014)3565 1658 y(\014)3565 1718 y(\014)3565 1778 y(\014)3615 1602 y(\023)3705 1743 y Fs(:)0 2042 y Fv(The)37 b(claim)e(on)i(the)g(distribution)g(of)g Fs(C)7 b Fv(\()p Fs(!)t Fv(\))36 b(is)h(pro)m(v)m(ed)g(as)g(in)g (Corollary)e(7.10.)53 b(This)37 b(giv)m(es)g(\(8.1\))0 2170 y(for)k Fs( )i Fk(2)d(F)456 2127 y Fr(+)446 2200 y Fq(\014)548 2170 y Fk(\\)27 b(L)710 2133 y Fp(K)771 2143 y Fh(!)710 2194 y Fp(1)827 2170 y Fv(.)65 b(F)-8 b(or)40 b(real-v)-6 b(alued)41 b(non)g(negativ)m(e)f Fs( )j Fk(2)d(F)2517 2125 y Fp(K)2578 2135 y Fh(!)2507 2200 y Fq(\014)2633 2170 y Fv(,)h(remark)f(that)3294 2143 y(~)3274 2170 y Fs( )3339 2185 y Fq(!)3436 2170 y Fv(=)g Fs( )3618 2185 y Fq(!)3702 2170 y Fv(+)0 2316 y(\()p Fs(C)110 2331 y Fq( )192 2316 y Fv(+)22 b(1\))p Fk(K)456 2331 y Fq(!)546 2316 y Fv(b)s(elongs)33 b(to)g Fk(F)1107 2273 y Fr(+)1097 2346 y Fq(\014)1194 2316 y Fk(\\)22 b(L)1351 2280 y Fp(K)1412 2290 y Fh(!)1351 2341 y Fp(1)1501 2316 y Fv(and)33 b(apply)g(the)g(ab)s(o)m(v)m(e)g(estimate) f(to)2962 2290 y(~)2942 2316 y Fs( )s Fv(.)44 b(Complex-v)-6 b(alued)0 2439 y(functions)35 b(are)e(decomp)s(osed)h(as)f(in)g (Corollary)f(7.10.)98 b Fe(\003)1599 2739 y Fw(References)0 2898 y Ft([A])176 b(Alv)n(es)31 b(J.F.,)g Fn(SRB)i(me)l(asur)l(es)g (for)g(nonhyp)l(erb)l(olic)f(systems)h(with)g(multidimensional)g(exp)l (ansion)p Ft(,)e(Ph.D.)287 2998 y(thesis)e(\(1997\),)g(to)f(app)r(ear)h (Annales)h(scien)n(t.)e(Ecole)g(normale)g(sup.)h(\(4\).)0 3098 y([Ba])137 b(Bahnm)r(\177)-45 b(uller)21 b(J.,)g Fn(Pesin)-9 b('s)24 b(entr)l(opy)e(formula)g(of)h(exp)l(anding)g(maps)p Ft(,)d(Random)g(Comput.)f(Dynam.)i Fd(4)g Ft(\(1996\),)287 3197 y(99{108.)0 3297 y([BKS])66 b(Baladi)25 b(V.,)f(Kondah)g(A.,)g (and)h(Sc)n(hmitt)d(B.,)i Fn(R)l(andom)i(c)l(orr)l(elations)g(for)g (smal)t(l)g(p)l(erturb)l(ations)h(of)f(exp)l(and-)287 3397 y(ing)31 b(maps)p Ft(,)c(Random)i(&)f(Computational)j(Dynamics)d Fd(4)g Ft(\(1996\),)h(179{204.)0 3496 y([BV])116 b(Baladi)39 b(V.)e(and)h(Viana)h(M.,)f Fn(Str)l(ong)h(sto)l(chastic)h(stability)g 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b(M.)f(and)i(Y)-7 b(oung)30 b(L.-S.,)e Fn(A)n(bsolutely)j(c)l(ontinuous)g(invariant)h(me)l(asur)l (es)e(and)h(r)l(andom)f(p)l(ertur-)287 4394 y(b)l(ations)g(for)f(c)l (ertain)i(one-dimensional)e(maps)p Ft(,)e(Ergo)r(dic)h(Theory)f(Dynam.) f(Systems)h Fd(12)g Ft(\(1992\),)h(13{37.)0 4493 y([BL])126 b(Blokh)39 b(A.M.)e(and)i(Lyubic)n(h)f(M.Y)-7 b(u.,)39 b Fn(Me)l(asur)l(able)g(dynamics)h(of)g(S-unimo)l(dal)e(maps)h(of)h (the)h(interval)p Ft(,)287 4599 y(Ann.)28 b(Scien)n(t.)779 4579 y(\023)771 4599 y(Ecole)g(Norm.)g(Sup.)g Fd(24)g Ft(\(1991\),)h(545{573.)0 4698 y([Bo])137 b(Bogensc)n(h)r(\177)-45 b(utz)27 b(T.,)g Fn(Sto)l(chastic)j(stability)g(of)f(e)l(quilibrium)i (states)p Ft(,)d(Random)f(Comput.)g(Dynam.)f Fd(4)h Ft(\(1996\),)287 4798 y(85{98.)0 4898 y([BoL])84 b(P)-7 b(.)39 b(Bougerol)h(and)h(J.)e (Lacroix,)k Fn(Pr)l(o)l(ducts)e(of)g(r)l(andom)f(matric)l(es)i(with)g (applic)l(ations)e(to)i(Scrh\177)-44 b(odinger)287 4997 y(op)l(er)l(ators)p Ft(,)27 b(Birkh\177)-42 b(auser,)28 b(Boston,)h(1985.)0 5097 y([BLS])79 b(Bruin)30 b(H.,)f(Luzzatto)g(S.,)g (and)h(v)-5 b(an)30 b(Strien)f(S.,)g Fn(De)l(c)l(ay)i(of)g(c)l(orr)l (elations)h(in)g(one-dimensional)f(dynamics)p Ft(,)287 5197 y(Preprin)n(t)c(\(1999\).)1847 5316 y(52)p eop %%Page: 53 53 53 52 bop 0 100 a Ft([Bu1])90 b(Buzzi)37 b(J.,)j Fn(SRB)g(me)l(asur)l (es)f(for)h(r)l(andom)e(L)l(asota-Y)-7 b(orke)39 b(maps)p Ft(,)g(Preprin)n(t)f(\(1998\),)j(to)d(app)r(ear)g(T)-7 b(rans.)287 199 y(Amer.)27 b(Math.)h(So)r(c.)0 299 y([Bu2])90 b(Buzzi)25 b(J.,)g Fn(Exp)l(onential)j(de)l(c)l(ay)f(of)h(c)l(orr)l (elations)f(for)h(r)l(andom)f(L)l(asota-Y)-7 b(orke)27 b(maps)p Ft(,)e(Preprin)n(t)g(\(1998\),)h(to)287 399 y(app)r(ear)j(Comm.)e(Math.)h(Ph)n(ys.)0 498 y([Co1])94 b(Collet)28 b(P)-7 b(.,)26 b Fn(Er)l(go)l(dic)j(pr)l(op)l(erties)h(of)g (some)f(unimo)l(dal)g(mappings)g(of)h(the)g(interval)p Ft(,)e(Preprin)n(t)f Fd(11)g Ft(Institute)287 598 y(Mittag)i(Le\017er)e (\(unpublished\))i(\(1984\).)0 697 y([Co2])94 b(Collet)29 b(P)-7 b(.,)27 b Fn(A)k(r)l(emark)g(ab)l(out)f(uniform)h(de-c)l(orr)l (elation)f(pr)l(efactors)p Ft(,)e(Preprin)n(t)g(\(1999\).)0 797 y([He])138 b(H.)24 b(Hennion,)h Fn(Limit)i(the)l(or)l(ems)g(for)f (pr)l(o)l(ducts)g(of)h(p)l(ositive)g(r)l(andom)f(matric)l(es)p Ft(,)f(Ann.)f(Probab.)g Fd(25)h Ft(\(1997\),)287 897 y(1545{1587.)0 996 y([KaK])65 b(Katok)35 b(A.)f(and)i(Kifer)e(Y.,)i Fn(R)l(andom)g(p)l(erturb)l(ations)h(of)g(tr)l(ansformations)f(of)h(an) g(interval)p Ft(,)g(J.)d(Analyse)287 1096 y(Math.)27 b Fd(47)h Ft(\(1986\),)h(193{237.)0 1196 y([KhK])60 b(Khanin)23 b(K.)e(and)h(Kifer)g(Y.,)h Fn(Thermo)l(dynamic)h(formalism)h(for)f(r)l (andom)g(tr)l(ansformations)h(and)g(statistic)l(al)287 1295 y(me)l(chanics)p Ft(,)42 b(Sinai's)e(Mosco)n(w)g(Seminar)f(on)h (Dynamical)f(Systems,)i(Math.)e(So)r(c.)g(T)-7 b(rans.)40 b(Ser.)e(2,)k(171,)287 1395 y(Amer.)27 b(Math.)h(So)r(c.,)f(Pro)n (vidence,)g(RI,)i(1996,)g(pp.)f(107{140.)0 1494 y([Ki])150 b(Kifer)29 b(Y.,)g Fn(Limit)k(the)l(or)l(ems)e(for)h(r)l(andom)f(tr)l (ansformations)g(and)h(pr)l(o)l(c)l(esses)f(in)h(r)l(andom)f(envir)l (onments)p Ft(,)287 1594 y(T)-7 b(rans.)28 b(Amer.)g(Math.)f(So)r(c.)i Fd(350)e Ft(\(1998\),)i(1481{1518.)0 1694 y([L)-7 b(Y])130 b(Ledrappier)27 b(F.)f(and)h(Y)-7 b(oung)27 b(L.-S.,)f Fn(Entr)l(opy)i(formula)h(for)g(r)l(andom)f(tr)l(ansformations)p Ft(,)e(Probab.)h(Theory)287 1793 y(Related)h(Fields)i Fd(80)e Ft(\(1988\),)h(217{240.)0 1893 y([PY])118 b(P)n(ollicott)23 b(M.)e(and)h(Y)-7 b(uri)23 b(M.,)e Fn(Dynamic)l(al)j(systems)h(and)f (er)l(go)l(dic)h(the)l(ory)p Ft(,)d(London)g(Mathematical)h(So)r(ciet)n (y)287 1993 y(Studen)n(t)k(T)-7 b(exts)28 b(40,)h(Cam)n(bridge)f(Univ)n (ersit)n(y)h(Press,)f(Cam)n(bridge,)g(1998.)0 2092 y([S])192 b(Seneta)31 b(E.,)g Fn(Nonne)l(gative)36 b(matric)l(es)f(and)e(Markov)h (chains)p Ft(,)e(Second)g(Edition)g(\(Springer)h(Series)f(in)g(Sta-)287 2192 y(tistics\),)d(Springer)g(V)-7 b(erlag,)29 b(New)f(Y)-7 b(ork-Berlin,)29 b(1981.)0 2291 y([V])176 b(Viana)27 b(M.,)d Fn(Multidimensional)k(nonhyp)l(erb)l(olic)f(attr)l(actors)p Ft(,)f(Inst.)g(Hautes)g(Etudes)f(Sci.)g(Publ.)g(Math.)g Fd(85)287 2391 y Ft(\(1997\),)k(63{96.)0 2491 y([Y)-7 b(o1])99 b(Y)-7 b(oung)22 b(L.-S.,)f Fn(Statistic)l(al)k(pr)l(op)l (erties)f(of)f(systems)h(with)h(some)e(hyp)l(erb)l(olicity)h(including) g(c)l(ertain)h(bil)t(liar)l(ds)p Ft(,)287 2590 y(Ann.)j(of)h(Math.)e (\(2\))i Fd(147)e Ft(\(1998\),)i(585{650.)0 2690 y([Y)-7 b(o2])99 b(Y)-7 b(oung)29 b(L.-S.,)e Fn(R)l(e)l(curr)l(enc)l(e)32 b(times)f(and)f(r)l(ates)h(of)g(mixing)p Ft(,)e(Israel)g(J.)g(Math.)e Fd(110)h Ft(\(1999\),)h(153{188.)120 2909 y Fu(V.)j(Baladi:)130 3009 y(Section)g(de)f(Ma)-5 b(th)841 3003 y(\023)841 3009 y(ema)g(tiques,)33 b(CH-1211)g(Genev)-7 b(a)33 b(24,)f (SWITZERLAND)130 3108 y(Current)g(address:)121 3208 y(Labora)-5 b(toire)21 b(de)h(Ma)-5 b(th)1015 3202 y(\023)1015 3208 y(ema)g(tiques,)26 b(UMR)e(8628)f(CNRS,)g(Universit)2631 3202 y(\023)2631 3208 y(e)h(P)-7 b(aris-Sud,)25 b(F-91405)f(Orsa)-5 b(y,)130 3308 y(FRANCE)120 3407 y Fn(E-mail)30 b(addr)l(ess)6 b Ft(:)41 b Fa(Viviane.Baladi@math.)q(u-p)q(su)q(d.f)q(r)120 3626 y Fu(M.)31 b(Benedicks:)130 3726 y(Dep)-5 b(ar)g(tment)31 b(of)g(Ma)-5 b(thema)g(tics,)34 b(R)n(o)n(y)-5 b(al)31 b(Institute)h(of)f(Technology,)h(S-100)h(44)e(Stockholm,)130 3826 y(SWEDEN)120 3925 y Fn(E-mail)f(addr)l(ess)6 b Ft(:)41 b Fa(michaelb@math.kth.se)120 4144 y Fu(V.)32 b(Ma)n(ume-Deschamps:)130 4244 y(Section)g(de)f(Ma)-5 b(th)841 4238 y(\023)841 4244 y(ema)g(tiques,)33 b(CH-1211)g(Genev)-7 b(a)33 b(24,)f (SWITZERLAND)130 4344 y(Current)g(address:)130 4443 y(Labora)-5 b(toire)31 b(de)g(Topologie,)f(Universit)1711 4437 y(\023)1711 4443 y(e)k(de)d(Bour)n(gogne,)g(F-21078)h(Dijon,)g(FRANCE)120 4543 y Fn(E-mail)e(addr)l(ess)6 b Ft(:)41 b Fa(Veronique.Maume@u-bo)q (urg)q(og)q(ne.)q(fr)1847 5316 y Ft(53)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9912020954103--