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b(example,)e(Chen,)g(Go) s(o)s(dman,)h(and)f(Zame)-20 4947 y([3)q(])j(exhibited)d(a)j(\014xed-p) s(oin)m(t)e(with)g(a)i(densit)m(y)e(on)i([0)p Fl(;)15 b Fo(1])32 b(that)f(is)e(not)i(con)m(tin)m(uous)f(on)g(a)h(dense)f (subset)g(of)g([0)p Fl(;)15 b Fo(1].)121 5060 y(The)32 b(presen)m(t)g(pap)s(er)e(deals)i(with)e(densit)m(y)h(appro)m(ximation) g(and)g(exact)i(sim)m(ulation)d(from)i(a)g(class)f(of)h(\014xed-)1938 5411 y(2)p eop %%Page: 3 3 3 2 bop -20 -21 a Fo(p)s(oin)m(ts)32 b(where)g(a)h(\014rst)e(general)i (restriction)e(is)h Fl(K)k Fm(\025)28 b Fo(2.)48 b(W)-8 b(e)33 b(hop)s(e)f(to)i(rep)s(ort)d(on)i(progress)f(in)f(the)i(case)h Fl(K)h Fo(=)29 b(1)-20 92 y(elsewhere.)67 b(W)-8 b(e)40 b(ha)m(v)m(e)h(to)f(in)m(tro)s(duce)e(a)i(few)f(restrictions)f(on)h (the)g(class)g(of)h(\014xed-p)s(oin)m(t)e(equations)h(in)f(order)-20 205 y(to)f(guaran)m(tee)h(algorithmic)d(tractabilit)m(y)-8 b(.)59 b(As)37 b(sho)m(wn)e(b)s(elo)m(w,)j(all)d(imp)s(ortan)m(t)g(kno) 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Fo(\026)-48 b Fl(g)s Fo(\()p Fl(v)s Fo(\))21 b(+)f Fl(\024)1234 1427 y Fj( )1322 1469 y Fk(K)5 b Fg(\000)p Fs(1)1333 1496 y Fj(X)1337 1691 y Fk(r)r Fs(=1)1476 1583 y Fo(\()p Fl(v)1555 1597 y Fk(r)1609 1583 y Fo(ln)14 b Fl(v)1744 1597 y Fk(r)1782 1583 y Fo(\))20 b(+)1928 1427 y Fj( )2000 1583 y Fo(1)h Fm(\000)2157 1469 y Fk(K)5 b Fg(\000)p Fs(1)2168 1496 y Fj(X)2172 1691 y Fk(r)r Fs(=1)2326 1583 y Fl(v)2370 1597 y Fk(r)2408 1427 y Fj(!)2495 1583 y Fo(ln)2586 1427 y Fj( )2658 1583 y Fo(1)21 b Fm(\000)2815 1469 y Fk(K)5 b Fg(\000)p Fs(1)2826 1496 y Fj(X)2830 1691 y Fk(r)r Fs(=1)2984 1583 y Fl(v)3028 1597 y Fk(r)3066 1427 y Fj(!!)3779 1583 y Fo(\(13\))-20 1869 y(where)41 b Fl(\024;)15 b(\024)398 1836 y Fg(0)467 1869 y Fl(>)43 b Fo(0)f(are)g(normalization)e(constan)m(ts)j(and)h (\026)-48 b Fl(g)t Fo(\()p Fl(v)s Fo(\))42 b(is)f(either)g(1)h(or)f Fl(v)k Fo(or)d Fl(v)s Fo(\(1)29 b Fm(\000)e Fl(v)s Fo(\))42 b(dep)s(ending)d(on)-20 1982 y(the)j(application.)71 b(W)-8 b(e)42 b(treat)g(the)f(cases)k(\026)-48 b Fl(g)t Fo(\()p Fl(v)s Fo(\))44 b(=)f(1)e(or)g(=)i Fl(v)s Fo(,)h(the)d(third)f (case)i(can)f(b)s(e)g(co)m(v)m(ered)h(with)e(sligh)m(t)-20 2095 y(mo)s(di\014cations.)f(W)-8 b(e)32 b(ha)m(v)m(e)1275 2326 y(Hess\()p Fl(g)s Fo(;)15 b Fl(v)s Fo(\))1658 2340 y Fk(ij)1746 2326 y Fo(=)25 b Fl(\024)1909 2198 y Fj(\022)2020 2265 y Fo(1)p 1987 2306 113 4 v 1987 2389 a Fl(v)2031 2403 y Fk(K)2129 2326 y Fo(+)20 b Fl(\016)2260 2340 y Fk(ij)2344 2265 y Fo(1)p 2331 2306 73 4 v 2331 2389 a Fl(v)2375 2403 y Fk(i)2413 2198 y Fj(\023)-20 2600 y Fo(with)32 b Fl(v)234 2614 y Fk(K)331 2600 y Fo(=)c(1)22 b Fm(\000)590 2532 y Fj(P)686 2558 y Fk(K)5 b Fg(\000)p Fs(1)686 2627 y Fk(r)r Fs(=1)859 2600 y Fl(v)903 2614 y Fk(r)974 2600 y Fo(and)32 b Fl(\016)1193 2614 y Fk(ij)1286 2600 y Fo(denoting)g(Kronec)m(k)m(er's)i(sym)m(b)s(ol.)45 b(Using)32 b(the)h(relation)3263 2532 y Fj(P)3358 2558 y Fk(K)3358 2627 y(r)r Fs(=1)3512 2559 y Fk(@)t(f)3587 2567 y Fe(r)p 3512 2579 110 4 v 3514 2631 a Fk(@)t(u)3596 2643 y Fe(l)3660 2600 y Fo(=)c(0)k(w)m(e)-20 2713 y(obtain)d(for)g(all) g(1)25 b Fm(\024)g Fl(l)i Fm(\024)e Fl(d)p Fo(:)978 2865 y Fj(\034)1057 2931 y Fl(@)5 b(f)1155 2945 y Fk(r)p 1057 2972 136 4 v 1059 3055 a Fl(@)g(u)1164 3070 y Fk(l)1202 2993 y Fl(;)15 b Fo(Hess)q(\()p Fl(g)s Fo(;)g Fl(f)10 b Fo(\()p Fm(\001)p Fo(\)\))1739 2931 y Fl(@)5 b(f)1837 2945 y Fk(r)p 1739 2972 V 1741 3055 a Fl(@)g(u)1846 3070 y Fk(l)1887 2865 y Fj(\035)1980 2993 y Fo(=)25 b Fl(\024)2177 2879 y Fk(K)2144 2906 y Fj(X)2147 3101 y Fk(r)r Fs(=1)2319 2931 y Fo(1)p 2300 2972 83 4 v 2300 3055 a Fl(f)2345 3069 y Fk(r)2408 2865 y Fj(\022)2484 2931 y Fl(@)5 b(f)2582 2945 y Fk(r)p 2484 2972 136 4 v 2486 3055 a Fl(@)g(u)2591 3070 y Fk(l)2630 2865 y Fj(\023)2697 2887 y Fs(2)2752 2993 y Fl(:)121 3278 y Fo(W)-8 b(e)25 b(pro)s(ceed)f(b)m(y)f(recalling) f(the)i(equations)f(\(4\))i(for)e(the)h(limit)e(la)m(ws)h(of)h(the)f (in)m(ternal)g(path)g(lengths)g(of)h(random)-20 3390 y Fl(m)p Fo(-ary)31 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Fs(\()p Fk(m)p Fg(\000)p Fs(1\))3770 4640 y Fo(:=)30 b(1)-20 4753 y(on)h Fm(f)6 b Fo(~)-51 b Fl(u)204 4772 y Fs(\()p Fk(j)t Fg(\000)p Fs(1\))411 4753 y Fl(<)25 b(u)559 4767 y Fs(1)624 4753 y Fl(<)30 b Fo(~)-50 b Fl(u)772 4772 y Fs(\()p Fk(j)t Fs(\))863 4753 y Fm(g)31 b Fo(w)m(e)g(ha)m(v)m(e)1375 5000 y Fl(@)5 b(f)1473 5014 y Fk(r)p 1371 5040 145 4 v 1371 5124 a Fl(@)g(u)1476 5138 y Fs(1)1551 5061 y Fo(=)1647 4875 y Fj(8)1647 4957 y(<)1647 5120 y(:)1840 4948 y Fo(1)83 b Fl(r)28 b Fo(=)d Fl(j)1769 5061 y Fm(\000)p Fo(1)83 b Fl(r)28 b Fo(=)d Fl(j)h Fo(+)20 b(1)1840 5174 y(0)83 b(otherwise)1938 5411 y(9)p eop %%Page: 10 10 10 9 bop -20 -21 a Fo(for)30 b Fl(j)h Fo(=)25 b(1)p Fl(;)15 b(:)g(:)g(:)33 b(;)15 b(m)20 b Fm(\000)g Fo(1.)41 b(This)29 b(implies)512 262 y Fl(\024)614 148 y Fk(m)580 175 y Fj(X)583 370 y Fk(r)r Fs(=1)754 200 y Fo(1)p 736 241 83 4 v 736 324 a Fl(f)781 338 y Fk(r)844 133 y Fj(\022)925 200 y Fl(@)5 b(f)1023 214 y Fk(r)p 920 241 145 4 v 920 324 a Fl(@)g(u)1025 338 y Fs(1)1075 133 y Fj(\023)1142 156 y Fs(2)1265 262 y Fo(=)82 b Fl(\024)1485 148 y Fk(m)p Fg(\000)p Fs(1)1496 175 y Fj(X)1501 371 y Fk(j)t Fs(=1)1654 262 y Fp(1)1706 280 y Fg(f)5 b Fs(~)-40 b Fk(u)1782 294 y Fi(\()p Fe(j)s Fd(\000)p Fi(1\))1941 280 y Fk()g(k)e Fo(+)d(1.)41 b(Th)m(us)29 b(w)m(e)i(c)m(ho)s(ose)825 1502 y Fl(p)871 1516 y Fs(0)936 1502 y Fo(:=)25 b Fl(k)e Fo(+)d(1)p Fl(;)107 b(D)1470 1516 y Fs(2)1510 1502 y Fo(\()p Fl(p;)15 b(c)p Fo(\))26 b(:=)2225 1441 y(2)p Fl(c)2309 1408 y Fk(k)r Fs(+1)p Fg(\000)p Fk(p)p 1862 1482 1033 4 v 1862 1565 a Fo(\()p Fl(k)e Fo(+)c(1)h Fm(\000)f Fl(p)p Fo(\))p Fl(B)5 b Fo(\()p Fl(k)23 b Fo(+)d(1)p Fl(;)15 b(k)24 b Fo(+)c(1\))2905 1502 y Fl(:)-20 1741 y Fo(Ad)30 b(\(7\):)42 b(Ev)-5 b(aluating)29 b(a)i(b)s(eta)g(in)m (tegral)f(w)m(e)h(easily)e(obtain)977 1980 y Fl(p)1023 1994 y Fs(1)1088 1980 y Fo(:=)c Fl(k)e Fo(+)d(1)p Fl(;)107 b(M)1635 1994 y Fk(p)1700 1980 y Fo(:=)1831 1918 y Fl(B)5 b Fo(\()p Fl(k)23 b Fo(+)d(1)h Fm(\000)f Fl(p;)15 b(k)23 b Fo(+)d(1)h Fm(\000)f Fl(p)p Fo(\))p 1831 1959 912 4 v 1988 2042 a Fl(B)5 b Fo(\()p Fl(k)23 b Fo(+)d(1)p Fl(;)15 b(k)24 b Fo(+)c(1\))2753 1980 y Fl(:)-20 2218 y Fo(Ad)30 b(\(12\):)43 b(Denote)1163 2409 y Fl(G)1235 2423 y Fk(n)1307 2409 y Fo(=)25 b Fm(f)p Fl(u)h Fm(2)e Fo([0)p Fl(;)15 b Fo(1])1791 2371 y Fs(2)p Fk(k)r Fs(+1)1987 2409 y Fo(:)25 b Fl(u)2089 2423 y Fk(n)2161 2409 y Fo(=)g(med\()p Fl(u)p Fo(\))p Fm(g)-20 2599 y Fo(for)30 b Fl(n)25 b Fo(=)g(1)p Fl(;)15 b(:)g(:)g(:)32 b(;)15 b Fo(2)p Fl(k)25 b Fo(+)20 b(1.)41 b(Then)29 b(with)g(the)i(notation)f(in)f(\(8\),)j(\(10\))g(w)m (e)f(obtain)f(on)g Fl(G)2925 2566 y Fg(0)2925 2622 y Fk(n;n)459 2886 y Fl(\015)5 b Fo(\()h(~)-51 b Fl(u)p Fo(\))26 b(=)160 b(inf)755 2951 y Fk(u)796 2959 y Fe(n)838 2951 y Fg(2)p Fk(G)940 2959 y Fe(n;n)1040 2951 y Fs(\()5 b(~)-40 b Fk(u)p Fs(\))1151 2886 y Fl(\024)1267 2772 y Fs(2)1219 2799 y Fj(X)1222 2994 y Fk(r)r Fs(=1)1393 2824 y Fo(1)p 1375 2865 83 4 v 1375 2948 a Fl(f)1420 2962 y Fk(r)1483 2757 y Fj(\022)1568 2824 y Fl(@)5 b(f)1666 2838 y Fk(r)p 1559 2865 153 4 v 1559 2948 a Fl(@)g(u)1664 2962 y Fk(n)1722 2757 y Fj(\023)1789 2780 y Fs(2)1853 2886 y Fo(=)160 b(inf)1949 2951 y Fk(u)1990 2959 y Fe(n)2033 2951 y Fg(2)p Fk(G)2135 2959 y Fe(n;n)2234 2951 y Fs(\()5 b(~)-40 b Fk(u)q Fs(\))2345 2886 y Fl(\024)2412 2757 y Fj(\022)2516 2824 y Fo(1)p 2490 2865 100 4 v 2490 2948 a Fl(u)2542 2962 y Fk(n)2619 2886 y Fo(+)2825 2824 y(1)p 2720 2865 256 4 v 2720 2948 a(1)20 b Fm(\000)g Fl(u)2928 2962 y Fk(n)2985 2757 y Fj(\023)3077 2886 y Fm(\025)25 b Fo(4)p Fl(\024;)-20 3152 y Fo(whic)m(h)k(implies)1182 3424 y(\000)c(=)1360 3310 y Fs(2)p Fk(k)r Fs(+1)1376 3337 y Fj(X)1375 3532 y Fk(n)p Fs(=1)1539 3300 y Fj(Z)1590 3506 y Fk(G)1645 3487 y Fd(0)1645 3523 y Fe(n;n)1887 3362 y Fo(1)p 1774 3403 273 4 v 1774 3490 a Fl(c)1813 3464 y Fs(1)p Fk(=)p Fs(2)1923 3490 y Fo(\()6 b(~)-51 b Fl(u)p Fo(\))2056 3424 y Fl(d)6 b Fo(~)-51 b Fl(u)25 b Fm(\024)2286 3362 y Fo(2)p Fl(k)f Fo(+)c(1)p 2286 3403 253 4 v 2325 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Fm(8)15 b Fl(")26 b(>)f Fo(0)15 b Fm(8)g Fl(d)25 b Fm(\025)g Fo(1)15 b Fm(8)g Fl(x)26 b Fm(\025)f Fo(1)h(:)f(\(ln)14 b Fl(x)p Fo(\))2267 5108 y Fk(d)2333 5146 y Fm(\024)2444 5084 y Fl(d)p Fo(!)p 2439 5125 83 4 v 2439 5208 a Fl(")2481 5182 y Fk(d)2532 5146 y Fl(x)2584 5108 y Fk(")2621 5146 y Fl(:)1133 b Fo(\(15\))1915 5411 y(11)p eop %%Page: 12 12 12 11 bop -20 -21 a Fo(Ad)30 b(\(5\):)42 b(Using)30 b(the)g(inequalit)m (y)f(\(15\))j(with)d Fl(")d Fo(=)f(1)p Fl(=d)31 b Fo(w)m(e)g(obtain)829 283 y Fl(\025)882 246 y Fk(d)937 100 y Fj(0)937 264 y(@)1027 197 y(\\)1017 394 y Fk(r)r Fg(6)p Fs(=)p Fk(j)1138 283 y Fm(f)p Fl(f)1228 297 y Fk(r)1291 283 y Fm(\024)25 b Fl(c=t)p Fm(g)1549 100 y Fj(1)1549 264 y(A)1713 283 y Fm(\024)83 b Fl(\025)1920 246 y Fk(d)1960 283 y Fo(\()p Fm(f)p Fl(f)2085 297 y Fk(r)2149 283 y Fm(\024)25 b Fl(c=t)p Fm(g)p Fo(\))1713 626 y(=)1914 512 y Fk(d)1867 540 y Fj(X)1871 735 y Fk(j)t Fs(=1)2148 565 y Fo(1)p 2023 605 296 4 v 2023 688 a(\()p Fl(j)h Fm(\000)20 b Fo(1\)!)2343 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4756 y(p)p 2001 4756 V 75 x Fo(32)p 1925 4872 167 4 v 1927 4959 a Fl(\015)1979 4933 y Fs(1)p Fk(=)p Fs(2)2102 4893 y Fm(j)p Fl(t)p Fm(j)2185 4855 y Fg(\000)p Fs(1)p Fk(=)p Fs(2)2350 4893 y Fl(;)108 b(t)25 b Fm(2)g Fo(R)p Fl(:)1060 b Fo(\(19\))1915 5411 y(13)p eop %%Page: 14 14 14 13 bop -20 -21 a Fp(Pro)s(of:)77 b Fo(Com)m(bine)29 b(Lemmas)h(2.2)i(and)d(2.3)j(in)d(Fill)f(and)i(Janson)g([11)q(].)p 3870 -21 71 71 v -20 225 a(Estimates)h(for)g(exp)s(onen)m(tial)f(in)m (tegrals)h(as)g(in)e(Lemma)i(3.2)h(are)g(w)m(ell-kno)m(wn)e(in)f (analytic)i(n)m(um)m(b)s(er)e(theory)j(The)-20 262 y Fm(p)p 56 262 91 4 v 76 x Fo(32)f(ma)m(y)g(b)s(e)f(replaced)g(b)m(y)g (8)h(\(T)-8 b(enen)m(baum)30 b([42)q(,)h(Lemma)f(4.4]\).)-20 563 y Fp(Pro)s(of)38 b(of)f(Theorem)f(3.1:)i Fo(Ad)32 b(\(16\):)46 b(With)31 b Fl(W)13 b Fo(\()p Fl(u)p Fo(\))29 b(:=)2030 495 y Fj(P)2126 521 y Fk(K)5 b Fg(\000)p Fs(1)2126 590 y Fk(r)r Fs(=1)2300 563 y Fl(x)2352 577 y Fk(r)2390 563 y Fl(f)2435 577 y Fk(r)2472 563 y Fo(\()p 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Fs(1)2743 921 y Fl(;)15 b(:)g(:)g(:)32 b(;)15 b(x)3012 935 y Fk(K)3080 921 y Fo(\))p Fl(:)639 b Fo(\(20\))-20 1167 y(It)31 b(is)e(su\016cien)m (t)h(to)h(obtain)f(a)h(b)s(ound)d(for)i(the)h(inner)d(in)m(tegral.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)618 1253 y Fj(\014)618 1307 y(\014)618 1362 y(\014)618 1416 y(\014)618 1471 y(\014)649 1288 y(Z)699 1494 y Fs([0)p Fk(;)p Fs(1])829 1475 y Fe(d)884 1412 y Fo(exp\()p Fl(itW)13 b Fo(\()p Fl(u)p Fo(\)\))i Fl(du)1492 1253 y Fj(\014)1492 1307 y(\014)1492 1362 y(\014)1492 1416 y(\014)1492 1471 y(\014)1549 1412 y Fm(\024)1676 1298 y Fg(1)1646 1325 y Fj(X)1645 1520 y Fk(n)p Fs(=1)1793 1288 y Fj(Z)1844 1494 y Fk(G)1899 1471 y Fd(0)1899 1518 y Fe(n;l)1999 1253 y Fj(\014)1999 1307 y(\014)1999 1362 y(\014)1999 1416 y(\014)1999 1471 y(\014)2030 1288 y(Z)2080 1494 y Fk(G)2135 1506 y Fe(n;l)2216 1494 y Fs(\()5 b(~)-40 b Fk(u)q Fs(\))2332 1412 y Fo(exp)o(\()p Fl(itW)13 b Fo(\()p Fl(u)p Fo(\)\))i Fl(du)2939 1427 y Fk(l)2967 1253 y Fj(\014)2967 1307 y(\014)2967 1362 y(\014)2967 1416 y(\014)2967 1471 y(\014)3012 1412 y Fl(d)6 b Fo(~)-51 b Fl(u)q(:)642 b Fo(\(21\))-20 1665 y(F)-8 b(or)36 b(the)g(inner)d(in)m(tegral)i(note)h(that)g Fl(u)1344 1680 y Fk(l)1404 1665 y Fm(7!)d Fl(f)1573 1679 y Fk(r)1611 1665 y Fo(\([)p Fl(u)1723 1680 y Fk(l)1749 1665 y Fl(;)22 b Fo(~)-52 b Fl(u)q Fo(]\))36 b(are)f(a\016ne)h(for)f (all)f Fl(r)i Fo(=)d(1)p Fl(;)15 b(:)g(:)g(:)33 b(;)15 b(K)7 b Fo(.)55 b(On)35 b Fl(G)3454 1680 y Fk(n;l)3565 1665 y Fm(\002)23 b(f)6 b Fo(~)-51 b Fl(u)q Fm(g)36 b Fo(w)m(e)-20 1778 y(ha)m(v)m(e)c(therefore)f Fl(@)625 1745 y Fs(2)664 1778 y Fl(f)10 b(=@)5 b(u)869 1745 y Fs(2)869 1806 y Fk(l)934 1778 y Fo(=)25 b(0.)41 b(This)29 b(yields)f(with)i(the)g(notation)h Fl(x)2384 1745 y Fg(\000)2468 1778 y Fo(:=)25 b(\()p Fl(x)2676 1792 y Fs(1)2736 1778 y Fm(\000)20 b Fl(x)2879 1792 y Fk(K)2947 1778 y Fl(;)15 b(:)g(:)g(:)32 b(;)15 b(x)3216 1792 y Fk(K)5 b Fg(\000)p Fs(1)3395 1778 y Fm(\000)20 b Fl(x)3538 1792 y Fk(K)3606 1778 y Fo(\))840 1945 y Fl(@)5 b(W)p 840 1985 152 4 v 850 2068 a(@)g(u)955 2083 y Fk(l)1085 2006 y Fo(=)1239 1878 y Fj(\034)1307 2006 y Fl(x)1359 1969 y Fg(\000)1438 2006 y Fm(\000)20 b Fo(\()p Fm(r)p Fl(g)s Fo(\))h Fm(\016)g Fl(f)5 b(;)1919 1945 y(@)g(f)p 1908 1985 132 4 v 1908 2068 a(@)g(u)2013 2083 y Fk(l)2049 1878 y Fj(\035)2132 2006 y Fl(;)730 2131 y Fj(\014)730 2186 y(\014)730 2240 y(\014)730 2295 y(\014)770 2201 y Fl(@)823 2168 y Fs(2)863 2201 y Fl(W)p 770 2242 192 4 v 794 2327 a(@)g(u)899 2296 y Fs(2)899 2356 y Fk(l)972 2131 y Fj(\014)972 2186 y(\014)972 2240 y(\014)972 2295 y(\014)1085 2263 y Fo(=)1239 2131 y Fj(\014)1239 2186 y(\014)1239 2240 y(\014)1239 2295 y(\014)1269 2135 y(\034)1337 2263 y Fl(x)1389 2225 y Fg(\000)1468 2263 y Fm(\000)20 b Fo(\()p Fm(r)p Fl(g)s Fo(\))i Fm(\016)e Fl(f)5 b(;)1938 2201 y(@)1991 2168 y Fs(2)2031 2201 y Fl(f)p 1938 2242 148 4 v 1939 2327 a(@)g(u)2044 2296 y Fs(2)2044 2356 y Fk(l)2095 2135 y Fj(\035)2183 2263 y Fo(+)2274 2135 y Fj(\034)2364 2201 y Fl(@)g(f)p 2352 2242 132 4 v 2352 2325 a(@)g(u)2457 2340 y Fk(l)2494 2263 y Fl(;)15 b Fo(Hess\()p Fl(g)s Fo(;)g Fl(f)10 b Fo(\))2948 2201 y Fl(@)5 b(f)p 2935 2242 V 2935 2325 a(@)g(u)3040 2340 y Fk(l)3078 2135 y Fj(\035)3146 2131 y(\014)3146 2186 y(\014)3146 2240 y(\014)3146 2295 y(\014)1085 2521 y Fo(=)1239 2390 y Fj(\014)1239 2444 y(\014)1239 2499 y(\014)1239 2553 y(\014)1269 2393 y(\034)1359 2460 y Fl(@)g(f)p 1347 2501 V 1347 2584 a(@)g(u)1452 2599 y Fk(l)1488 2521 y Fl(;)15 b Fo(Hess)q(\()p Fl(g)s Fo(;)g Fl(f)10 b Fo(\))1943 2460 y Fl(@)5 b(f)p 1930 2501 V 1930 2584 a(@)g(u)2035 2599 y Fk(l)2072 2393 y Fj(\035)2141 2390 y(\014)2141 2444 y(\014)2141 2499 y(\014)2141 2553 y(\014)1085 2699 y Fm(\025)83 b Fl(\015)5 b(;)-20 2872 y Fo(with)29 b Fl(\015)36 b Fo(de\014ned)29 b(in)g(\(11\).)42 b(Application)29 b(of)h(Lemma)h(3.2)g(implies)1088 2963 y Fj(\014)1088 3017 y(\014)1088 3072 y(\014)1088 3127 y(\014)1088 3181 y(\014)1118 2998 y(Z)1169 3204 y Fk(G)1224 3182 y Fd(0)1224 3228 y Fe(n;l)1305 3204 y Fs(\()5 b(~)-40 b Fk(u)q Fs(\))1420 3122 y Fo(exp\()p Fl(itW)13 b Fo(\()p Fl(u)p Fo(\)\))i Fl(du)2028 3137 y Fk(l)2056 2963 y Fj(\014)2056 3017 y(\014)2056 3072 y(\014)2056 3127 y(\014)2056 3181 y(\014)2111 3122 y Fm(\024)2217 2985 y(p)p 2293 2985 91 4 v 76 x Fo(32)p 2217 3101 167 4 v 2219 3188 a Fl(\015)2271 3162 y Fs(1)p Fk(=)p Fs(2)2394 3122 y Fm(j)p Fl(t)p Fm(j)2477 3084 y Fg(\000)p Fs(1)p Fk(=)p Fs(2)2642 3122 y Fl(:)-20 3374 y Fo(and)30 b(with)f(the)i(outer)f(in)m(tegrations)h(and)e (summation)h(in)f(\(20\),)j(\(21\),)g(and)e(with)f(\(12\))j(it)e(follo) m(ws)f(that)1465 3547 y Fm(j)p Fl(\036)p Fo(\()p Fl(t)p Fo(\))p Fm(j)d(\024)1794 3468 y(p)p 1870 3468 91 4 v 79 x Fo(32)q(\000)p Fm(j)p Fl(t)p Fm(j)2101 3510 y Fg(\000)p Fs(1)p Fk(=)p Fs(2)2266 3547 y Fl(;)-20 3720 y Fo(th)m(us)k Fl(c)219 3739 y Fs(1)p Fk(=)p Fs(2)355 3720 y Fm(\024)451 3645 y(p)p 527 3645 V 75 x Fo(32)q(\000.)-20 3833 y(Ad)g(\(17\):)43 b(F)-8 b(or)31 b(0)25 b Fl(<)g(p)g(<)g(p)899 3847 y Fs(1)939 3833 y Fo(,)30 b(using)f(\(7\),)j(w)m(e)e(ha)m(v)m(e)490 4092 y Fm(j)p Fl(\036)p Fo(\()p Fl(t)p Fo(\))p Fm(j)84 b(\024)935 3969 y Fj(Z)986 4175 y Fs([0)p Fk(;)p Fs(1])1116 4156 y Fe(d)1200 3979 y Fk(K)1174 4006 y Fj(Y)1170 4200 y Fk(r)r Fs(=1)1309 4092 y Fm(j)p Fl(\036)p Fo(\()p Fl(f)1468 4106 y Fk(r)1507 4092 y Fo(\()p Fl(u)p Fo(\))p Fl(t)p Fo(\))p Fm(j)15 b Fl(du)27 b Fm(\024)1959 3969 y Fj(Z)2009 4175 y Fs([0)p Fk(;)p Fs(1])2139 4156 y Fe(d)2224 3979 y Fk(K)2198 4006 y Fj(Y)2194 4200 y Fk(r)r Fs(=1)2473 4030 y Fl(c)2512 4044 y Fk(p)p 2343 4072 340 4 v 2343 4155 a Fl(f)2398 4111 y Fk(p)2388 4167 y(r)2437 4155 y Fo(\()p Fl(u)p Fo(\))p Fm(j)p Fl(t)p Fm(j)2642 4129 y Fk(p)2708 4092 y Fl(du)e Fm(\024)g Fl(c)2967 4055 y Fk(K)2967 4115 y(p)3036 4092 y Fl(M)3124 4106 y Fk(p)3164 4092 y Fm(j)p Fl(t)p Fm(j)3247 4055 y Fg(\000)p Fk(K)5 b(p)3406 4092 y Fl(:)-20 4376 y Fo(Ad)29 b(\(18\):)42 b(W)-8 b(e)30 b(assume)e Fl(c)861 4390 y Fk(p)927 4376 y Fl(<)d Fm(1)k Fo(for)f(a)i Fl(p)25 b(>)g(p)1568 4390 y Fs(0)1636 4376 y Fo(and)j Fl(t)d(>)g(K)7 b(c)2088 4328 y Fs(1)p Fk(=p)2088 4387 y(p)2199 4376 y Fo(;)29 b(in)f(the)h(case)h(0) c Fl(<)f(t)g(<)g(K)7 b(c)3148 4328 y Fs(1)p Fk(=p)3148 4387 y(p)3287 4376 y Fo(w)m(e)29 b(ha)m(v)m(e)i(trivially)-20 4508 y Fm(j)p Fl(\036)p Fo(\()p Fl(t)p Fo(\))p Fm(j)c(\024)e Fl(c)349 4522 y Fk(p)p Fs(+)p Fk(s)476 4508 y Fm(j)p Fl(t)p Fm(j)559 4475 y Fg(\000)p Fs(\()p Fk(p)p Fs(+)p Fk(s)p Fs(\))826 4508 y Fo(since)k Fm(j)p Fl(\036)p Fo(\()p Fl(t)p Fo(\))p Fm(j)d(\024)f Fo(1.)41 b(F)-8 b(or)30 b Fl(t)25 b(>)g(K)7 b(c)1927 4460 y Fs(1)p Fk(=p)1927 4520 y(p)2067 4508 y Fo(w)m(e)30 b(cannot)g(ha)m(v)m(e)h Fl(f)2751 4522 y Fk(r)2813 4508 y Fm(\024)25 b Fl(c)2948 4460 y Fs(1)p Fk(=p)2948 4520 y(p)3059 4508 y Fl(=t)k Fo(for)h(all)e Fl(r)g Fo(=)d(1)p Fl(;)15 b(:)g(:)g(:)32 b(;)15 b(K)-20 4641 y Fo(since)208 4573 y Fj(P)319 4641 y Fl(f)364 4655 y Fk(r)435 4641 y Fo(=)34 b(1.)56 b(Th)m(us)34 b(w)m(e)i(ha)m(v)m(e)h(only)d(the)i(t)m(w)m(o)h(cases)f(\\all)f(but)f (one)i Fl(f)2596 4655 y Fk(r)2669 4641 y Fo(are)g Fm(\024)d Fl(c)2969 4593 y Fs(1)p Fk(=p)2969 4653 y(p)3080 4641 y Fl(=t)p Fo(")j(and)e(\\at)j(least)f(t)m(w)m(o)-20 4772 y Fl(f)25 4786 y Fk(r)63 4772 y Fl(;)15 b(f)148 4786 y Fk(q)216 4772 y Fo(are)31 b Fl(>)25 b(c)503 4724 y Fs(1)p Fk(=p)503 4783 y(p)613 4772 y Fl(=t)p Fo(".)42 b(This)28 b(yields)537 5035 y([0)p Fl(;)15 b Fo(1])717 4997 y Fk(d)785 5035 y Fo(=)881 4879 y Fj( )997 4921 y Fk(K)979 4948 y Fj([)968 5144 y Fk(j)t Fs(=1)1137 4921 y Fk(K)1119 4948 y Fj(\\)1115 5132 y Fe(r)r Fi(=1)1116 5188 y Fe(r)r Fd(6)p Fi(=)p Fe(j)1248 4879 y Fj(\()1321 5035 y Fl(f)1366 5049 y Fk(r)1429 5035 y Fm(\024)1535 4973 y Fl(c)1574 4925 y Fs(1)p Fk(=p)1574 4985 y(p)p 1535 5014 150 4 v 1593 5097 a Fl(t)1694 4879 y Fj(\))q(!)1860 5035 y Fm([)1941 4879 y Fj( )2081 4921 y Fk(K)2063 4948 y Fj([)2038 5133 y Fe(r)n(;j)s Fi(=1)2061 5189 y Fe(r)r Fd(6)p Fi(=)p Fe(j)2214 4879 y Fj(\()2288 5035 y Fl(f)2333 5049 y Fk(r)2395 5035 y Fl(>)2501 4973 y(c)2540 4925 y Fs(1)p Fk(=p)2540 4985 y(p)p 2501 5014 V 2560 5097 a Fl(t)2661 5035 y(;)g(f)2746 5049 y Fk(j)2808 5035 y Fl(>)2913 4973 y(c)2952 4925 y Fs(1)p Fk(=p)2952 4985 y(p)p 2913 5014 V 2972 5097 a Fl(t)3073 4879 y Fj(\)!)1915 5411 y Fo(14)p eop %%Page: 15 15 15 14 bop -20 -21 a Fo(W)-8 b(e)35 b(denote)f(the)f(\014rst)g(of)h (these)g(t)m(w)m(o)g(sets)g(b)m(y)f Fl(B)1682 -7 y Fs(1)1722 -21 y Fo(.)50 b(The)33 b(second)g(one)h(w)m(e)g(in)m(tersect)g(with)e ([0)p Fl(;)15 b Fo(1])3350 -54 y Fk(d)3422 -21 y Fo(=)30 b Fm([)3584 -54 y Fk(K)3584 3 y(q)r Fs(=1)3712 -21 y Fm(f)p Fl(f)3802 -7 y Fk(q)3870 -21 y Fm(\025)-20 92 y Fo(1)p Fl(=K)7 b Fm(g)p Fo(.)42 b(It)31 b(is)e(easily)g(seen)i(that)g (the)f(second)h(set)g(is)e(then)h(a)h(subset)f(of)1208 368 y Fl(B)1277 382 y Fs(2)1342 368 y Fo(:=)1519 255 y Fk(K)1501 282 y Fj([)1473 465 y Fe(q)r(;r)r Fi(=1)1498 522 y Fe(q)r Fd(6)p Fi(=)p Fe(r)1655 213 y Fj(\()1728 368 y Fl(f)1773 382 y Fk(q)1836 368 y Fm(\025)1961 307 y Fo(1)p 1942 347 84 4 v 1942 431 a Fl(K)2035 368 y(;)15 b(f)2120 382 y Fk(r)2183 368 y Fl(>)2289 307 y(c)2328 259 y Fs(1)p Fk(=p)2328 318 y(p)p 2289 347 150 4 v 2348 431 a Fl(t)2449 213 y Fj(\))2522 368 y Fl(;)-20 705 y Fo(th)m(us)30 b([0)p Fl(;)15 b Fo(1])360 672 y Fk(d)428 705 y Fo(=)25 b Fl(B)593 719 y Fs(1)652 705 y Fm([)20 b Fl(B)802 719 y Fs(2)842 705 y Fo(.)40 b(Therefore,)31 b(w)m(e)f(ha)m(v)m(e)594 982 y Fm(j)p Fl(\036)p Fo(\()p Fl(t)p Fo(\))p Fm(j)c(\024)923 858 y Fj(Z)974 1064 y Fs([0)p Fk(;)p Fs(1])1104 1045 y Fe(d)1189 868 y Fk(K)1163 895 y Fj(Y)1159 1090 y Fk(r)r Fs(=1)1298 982 y Fo(min)1464 826 y Fj(\()1708 920 y Fl(c)1747 934 y Fk(p)p 1548 961 399 4 v 1548 1044 a Fo(\()p Fl(f)1628 1058 y Fk(r)1665 1044 y Fo(\()p Fl(u)p Fo(\))p Fm(j)p Fl(t)p Fm(j)p Fo(\))1905 1018 y Fk(p)1957 982 y Fl(;)15 b Fo(1)2042 826 y Fj(\))2131 982 y Fl(du)25 b Fm(\024)2351 858 y Fj(Z)2402 1064 y Fk(B)2455 1073 y Fi(1)2509 982 y Fo(+)2595 858 y Fj(Z)2645 1064 y Fk(B)2698 1073 y Fi(2)2763 982 y Fo(=:)g Fl(I)i Fo(+)20 b Fl(I)7 b(I)g(:)-20 1282 y Fo(F)-8 b(or)28 b(the)f(estimate)h (of)f Fl(I)34 b Fo(w)m(e)28 b(note)g(that)f Fl(f)1396 1296 y Fk(j)1432 1282 y Fo(\()p Fl(u)p Fo(\))f Fm(\025)f Fo(1)14 b Fm(\000)g Fo(\()p Fl(K)20 b Fm(\000)14 b Fo(1\))p Fl(c)2156 1234 y Fs(1)p Fk(=p)2156 1294 y(p)2267 1282 y Fl(=t)27 b Fo(on)g Fm(\\)2556 1297 y Fk(r)r Fg(6)p Fs(=)p Fk(j)2681 1282 y Fm(f)p Fl(f)2771 1296 y Fk(r)2834 1282 y Fm(\024)e Fl(c)2969 1234 y Fs(1)p Fk(=p)2969 1294 y(p)3080 1282 y Fl(=t)p Fm(g)p Fo(,)j(so)f(that)h(w)m(e)g(obtain)-20 1395 y Fl(f)25 1409 y Fk(j)61 1395 y Fo(\()p Fl(u)p Fo(\))e Fm(\025)f Fo(1)p Fl(=K)38 b Fo(on)30 b(this)g(set.)41 b(With)30 b(\(5\),)i(this)d(yields)1116 1675 y Fl(I)90 b Fm(\024)1433 1561 y Fk(K)1400 1589 y Fj(X)1404 1784 y Fk(j)t Fs(=1)1546 1551 y Fj(Z)1596 1758 y Fg(\\)1643 1770 y Fe(r)r Fd(6)p Fi(=)p Fe(j)1753 1758 y Fg(f)p Fk(f)1822 1766 y Fe(r)1857 1758 y Fg(\024)p Fk(c)1943 1722 y Fi(1)p Fe(=p)1943 1768 y(p)2041 1758 y Fk(=t)p Fg(g)2300 1613 y Fl(c)2339 1627 y Fk(p)p 2166 1654 347 4 v 2166 1737 a Fo(\()p Fl(f)2246 1751 y Fk(j)2283 1737 y Fo(\()p Fl(u)p Fo(\))p Fl(t)p Fo(\))2473 1711 y Fk(p)2523 1675 y Fl(du)1246 2008 y Fm(\024)83 b Fl(c)1439 2022 y Fk(p)1479 2008 y Fl(K)1563 1970 y Fk(p)1602 2008 y Fl(t)1635 1970 y Fg(\000)p Fk(p)1778 1894 y(K)1744 1921 y Fj(X)1749 2117 y Fk(j)t Fs(=1)1891 2008 y Fl(\025)1944 1970 y Fk(d)1984 1852 y Fj( )2103 1894 y Fk(K)2085 1921 y Fj(\\)2082 2105 y Fe(r)r Fi(=1)2083 2161 y Fe(r)r Fd(6)p Fi(=)p Fe(j)2214 1852 y Fj(\()2287 2008 y Fl(f)2332 2022 y Fk(r)2395 2008 y Fm(\024)2501 1946 y Fl(c)2540 1898 y Fs(1)p Fk(=p)2540 1958 y(p)p 2501 1987 150 4 v 2559 2070 a Fl(t)2661 1852 y Fj(\)!)1246 2301 y Fm(\024)g Fl(c)1439 2315 y Fk(p)1479 2301 y Fl(K)1563 2263 y Fk(p)1602 2301 y Fl(D)1677 2315 y Fs(1)1717 2301 y Fo(\()p Fl(c)1791 2263 y Fs(1)p Fk(=p)1791 2323 y(p)1902 2301 y Fo(\))p Fl(t)1970 2263 y Fg(\000)p Fs(\()p Fk(p)p Fs(+)p Fk(s)p Fs(\))2207 2301 y Fl(:)-20 2491 y Fo(F)-8 b(or)31 b Fl(I)7 b(I)38 b Fo(w)m(e)31 b(estimate)g(\014rst)467 2599 y Fj(Z)518 2805 y Fg(f)p Fk(f)587 2813 y Fe(q)621 2805 y Fg(\025)p Fs(1)p Fk(=K)q(;f)861 2813 y Fe(r)895 2805 y Fk(>c)981 2770 y Fi(1)p Fe(=p)981 2816 y(p)1079 2805 y Fk(=t)p Fg(g)1461 2652 y Fl(c)1500 2619 y Fs(2)1500 2675 y Fk(p)p 1204 2702 594 4 v 1204 2785 a Fo(\()p Fl(f)1284 2799 y Fk(q)1322 2785 y Fo(\()p Fl(u)p Fo(\))p Fl(f)1489 2799 y Fk(r)1527 2785 y Fo(\()p Fl(u)p Fo(\)\))1684 2759 y Fk(p)1725 2785 y Fl(t)1758 2759 y Fs(2)1823 2723 y Fl(du)25 b Fm(\024)g Fl(c)2082 2685 y Fs(2)2082 2745 y Fk(p)2122 2723 y Fl(K)2206 2685 y Fk(p)2246 2723 y Fl(t)2279 2685 y Fg(\000)p Fs(2)p Fk(p)2423 2599 y Fj(Z)2474 2805 y Fg(f)p Fk(f)2543 2813 y Fe(r)2578 2805 y Fk(>c)2664 2770 y Fi(1)p Fe(=p)2664 2816 y(p)2762 2805 y Fk(=t)p Fg(g)2877 2723 y Fl(f)2932 2685 y Fg(\000)p Fk(p)2922 2745 y(r)3026 2723 y Fo(\()p Fl(u)p Fo(\))15 b Fl(du:)-20 2979 y Fo(This)29 b(yields,)g(using)g (\(6\),)957 3256 y Fl(I)7 b(I)90 b Fm(\024)83 b Fo(\()p Fl(K)27 b Fm(\000)20 b Fo(1\))p Fl(c)1637 3218 y Fs(2)1637 3278 y Fk(p)1678 3256 y Fl(K)1762 3218 y Fk(p)1801 3256 y Fl(t)1834 3218 y Fg(\000)p Fs(2)p Fk(p)2012 3142 y(K)1979 3169 y Fj(X)1982 3364 y Fk(r)r Fs(=1)2125 3132 y Fj(Z)2176 3338 y Fg(f)p Fk(f)2245 3346 y Fe(r)2280 3338 y Fk(>c)2366 3303 y Fi(1)p Fe(=p)2366 3349 y(p)2463 3338 y Fk(=t)p Fg(g)2579 3256 y Fl(f)2634 3218 y Fg(\000)p Fk(p)2624 3278 y(r)2728 3256 y Fo(\()p Fl(u)p Fo(\))15 b Fl(du)1134 3493 y Fm(\024)83 b Fo(\()p Fl(K)27 b Fm(\000)20 b Fo(1\))p Fl(c)1637 3456 y Fs(2)1637 3516 y Fk(p)1678 3493 y Fl(K)1762 3456 y Fs(2)1801 3493 y Fl(D)1876 3507 y Fs(2)1915 3493 y Fo(\()p Fl(c)1989 3456 y Fs(1)p Fk(=p)1989 3516 y(p)2100 3493 y Fo(\))p Fl(t)2168 3456 y Fg(\000)p Fs(\()p Fk(p)p Fs(+)p Fk(s)p Fs(\))2405 3493 y Fl(:)-20 3683 y Fo(The)30 b(assertion)g(follo)m(ws.)p 3870 3683 71 71 v -20 3932 a Ff(3.2)113 b(The)37 b(dominating)f(curv)m(e)-20 4104 y Fo(F)-8 b(or)34 b(a)f(rejection)g(algorithm)f(a)i(dominating,)e(in)m (tegrable)g(curv)m(e)i Fl(q)h Fo(for)e(the)g(densit)m(y)g Fl(w)i Fo(to)f(b)s(e)e(sampled)g(from)g(is)-20 4217 y(necessary)-8 b(,)30 b(suc)m(h)e(that)h(from)f(the)g(distribution)d(with)i(densit)m (y)g Fl(q)s(=)p Fm(k)p Fl(q)s Fm(k)2397 4231 y Fs(1)2466 4217 y Fo(it)h(is)f(easy)i(to)g(sample.)39 b(If)28 b(Lipsc)m(hitz-)g (and)-20 4330 y(momen)m(t-information)g(on)g Fl(w)j Fo(is)d(a)m(v)-5 b(ailable)28 b(a)h(curv)m(e)f Fl(q)k Fo(can)c(b)s(e)g(constructed)h(on) f(the)h(basis)e(of)i(Theorem)f(3.3)i(and)-20 4443 y(Theorem)g(3.5)i(in) d(Devro)m(y)m(e)j([4)q(,)f(p.)40 b(315,)32 b(p.)40 b(320].)i(F)-8 b(or)31 b(this)f(w)m(e)g(denote)h(b)m(y)g Fl(K)2718 4457 y Fs(1)2757 4443 y Fl(;)15 b(K)2874 4457 y Fs(2)2914 4443 y Fl(;)g(K)3031 4457 y Fs(3)3097 4443 y Fl(>)25 b Fo(0)30 b(constan)m(ts)i(with)1025 4633 y Fm(k)p Fl(w)r Fm(k)1182 4647 y Fg(1)1284 4633 y Fm(\024)25 b Fl(K)1457 4647 y Fs(1)1496 4633 y Fl(;)106 b Fm(k)p Fl(w)1739 4595 y Fg(0)1764 4633 y Fm(k)1809 4647 y Fg(1)1909 4633 y Fm(\024)25 b Fl(K)2082 4647 y Fs(2)2122 4633 y Fl(;)121 b Fo(E)15 b Fl(X)2427 4595 y Fs(4)2492 4633 y Fm(\024)25 b Fl(K)2665 4647 y Fs(3)2705 4633 y Fl(:)1049 b Fo(\(22\))-20 4823 y(The)33 b(existence)h(of)f(momen)m(ts)h(of)g(all)e(orders)g(of)i Fl(X)40 b Fo(follo)m(ws)33 b(since)f(the)i(Laplace)f(transform)g(of)g Fl(X)41 b Fo(is)32 b(\014nite)h(in)f(a)-20 4936 y(neigh)m(b)s(orho)s(o) s(d)c(of)j(0,)g(see)g(R\177)-45 b(osler)30 b([35)q(].)41 b(Then)29 b(a)i(dominating,)e(in)m(tegrable)h(curv)m(e)g(for)h Fl(w)h Fo(is)e(giv)m(en)g(b)m(y)1047 5149 y Fl(q)s Fo(\()p Fl(x)p Fo(\))c(:=)f(min)1527 5048 y Fj(n)1588 5149 y Fl(K)1665 5163 y Fs(1)1704 5149 y Fl(;)1744 5065 y Fj(p)p 1836 5065 279 4 v 1836 5149 a Fo(2)p Fl(K)1958 5163 y Fs(2)1998 5149 y Fl(K)2075 5163 y Fs(3)2114 5149 y Fl(x)2166 5111 y Fg(\000)p Fs(2)2261 5048 y Fj(o)2321 5149 y Fl(;)107 b(x)25 b Fm(2)g Fo(R)p Fl(:)1071 b Fo(\(23\))1915 5411 y(15)p eop %%Page: 16 16 16 15 bop -20 -21 a Fo(This)33 b(follo)m(ws)h(from)h(the)g(general)f (inequalit)m(y)g Fl(w)r Fo(\()p Fl(x)p Fo(\))f Fm(\024)g Fo(\(2)p Fl(K)2103 -7 y Fs(2)2158 -21 y Fo(min)o Fm(f)p Fl(F)13 b Fo(\()p Fl(x)p Fo(\))p Fl(;)i Fo(1)25 b Fm(\000)e Fl(F)13 b Fo(\()p Fl(x)p Fo(\))p Fm(g)p Fo(\))3025 -54 y Fs(1)p Fk(=)p Fs(2)3137 -21 y Fo(,)36 b(cf.)f(Theorem)f(3.5)i(in)-20 92 y(Devro)m(y)m(e)e([4)q(],)f(where)e Fl(F)45 b Fo(is)31 b(the)h(distribution)c(function)j(of)h Fl(X)7 b Fo(,)33 b(and,)f(b)m(y)g(Mark)m(o)m(v's)h(inequalit)m(y)-8 b(,)32 b(min)n Fm(f)p Fl(F)13 b Fo(\()p Fl(x)p Fo(\))p Fl(;)i Fo(1)23 b Fm(\000)-20 205 y Fl(F)13 b Fo(\()p Fl(x)p Fo(\))p Fm(g)27 b(\024)40 b Fo(E)15 b Fl(X)515 172 y Fs(4)555 205 y Fl(=x)652 172 y Fs(4)692 205 y Fo(.)121 318 y(A)31 b(random)f(v)-5 b(ariate)30 b(with)f(densit)m(y)h Fl(q)s(=)p Fm(k)p Fl(q)s Fm(k)1598 332 y Fs(1)1669 318 y Fo(is)f(giv)m(en)i(b)m(y)1527 555 y Fl(S)1598 493 y Fo(\(2)p Fl(K)1755 507 y Fs(2)1795 493 y Fl(K)1872 507 y Fs(3)1912 493 y Fo(\))1947 460 y Fs(1)p Fk(=)p Fs(4)p 1598 534 460 4 v 1730 643 a Fl(K)1814 595 y Fs(1)p Fk(=)p Fs(2)1807 669 y(1)2092 493 y Fl(U)2154 507 y Fs(1)p 2092 534 102 4 v 2092 617 a Fl(U)2154 631 y Fs(2)2204 555 y Fl(;)1550 b Fo(\(24\))-20 807 y(with)30 b Fl(U)250 821 y Fs(1)290 807 y Fl(;)15 b(U)392 821 y Fs(2)432 807 y Fl(;)g(S)36 b Fo(b)s(eing)30 b(indep)s(enden)m(t,)f Fl(U)1413 821 y Fs(1)1453 807 y Fl(;)15 b(U)1555 821 y Fs(2)1622 807 y Fm(\030)26 b Fo(uniform)n([0)p Fl(;)15 b Fo(1])33 b(and)d Fl(S)36 b Fo(b)s(eing)30 b(an)h(equiprobable)e (random)i(sign,)-20 920 y(cf.)g(Theorem)f(3.3)i(in)d(Devro)m(y)m(e)j ([4)q(].)41 b(In)29 b(our)h(situation)g(the)g(follo)m(wing)f(c)m (hoices)i(for)f Fl(K)2972 934 y Fs(1)3012 920 y Fl(;)15 b(K)3129 934 y Fs(2)3169 920 y Fl(;)g(K)3286 934 y Fs(3)3356 920 y Fo(are)31 b(p)s(ossible:)-20 1083 y Fp(Lemma)i(3.3)46 b Fc(De\014ne)38 b Fl(\030)j Fc(as)e(in)f(\(2\))h(and)g Fl(\030)1524 1097 y Fs(3)1598 1083 y Fo(:=)1729 1015 y Fj(P)1825 1041 y Fk(K)1825 1110 y(r)r Fs(=1)1984 1083 y Fo(E)15 b Fl(A)2129 1050 y Fs(3)2129 1106 y Fk(r)2168 1083 y Fc(,)39 b Fl(\030)2275 1097 y Fs(4)2350 1083 y Fo(:=)2481 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1777 y Fl(c)1421 1729 y Fs(2)p Fk(=p)1421 1789 y(p)p 1356 1818 203 4 v 1356 1901 a Fl(p)20 b Fm(\000)g Fo(2)1568 1683 y Fj(!)1640 1839 y Fl(;)109 b(p)25 b(>)f Fo(2)560 2123 y Fl(K)637 2137 y Fs(3)759 2123 y Fo(:=)960 2061 y Fm(k)p Fl(g)s Fm(k)1096 2028 y Fs(4)1096 2084 y Fg(1)p 948 2102 236 4 v 948 2185 a Fo(1)d Fm(\000)f Fl(\030)1145 2199 y Fs(4)1209 1995 y Fj(\022)1276 2123 y Fo(1)h(+)1520 2061 y(1)p 1443 2102 201 4 v 1443 2185 a(1)f Fm(\000)g Fl(\030)1673 2123 y Fo(+)1869 2061 y(1)p 1774 2102 236 4 v 1774 2185 a(1)h Fm(\000)f Fl(\030)1971 2199 y Fs(3)2040 2123 y Fo(+)2388 2061 y Fl(K)p 2141 2102 578 4 v 2141 2185 a Fo(\(1)h Fm(\000)f Fl(\030)t Fo(\)\(1)h Fm(\000)f Fl(\030)2644 2199 y Fs(3)2683 2185 y Fo(\))2749 2123 y(+)2850 2061 y Fl(K)7 b Fo(\()p Fl(K)27 b Fm(\000)20 b Fo(1\))p 2850 2102 395 4 v 2892 2185 a(\(1)h Fm(\000)f Fl(\030)t Fo(\))3163 2159 y Fs(2)3254 1995 y Fj(\023)3336 2123 y Fl(:)-20 2343 y Fc(Mor)-5 b(e)g(over)34 b(we)f(have)996 2572 y Fm(k)p Fl(w)1108 2535 y Fg(0)q(0)1152 2572 y Fm(k)1197 2586 y Fg(1)1297 2572 y Fm(\024)25 b Fl(K)1470 2586 y Fs(4)1535 2572 y Fo(:=)1671 2511 y(1)p 1666 2551 56 4 v 1666 2635 a Fl(\031)1746 2417 y Fj( )1818 2572 y Fl(c)1857 2535 y Fs(1)p Fk(=p)1857 2595 y(p)1988 2572 y Fo(+)2115 2511 y Fl(c)2154 2463 y Fs(3)p Fk(=p)2154 2523 y(p)p 2089 2551 203 4 v 2089 2635 a Fl(p)20 b Fm(\000)g Fo(3)2301 2417 y Fj(!)2388 2572 y Fl(;)109 b(p)25 b(>)g Fo(3)p Fl(:)-20 2839 y Fp(Pro)s(of:)77 b Fo(By)30 b(the)h(F)-8 b(ourier)30 b(in)m(v)m(ersion)f(form)m(ula)g(the)i Fl(k)s Fo(-th)g(deriv)-5 b(ativ)m(e)29 b Fl(w)2505 2806 y Fs(\()p Fk(k)r Fs(\))2634 2839 y Fo(satis\014es)1066 3053 y Fm(k)p Fl(w)1178 3016 y Fs(\()p Fk(k)r Fs(\))1277 3053 y Fm(k)1322 3067 y Fg(1)1422 3053 y Fm(\024)1556 2992 y Fo(1)p 1528 3032 101 4 v 1528 3115 a(2)p Fl(\031)1654 2929 y Fj(Z)1745 2956 y Fg(1)1704 3135 y(\0001)1849 3053 y Fm(j)p Fl(t)p Fm(j)1932 3016 y Fk(k)1975 3053 y Fm(j)p Fl(\036)p Fo(\()p Fl(t)p Fo(\))p Fm(j)p Fl(dt;)108 b(k)28 b Fm(2)d 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y Fs(\()p Fk(r)r Fs(\))2425 3536 y Fj(\021)c(\020)2549 3637 y Fl(A)2617 3599 y Fs(\()p Fk(n)p Fs(\))2617 3659 y Fk(s)2719 3637 y Fl(X)2801 3589 y Fs(\()p Fk(s)p Fs(\))2794 3663 y Fk(n)p Fg(\000)p Fs(1)2952 3637 y Fm(\000)k Fl(A)3111 3651 y Fk(s)3148 3637 y Fl(X)3230 3599 y Fs(\()p Fk(s)p Fs(\))3322 3536 y Fj(\021)1160 3905 y Fo(=:)25 b Fl(I)j Fo(+)19 b Fl(I)7 b(I)28 b Fo(+)20 b Fl(I)7 b(I)g(I)27 b Fo(+)20 b Fl(I)7 b(V)e(:)-20 4060 y Fo(W)-8 b(e)32 b(ha)m(v)m(e)f Fl(I)7 b(I)33 b Fm(\024)25 b Fl(R)632 4027 y Fs(2)631 4088 y Fk(b)671 4060 y Fo(\()p Fl(n)p Fo(\),)31 b(and)992 4307 y Fl(I)7 b(I)g(I)90 b Fo(=)83 b(2)15 b(E)1556 4194 y Fk(K)1523 4221 y Fj(X)1526 4415 y Fk(r)r Fs(=1)1669 4206 y Fj(\020)1723 4307 y Fl(A)1791 4270 y Fs(\()p Fk(n)p Fs(\))1791 4330 y Fk(r)1893 4307 y Fl(X)1975 4259 y Fs(\()p Fk(r)r Fs(\))1968 4333 y Fk(n)p Fg(\000)p Fs(1)2126 4307 y Fm(\000)20 b Fl(A)2285 4321 y Fk(r)2323 4307 y Fl(X)2405 4270 y Fs(\()p Fk(r)r Fs(\))2498 4206 y Fj(\021)2568 4307 y Fo(\()p Fl(b)2642 4270 y Fs(\()p Fk(n)p Fs(\))2764 4307 y Fm(\000)g Fl(b)p Fo(\))1216 4626 y(=)83 b(2)15 b(E)1556 4512 y Fk(K)1523 4539 y Fj(X)1526 4733 y Fk(r)r Fs(=1)1669 4626 y Fl(A)1737 4588 y Fs(\()p Fk(n)p Fs(\))1737 4648 y Fk(r)1839 4626 y Fl(X)1921 4578 y Fs(\()p Fk(r)r Fs(\))1914 4652 y Fk(n)p Fg(\000)p Fs(1)2052 4626 y Fo(\()p Fl(b)2126 4588 y Fs(\()p Fk(n)p Fs(\))2248 4626 y Fm(\000)20 b Fl(b)p Fo(\))1216 4944 y Fm(\024)83 b Fo(2)1464 4830 y Fk(K)1430 4858 y Fj(X)1434 5052 y Fk(r)r Fs(=1)1577 4944 y Fm(k)p Fl(A)1690 4906 y Fs(\()p Fk(n)p Fs(\))1690 4966 y Fk(r)1792 4944 y Fm(k)1837 4958 y Fs(2)1877 4944 y Fm(k)p Fl(b)1961 4906 y Fs(\()p Fk(n)p Fs(\))2084 4944 y Fm(\000)20 b Fl(b)p Fm(k)2259 4958 y Fs(2)2314 4944 y Fo(E)15 b Fl(X)2466 4958 y Fk(n)p Fg(\000)p Fs(1)1216 5162 y Fm(\024)83 b Fo(2)p Fl(C)1480 5176 y Fk(A)1537 5162 y Fl(R)1606 5177 y Fk(b)1641 5162 y Fo(\()p Fl(n)p Fo(\))p Fl(M)10 b Fo(\()p Fl(n)20 b Fm(\000)g Fo(1\))p Fl(:)1915 5411 y Fo(18)p eop %%Page: 19 19 19 18 bop -20 -21 a Fo(Analogously)737 240 y Fl(I)7 b(V)104 b Fo(=)97 b(E)1256 126 y Fk(K)1222 153 y Fj(X)1212 337 y Fe(r)n(;s)p Fi(=1)1235 393 y Fe(r)r Fd(6)p Fi(=)p Fe(s)1389 139 y Fj(\020)1443 240 y Fl(A)1511 202 y Fs(\()p Fk(n)p Fs(\))1511 262 y Fk(r)1613 240 y Fl(X)1695 192 y Fs(\()p Fk(r)r Fs(\))1688 266 y Fk(n)p Fg(\000)p Fs(1)1846 240 y Fm(\000)20 b Fl(A)2005 254 y Fk(r)2043 240 y Fl(X)2125 202 y Fs(\()p Fk(r)r Fs(\))2218 139 y Fj(\021)c(\020)2342 240 y Fl(A)2410 202 y Fs(\()p Fk(n)p Fs(\))2410 262 y Fk(s)2512 240 y Fl(X)2594 192 y Fs(\()p Fk(s)p Fs(\))2587 266 y Fk(n)p Fg(\000)p Fs(1)2745 240 y Fm(\000)k Fl(A)2904 254 y Fk(s)2941 240 y Fl(X)3023 202 y Fs(\()p Fk(s)p Fs(\))3115 139 y Fj(\021)941 613 y Fo(=)1148 500 y Fk(K)1115 527 y Fj(X)1104 711 y Fe(r)n(;s)p Fi(=1)1127 767 y Fe(r)r Fd(6)p Fi(=)p Fe(s)1297 613 y Fo(E)15 b([)p Fl(A)1467 576 y Fs(\()p Fk(n)p Fs(\))1467 636 y Fk(r)1569 613 y Fl(A)1637 576 y Fs(\()p Fk(n)p Fs(\))1637 636 y Fk(s)1739 613 y Fo(])g(E)g([)p Fl(X)1956 627 y Fk(n)p Fg(\000)p Fs(1)2094 613 y Fo(])2119 576 y Fs(2)941 899 y Fm(\024)82 b Fl(C)1166 862 y Fg(0)1159 922 y Fk(A)1216 899 y Fl(M)1314 862 y Fs(2)1354 899 y Fo(\()p Fl(n)20 b Fm(\000)g Fo(1\))p Fl(:)-20 1079 y Fo(Finally)-8 b(,)29 b(b)m(y)i(the)f(Cauc)m(h)m(y-Sc)m (h)m(w)m(arz)i(inequalit)m(y)431 1340 y Fl(I)90 b Fo(=)98 b(E)856 1226 y Fk(K)822 1254 y Fj(X)826 1448 y Fk(r)r Fs(=1)969 1239 y Fj(\020)1023 1340 y Fl(A)1091 1302 y Fs(\()p Fk(n)p Fs(\))1091 1362 y Fk(r)1193 1340 y Fl(X)1275 1292 y Fs(\()p Fk(r)r Fs(\))1268 1366 y Fk(n)p Fg(\000)p Fs(1)1426 1340 y Fm(\000)20 b Fl(A)1585 1354 y Fk(r)1623 1340 y Fl(X)1705 1302 y Fs(\()p Fk(r)r Fs(\))1798 1239 y Fj(\021)1852 1262 y Fs(2)561 1658 y Fo(=)98 b(E)856 1545 y Fk(K)822 1572 y Fj(X)826 1766 y Fk(r)r Fs(=1)969 1557 y Fj(\020)1023 1658 y Fl(A)1091 1621 y Fs(\()p Fk(n)p Fs(\))1091 1681 y Fk(r)1193 1658 y Fo(\()p Fl(X)1310 1610 y Fs(\()p Fk(r)r Fs(\))1303 1684 y Fk(n)p Fg(\000)p Fs(1)1461 1658 y Fm(\000)20 b Fl(X)1634 1621 y Fs(\()p Fk(r)r Fs(\))1727 1658 y Fo(\))h Fm(\000)f Fo(\()p Fl(A)1977 1621 y Fs(\()p Fk(n)p Fs(\))1977 1681 y Fk(r)2099 1658 y Fm(\000)g Fl(A)2258 1672 y Fk(r)2296 1658 y Fo(\))p Fl(X)2413 1621 y Fs(\()p Fk(r)r Fs(\))2507 1557 y Fj(\021)2561 1580 y Fs(2)561 1977 y Fo(=)749 1863 y Fk(K)715 1890 y Fj(X)719 2085 y Fk(r)r Fs(=1)862 1821 y Fj( )949 1977 y Fo(E)15 b(\()p Fl(A)1129 1939 y Fs(\()p Fk(n)p Fs(\))1129 1999 y Fk(r)1231 1977 y Fo(\))1266 1939 y Fs(2)1306 1977 y Fl(`)1344 1939 y Fs(2)1344 1999 y(2)1383 1977 y Fo(\()p Fl(X)1493 1991 y Fk(n)p Fg(\000)p Fs(1)1631 1977 y Fl(;)g(X)7 b Fo(\))21 b(+)f Fm(k)p Fl(A)2013 1939 y Fs(\()p Fk(n)p Fs(\))2013 1999 y Fk(r)2136 1977 y Fm(\000)g Fl(A)2295 1991 y Fk(r)2333 1977 y Fm(k)2378 1939 y Fs(2)2378 1999 y(2)2433 1977 y Fo(E)15 b Fl(X)2592 1939 y Fs(2)766 2286 y Fo(+)k(2)c(E)g([)p Fl(A)1086 2249 y Fs(\()p Fk(n)p Fs(\))1086 2309 y Fk(r)1189 2286 y Fo(\()p Fl(A)1292 2249 y Fs(\()p Fk(n)p Fs(\))1292 2309 y Fk(r)1415 2286 y Fm(\000)20 b Fl(A)1574 2300 y Fk(r)1612 2286 y Fo(\)\()p Fl(X)1764 2238 y Fs(\()p Fk(r)r Fs(\))1757 2312 y Fk(n)p Fg(\000)p Fs(1)1915 2286 y Fm(\000)g Fl(X)2088 2249 y Fs(\()p Fk(r)r Fs(\))2182 2286 y Fo(\))p Fl(X)2299 2249 y Fs(\()p Fk(r)r Fs(\))2392 2286 y Fo(])2417 2131 y Fj(!)561 2531 y Fm(\024)83 b Fl(\030)759 2493 y Fs(2)798 2531 y Fo(\()p Fl(n)p Fo(\))p Fl(`)961 2493 y Fs(2)961 2553 y(2)1001 2531 y Fo(\()p Fl(X)1111 2545 y Fk(n)p Fg(\000)p Fs(1)1249 2531 y Fl(;)15 b(X)7 b Fo(\))21 b(+)f Fl(R)1588 2483 y Fs(\(2\))1587 2559 y(\006)1682 2531 y Fo(\()p Fl(n)p Fo(\))15 b(E)g Fl(X)1981 2493 y Fs(2)766 2761 y Fo(+)k(2)951 2647 y Fk(K)917 2674 y Fj(X)921 2869 y Fk(r)r Fs(=1)1064 2761 y Fm(k)p Fl(A)1177 2723 y Fs(\()p Fk(n)p Fs(\))1177 2783 y Fk(r)1299 2761 y Fm(\000)h Fl(A)1458 2775 y Fk(r)1496 2761 y Fm(k)1541 2775 y Fs(2)1581 2761 y Fm(k)p Fl(X)1708 2723 y Fs(\()p Fk(r)r Fs(\))1802 2761 y Fm(k)1847 2775 y Fs(2)1887 2761 y Fm(k)p Fl(A)2000 2723 y Fs(\()p Fk(n)p Fs(\))2000 2783 y Fk(r)2102 2761 y Fo(\()p Fl(X)2219 2713 y Fs(\()p Fk(r)r Fs(\))2212 2787 y Fk(n)p Fg(\000)p Fs(1)2370 2761 y Fm(\000)g Fl(X)2543 2723 y Fs(\()p Fk(r)r Fs(\))2637 2761 y Fo(\))p Fm(k)2717 2775 y Fs(2)561 3009 y Fo(=)83 b Fl(\030)759 2971 y Fs(2)798 3009 y Fo(\()p Fl(n)p Fo(\))p Fl(`)961 2971 y Fs(2)961 3031 y(2)1001 3009 y Fo(\()p Fl(X)1111 3023 y Fk(n)p Fg(\000)p Fs(1)1249 3009 y Fl(;)15 b(X)7 b Fo(\))21 b(+)f Fl(R)1588 2961 y Fs(\(2\))1587 3037 y(\006)1682 3009 y Fo(\()p Fl(n)p Fo(\))p Fm(k)p Fl(X)7 b Fm(k)1979 2971 y Fs(2)1979 3031 y(2)2040 3009 y Fo(+)20 b(2\()p Fl(R)2281 2961 y Fs(\(2\))2280 3037 y(\006)2376 3009 y Fo(\()p Fl(n)p Fo(\)\))2536 2971 y Fs(1)p Fk(=)p Fs(2)2647 3009 y Fm(k)p Fl(X)7 b Fm(k)2819 3023 y Fs(2)2860 3009 y Fl(C)2925 3023 y Fk(A)2982 3009 y Fl(`)3020 3023 y Fs(2)3059 3009 y Fo(\()p Fl(X)3169 3023 y Fk(n)p Fg(\000)p Fs(1)3307 3009 y Fl(;)15 b(X)7 b Fo(\))p Fl(:)-20 3188 y Fo(W)-8 b(e)32 b(denote)f(the)f(prefactors)h(and)f(a)h(constan)m(t)g(used)f (later)g(b)m(y)577 3379 y Fl(b)616 3393 y Fk(n)746 3379 y Fo(:=)83 b(2)p Fl(C)1035 3393 y Fk(A)1092 3379 y Fm(k)p Fl(X)7 b Fm(k)1264 3393 y Fs(2)1305 3379 y Fo(\()p Fl(R)1410 3331 y Fs(\(2\))1409 3407 y(\006)1505 3379 y Fo(\()p Fl(n)p Fo(\)\))1665 3341 y Fs(1)p Fk(=)p Fs(2)1775 3379 y Fl(;)577 3538 y(c)616 3552 y Fk(n)746 3538 y Fo(:=)83 b Fl(R)995 3501 y Fs(2)994 3561 y Fk(b)1034 3538 y Fo(\()p Fl(n)p Fo(\))21 b(+)f(2)p Fl(C)1381 3552 y Fk(A)1438 3538 y Fl(R)1507 3553 y Fk(b)1541 3538 y Fo(\()p Fl(n)p Fo(\))p Fl(M)10 b Fo(\()p Fl(n)21 b Fm(\000)f Fo(1\))h(+)f Fl(C)2230 3501 y Fg(0)2223 3561 y Fk(A)2280 3538 y Fl(M)2378 3501 y Fs(2)2417 3538 y Fo(\()p Fl(n)g Fm(\000)g Fo(1\))h(+)f Fl(R)2880 3490 y Fs(\(2\))2879 3567 y(\006)2974 3538 y Fo(\()p Fl(n)p Fo(\))p Fm(k)p Fl(X)7 b Fm(k)3271 3501 y Fs(2)3271 3561 y(2)3312 3538 y Fl(;)569 3705 y(d)616 3719 y Fk(n)746 3705 y Fo(:=)83 b(max)1109 3604 y Fj(n)1170 3705 y Fl(b)1209 3719 y Fk(n)1256 3705 y Fl(=\030)t(;)15 b(c)1424 3667 y Fs(1)p Fk(=)p Fs(2)1424 3727 y Fk(n)1535 3604 y Fj(o)1611 3705 y Fl(:)-20 3914 y Fo(Assume)21 b(that)g(there)g(exists)g(an)g Fl(`)k Fm(2)g Fo(N)c(suc)m(h)f(that)i (for)e(all)g Fl(n)25 b Fm(\025)g Fl(`)p Fo(,)e Fl(\030)t Fo(\()p Fl(n)p Fo(\))i Fm(2)g Fo([)p Fl(\030)t(=)p Fo(2)p Fl(;)15 b Fo(\(1)q(+)q Fl(\030)t Fo(\))p Fl(=)p Fo(2].)42 b(Denote)3411 3890 y(\026)3401 3914 y Fl(\030)29 b Fo(:=)d(\(1)q(+)q Fl(\030)t Fo(\))p Fl(=)p Fo(2.)-20 4027 y(Then)k(w)m(e)h(obtain)e (altogether)629 4238 y Fl(`)667 4252 y Fs(2)706 4238 y Fo(\()p Fl(X)816 4252 y Fk(n)864 4238 y Fl(;)15 b(X)7 b Fo(\))84 b Fm(\024)f Fl(R)1328 4252 y Fk(X)1395 4238 y Fo(\()p Fl(n)p Fo(\))21 b(+)1632 4127 y Fj(q)p 1723 4127 1570 4 v 111 x Fl(\030)1767 4212 y Fs(2)1806 4238 y Fo(\()p Fl(n)p Fo(\))p Fl(`)1969 4207 y Fs(2)1969 4264 y(2)2009 4238 y Fo(\()p Fl(X)2119 4252 y Fk(n)p Fg(\000)p Fs(1)2256 4238 y Fl(;)15 b(X)7 b Fo(\))22 b(+)e Fl(b)2565 4252 y Fk(n)2612 4238 y Fl(`)2650 4252 y Fs(2)2689 4238 y Fo(\()p Fl(X)2799 4252 y Fk(n)p Fg(\000)p Fs(1)2937 4238 y Fl(;)15 b(X)7 b Fo(\))21 b(+)f Fl(c)3245 4252 y Fk(n)1105 4410 y Fm(\024)83 b Fl(R)1328 4424 y Fk(X)1395 4410 y Fo(\()p Fl(n)p Fo(\))21 b(+)1632 4328 y Fj(p)p 1723 4328 969 4 v 82 x Fo(\()p Fl(\030)t Fo(\()p Fl(n)p Fo(\))p Fl(`)1965 4424 y Fs(2)2004 4410 y Fo(\()p Fl(X)2114 4424 y Fk(n)p Fg(\000)p Fs(1)2252 4410 y Fl(;)15 b(X)7 b Fo(\))22 b(+)e Fl(d)2569 4424 y Fk(n)2616 4410 y Fo(\))2651 4384 y Fs(2)1105 4548 y Fo(=)83 b Fl(R)1328 4562 y Fk(X)1395 4548 y Fo(\()p Fl(n)p Fo(\))21 b(+)f Fl(d)1679 4562 y Fk(n)1746 4548 y Fo(+)g Fl(\030)t Fo(\()p Fl(n)p Fo(\))p Fl(`)2044 4562 y Fs(2)2084 4548 y Fo(\()p Fl(X)2194 4562 y Fk(n)p Fg(\000)p Fs(1)2331 4548 y Fl(;)15 b(X)7 b Fo(\))1105 4773 y Fm(\024)1268 4749 y Fo(\026)1259 4773 y Fl(\030)1303 4735 y Fk(n)p Fg(\000)p Fk(`)1434 4773 y Fl(`)1472 4787 y Fs(2)1511 4773 y Fo(\()p Fl(X)1621 4788 y Fk(`)1655 4773 y Fl(;)15 b(X)7 b Fo(\))21 b(+)1924 4659 y Fk(n)p Fg(\000)p Fs(1)p Fg(\000)p Fk(`)1967 4687 y Fj(X)1975 4882 y Fk(i)p Fs(=0)2165 4749 y Fo(\026)2156 4773 y Fl(\030)2200 4735 y Fk(i)2228 4773 y Fo(\()p Fl(R)2332 4787 y Fk(X)2400 4773 y Fo(\()p Fl(n)f Fm(\000)g Fl(i)p Fo(\))g(+)g Fl(d)2825 4787 y Fk(n)p Fg(\000)p Fk(i)2952 4773 y Fo(\))1105 5090 y Fm(\024)1268 5066 y Fo(\026)1259 5090 y Fl(\030)1303 5052 y Fk(n)1359 5066 y Fo(\026)1350 5090 y Fl(\030)1394 5052 y Fg(\000)p Fk(`)1482 5090 y Fo(\()p Fm(k)p Fl(X)7 b Fm(k)1689 5104 y Fs(2)1750 5090 y Fo(+)20 b Fm(k)p Fl(X)1961 5105 y Fk(`)1995 5090 y Fm(k)2040 5104 y Fs(2)2080 5090 y Fo(\))g(+)2226 4976 y Fk(n)p Fg(\000)p Fs(1)2227 5003 y Fj(X)2235 5199 y Fk(i)p Fs(=0)2384 5066 y Fo(\026)2374 5090 y Fl(\030)2418 5052 y Fk(i)2446 5090 y Fo(\()p Fl(R)2550 5104 y Fk(X)2618 5090 y Fo(\()p Fl(n)g Fm(\000)g Fl(i)p Fo(\))h(+)f Fl(d)3044 5104 y Fk(n)p Fg(\000)p Fk(i)3170 5090 y Fo(\))p Fl(:)549 b Fo(\(36\))1915 5411 y(19)p eop %%Page: 20 20 20 19 bop -20 -15 a Fo(In)32 b(order)g(to)i(obtain)e(explicit)f (estimates)i(w)m(e)g(ha)m(v)m(e)h(to)f(sp)s(ecify)e(the)i(functions)e Fl(R)2846 -1 y Fs(\006)2901 -15 y Fl(;)15 b(R)3011 -63 y Fs(\(2\))3010 14 y(\006)3106 -15 y Fl(;)g(R)3215 0 y Fk(b)3249 -15 y Fl(;)g(R)3358 -1 y Fk(X)3426 -15 y Fo(.)47 b(W)-8 b(e)34 b(assume)-20 98 y(that)d(for)f(all)g Fl(n)25 b Fm(\025)f Fo(1,)753 313 y Fl(R)822 327 y Fs(\001)885 313 y Fo(\()p Fl(n)p Fo(\))h Fm(\025)1146 251 y Fo(1)p 1141 292 55 4 v 1141 375 a Fl(n)1206 313 y(;)106 b(R)1406 327 y Fs(\006)1461 313 y Fo(\()p Fl(n)p Fo(\))26 b Fm(\024)f Fl(C)1773 327 y Fs(\006)1838 251 y Fo(ln)o(\()p Fl(n)p Fo(\))p 1838 292 202 4 v 1911 375 a Fl(n)2049 313 y(;)106 b(R)2250 265 y Fs(\(2\))2249 342 y(\006)2344 313 y Fo(\()p Fl(n)p Fo(\))26 b Fm(\024)f Fl(C)2663 265 y Fs(\(2\))2656 342 y(\006)2766 251 y Fo(ln)o(\()p Fl(n)p Fo(\))p 2766 292 V 2820 375 a Fl(n)2875 349 y Fs(2)2977 313 y Fl(;)777 b Fo(\(37\))935 627 y Fl(R)1004 642 y Fk(b)1038 627 y Fo(\()p Fl(n)p Fo(\))26 b Fm(\024)1295 566 y Fl(C)1360 581 y Fk(b)p 1295 606 100 4 v 1298 689 a Fl(n)1353 663 y Fs(2)1404 627 y Fl(;)106 b(R)1604 641 y Fk(X)1672 627 y Fo(\()p Fl(n)p Fo(\))25 b Fm(\024)1928 566 y Fl(C)1993 580 y Fk(X)p 1928 606 133 4 v 1947 689 a Fl(n)2002 663 y Fs(2)2070 627 y Fl(;)107 b Fm(j)p Fl(\030)t Fo(\()p Fl(n)p Fo(\))20 b Fm(\000)g Fl(\030)t Fm(j)26 b(\024)2708 564 y Fl(C)2773 579 y Fk(\030)p 2708 606 103 4 v 2732 689 a Fl(n)-20 855 y Fo(with)j(constan)m(ts)j Fl(C)655 869 y Fs(\006)710 855 y Fl(;)15 b(C)822 807 y Fs(\(2\))815 883 y(\006)916 855 y Fl(;)g(C)1021 870 y Fk(b)1056 855 y Fl(;)g(C)1161 869 y Fk(X)1229 855 y Fl(;)g(C)1334 870 y Fk(\030)1397 855 y Fl(>)25 b Fo(0)31 b(and)f(the)g(con)m(traction)i (factor)f Fl(\030)j Fo(giv)m(en)d(in)e(\(2\).)121 968 y(In)h(order)g(to)h(mak)m(e)g(the)g(previous)e(estimates)i(explicit)e (w)m(e)h(start)h(with)e(t)m(w)m(o)j(Lemmas:)-20 1128 y Fp(Lemma)h(4.1)46 b Fc(F)-7 b(or)34 b(al)5 b(l)33 b Fl(n)25 b Fm(2)g Fo(N)32 b Fc(we)h(have)332 1372 y Fm(k)p Fl(X)452 1386 y Fk(n)499 1372 y Fm(k)544 1386 y Fg(1)645 1372 y Fm(\024)25 b Fl(Q)813 1386 y Fk(n)885 1372 y Fo(:=)1006 1217 y Fj(\()1176 1278 y Fo(\()p Fl(C)1276 1292 y Fk(X)1364 1278 y Fo(+)20 b Fl(C)1520 1293 y Fk(b)1574 1278 y Fo(+)g Fm(k)p Fl(g)s Fm(k)1801 1292 y Fg(1)1877 1278 y Fo(\)\()p Fl(n)h Fo(+)f Fl(C)2179 1292 y Fs(\006)2234 1278 y Fo(\))15 b(ln)o(\()p Fl(n)20 b Fo(+)g(1\))p Fl(;)139 b Fc(if)33 b Fo(0)25 b Fl(<)g(C)3125 1292 y Fs(\006)3206 1278 y Fm(\024)f Fo(1)p Fl(;)1121 1466 y(\020)7 b Fo(\()p Fm(d)p Fl(C)1308 1480 y Fs(\006)1363 1466 y Fm(e)p Fo(\)\()p Fl(C)1538 1480 y Fk(X)1627 1466 y Fo(+)20 b Fl(C)1783 1481 y Fk(b)1837 1466 y Fo(+)g Fm(k)p Fl(g)s Fm(k)2064 1480 y Fg(1)2140 1466 y Fo(\)\()p Fl(n)h Fo(+)f Fl(C)2442 1480 y Fs(\006)2497 1466 y Fo(\))2532 1433 y Fg(d)p Fk(C)2613 1444 y Fi(\006)2661 1433 y Fg(e)2697 1466 y Fl(;)83 b Fc(if)33 b Fl(C)2959 1480 y Fs(\006)3039 1466 y Fl(>)25 b Fo(1)p Fl(;)-20 1640 y Fc(wher)-5 b(e)34 b Fl(\020)7 b Fo(\()p Fm(\001)p Fo(\))33 b Fc(denotes)h(the)f(R)n(iemannian)g Fl(\020)7 b Fc(-function,)32 b Fl(\020)7 b Fo(\()p Fl(s)p Fo(\))25 b(:=)2176 1572 y Fj(P)2272 1667 y Fk(n)p Fg(\025)p Fs(1)2424 1640 y Fl(n)2479 1607 y Fg(\000)p Fk(s)2571 1640 y Fc(.)-20 1801 y Fp(Pro)s(of:)77 b Fo(By)30 b(de\014nition)e(of)j Fl(X)1064 1815 y Fk(n)1111 1801 y Fo(,)262 1973 y Fm(k)p Fl(X)382 1987 y Fk(n)430 1973 y Fm(k)475 1987 y Fg(1)633 1973 y Fm(\024)83 b(k)p Fl(X)907 1987 y Fk(n)975 1973 y Fm(\000)1092 1950 y Fo(~)1065 1973 y Fl(X)1140 1987 y Fk(n)1188 1973 y Fm(k)1233 1987 y Fg(1)1328 1973 y Fo(+)20 b Fm(k)1491 1950 y Fo(~)1464 1973 y Fl(X)1539 1987 y Fk(n)1587 1973 y Fm(k)1632 1987 y Fg(1)633 2197 y Fm(\024)83 b Fl(R)856 2211 y Fk(X)923 2197 y Fo(\()p Fl(n)p Fo(\))21 b(+)1159 2038 y Fj(\015)1159 2093 y(\015)1159 2147 y(\015)1159 2202 y(\015)1159 2256 y(\015)1243 2084 y Fk(K)1210 2111 y Fj(X)1214 2305 y Fk(r)r Fs(=1)1356 2197 y Fl(A)1424 2160 y Fs(\()p Fk(n)p Fs(\))1424 2220 y Fk(r)1526 2197 y Fl(X)1608 2149 y Fs(\()p Fk(r)r Fs(\))1601 2223 y Fk(n)p Fg(\000)p Fs(1)1759 2197 y Fo(+)f Fl(b)1889 2160 y Fs(\()p Fk(n)p Fs(\))1991 2038 y Fj(\015)1991 2093 y(\015)1991 2147 y(\015)1991 2202 y(\015)1991 2256 y(\015)2041 2318 y Fg(1)633 2520 y Fm(\024)83 b Fl(R)856 2534 y Fk(X)923 2520 y Fo(\()p Fl(n)p Fo(\))21 b(+)e Fl(R)1228 2535 y Fk(b)1263 2520 y Fo(\()p Fl(n)p Fo(\))h(+)g Fm(k)p Fl(b)p Fm(k)1628 2534 y Fg(1)1724 2520 y Fo(+)1848 2406 y Fk(K)1815 2433 y Fj(X)1819 2628 y Fk(r)r Fs(=1)1961 2520 y Fm(k)p Fl(A)2074 2482 y Fs(\()p Fk(n)p Fs(\))2074 2542 y Fk(r)2197 2520 y Fm(\000)g Fl(A)2356 2534 y Fk(r)2394 2520 y Fm(k)2439 2534 y Fg(1)2514 2520 y Fm(k)p Fl(X)2641 2472 y Fs(\()p Fk(r)r Fs(\))2634 2546 y Fk(n)p Fg(\000)p Fs(1)2772 2520 y Fm(k)2817 2534 y Fg(1)2913 2520 y Fo(+)3004 2361 y Fj(\015)3004 2415 y(\015)3004 2470 y(\015)3004 2524 y(\015)3004 2579 y(\015)3088 2406 y Fk(K)3054 2433 y Fj(X)3058 2628 y Fk(r)r Fs(=1)3200 2520 y Fl(A)3268 2534 y Fk(r)3307 2520 y Fl(X)3389 2472 y Fs(\()p Fk(r)r Fs(\))3382 2546 y Fk(n)p Fg(\000)p Fs(1)3519 2361 y Fj(\015)3519 2415 y(\015)3519 2470 y(\015)3519 2524 y(\015)3519 2579 y(\015)3570 2641 y Fg(1)633 2772 y Fm(\024)83 b Fl(C)852 2786 y Fk(X)939 2772 y Fo(+)20 b Fl(C)1095 2787 y Fk(b)1150 2772 y Fo(+)g Fm(k)p Fl(g)s Fm(k)1377 2786 y Fg(1)1473 2772 y Fo(+)g(\(1)h(+)f Fl(R)1825 2786 y Fs(\006)1880 2772 y Fo(\()p Fl(n)p Fo(\)\))p Fm(k)p Fl(X)2167 2724 y Fs(\()p Fk(r)r Fs(\))2160 2798 y Fk(n)p Fg(\000)p Fs(1)2298 2772 y Fm(k)633 3024 y(\024)787 2923 y Fj(\020)841 3024 y Fl(R)910 3038 y Fk(X)977 3024 y Fo(\()p Fl(n)p Fo(\))h(+)f Fl(R)1283 3039 y Fk(b)1317 3024 y Fo(\()p Fl(n)p Fo(\))h(+)e Fm(k)p Fl(g)s Fm(k)1689 3038 y Fg(1)1766 2923 y Fj(\021)1879 2911 y Fk(n)1835 2938 y Fj(X)1840 3133 y Fk(j)t Fs(=1)1982 2842 y Fj(0)1982 3005 y(@)2061 2911 y Fk(n)p Fg(\000)p Fs(1)2070 2938 y Fj(Y)2072 3133 y Fk(i)p Fs(=)p Fk(j)2194 3024 y Fo(\(1)i(+)f Fl(R)2455 3038 y Fs(\006)2510 3024 y Fo(\()p Fl(i)h Fo(+)f(1\)\))2803 2842 y Fj(1)2803 3005 y(A)2899 3024 y Fl(:)-20 3301 y Fo(With)30 b Fl(R)284 3315 y Fs(\006)339 3301 y Fo(\()p Fl(n)p Fo(\))c Fm(\024)f Fl(C)651 3315 y Fs(\006)706 3301 y Fl(=n)p Fo(,)30 b(w)m(e)h(obtain) 1162 3443 y Fk(n)p Fg(\000)p Fs(1)1171 3470 y Fj(Y)1173 3666 y Fk(i)p Fs(=)p Fk(j)1295 3556 y Fo(\(1)21 b(+)f Fl(R)1556 3570 y Fs(\006)1611 3556 y Fo(\()p Fl(i)h Fo(+)f(1\)\))27 b Fm(\024)2037 3495 y Fo(\()p Fl(n)20 b Fo(+)g Fl(C)2303 3509 y Fs(\006)2358 3495 y Fo(\))2393 3462 y Fg(d)p Fk(C)2474 3473 y Fi(\006)2522 3462 y Fg(e)p 2037 3536 522 4 v 2080 3623 a Fo(\()p Fl(j)26 b Fo(+)20 b(1\))2349 3596 y Fg(d)p Fk(C)2430 3607 y Fi(\006)2479 3596 y Fg(e)2568 3556 y Fl(:)-20 3818 y Fo(Th)m(us,)695 4055 y Fm(k)p Fl(X)815 4069 y Fk(n)862 4055 y Fm(k)907 4069 y Fg(1)1008 4055 y Fm(\024)1104 3954 y Fj(\020)1158 4055 y Fl(C)1223 4069 y Fk(X)1311 4055 y Fo(+)f Fl(C)1466 4070 y Fk(b)1521 4055 y Fo(+)h Fm(k)p Fl(g)s Fm(k)1748 4069 y Fg(1)1824 3954 y Fj(\021)1878 4055 y Fo(\()p Fl(n)h Fo(+)e Fl(C)2144 4069 y Fs(\006)2200 4055 y Fo(\))2235 4017 y Fg(d)p Fk(C)2316 4028 y Fi(\006)2364 4017 y Fg(e)2459 3941 y Fk(n)2415 3968 y Fj(X)2419 4164 y Fk(j)t Fs(=1)2546 4055 y Fo(\()p Fl(j)26 b Fo(+)20 b(1\))2815 4017 y Fg(\000d)p Fk(C)2951 4028 y Fi(\006)3000 4017 y Fg(e)3036 4055 y Fl(;)-20 4324 y Fo(whic)m(h)29 b(leads)h(to)h(the)g(assertion.)p 3870 4324 71 71 v -20 4570 a Fp(Lemma)i(4.2)46 b Fc(We)32 b(have)1160 4803 y Fm(8)15 b Fo(0)25 b Fl(<)1402 4779 y Fo(\026)1392 4803 y Fl(\030)k(<)c Fo(1)p Fl(;)31 b Fm(8)15 b Fl(n)25 b Fm(\025)g Fo(1)h(:)2021 4689 y Fk(n)p Fg(\000)p Fs(1)2022 4716 y Fj(X)2030 4912 y Fk(i)p Fs(=0)2251 4717 y Fo(\026)2242 4741 y Fl(\030)2286 4708 y Fk(i)p 2179 4782 197 4 v 2179 4865 a Fl(n)20 b Fm(\000)g Fl(i)2411 4803 y Fm(\024)2650 4741 y Fo(1)p 2517 4782 311 4 v 2517 4868 a(\(1)h Fm(\000)2719 4844 y Fo(\026)2709 4868 y Fl(\030)t Fo(\))2788 4842 y Fs(2)2868 4741 y Fo(1)p 2863 4782 55 4 v 2863 4865 a Fl(n)3779 4803 y Fo(\(38\))1160 5074 y Fm(8)15 b Fo(0)25 b Fl(<)1402 5050 y Fo(\026)1392 5074 y Fl(\030)k(<)c Fo(1)p Fl(;)31 b Fm(8)15 b Fl(n)25 b Fm(\025)g Fo(1)h(:)2030 5050 y(\026)2021 5074 y Fl(\030)2065 5036 y Fk(n)2137 5074 y Fm(\024)2390 5013 y Fo(1)p 2243 5053 339 4 v 2243 5140 a Fl(e)15 b Fo(ln)o(\(1)p Fl(=)2510 5116 y Fo(\026)2501 5140 y Fl(\030)6 b Fo(\))2622 5013 y(1)p 2617 5053 55 4 v 2617 5136 a Fl(n)2682 5074 y(:)1072 b Fo(\(39\))1915 5411 y(20)p eop %%Page: 21 21 21 20 bop -20 -21 a Fp(Pro)s(of:)77 b Fo(F)-8 b(or)31 b(\(38\))h(note)f(that)g(1)p Fl(=)p Fo(\()p Fl(n)21 b Fm(\000)f Fl(i)p Fo(\))25 b Fm(\024)g Fo(\()p Fl(i)c Fo(+)f(1\))p Fl(=n)31 b Fo(for)f(all)g Fl(n)25 b Fm(\025)f Fo(1)31 b(and)f(0)c Fm(\024)f Fl(i)g Fm(\024)g Fl(n)20 b Fm(\000)g Fo(1.)41 b(This)29 b(implies)746 153 y Fk(n)p Fg(\000)p Fs(1)746 180 y Fj(X)755 376 y Fk(i)p Fs(=0)976 181 y Fo(\026)966 205 y Fl(\030)1010 172 y Fk(i)p 904 246 197 4 v 904 329 a Fl(n)20 b Fm(\000)f Fl(i)1136 267 y Fm(\024)1246 205 y Fo(1)p 1242 246 55 4 v 1242 329 a Fl(n)1322 153 y Fk(n)p Fg(\000)p Fs(1)1322 180 y Fj(X)1331 376 y Fk(i)p Fs(=0)1455 267 y Fo(\()p Fl(i)h Fo(+)g(1\))1721 243 y(\026)1712 267 y Fl(\030)1756 229 y Fk(i)1811 267 y Fm(\024)1921 205 y Fo(1)p 1916 246 V 1916 329 a Fl(n)2027 153 y Fg(1)1996 180 y Fj(X)2005 376 y Fk(i)p Fs(=0)2127 267 y Fo(\()p Fl(i)h Fo(+)f(1\))2394 243 y(\026)2385 267 y Fl(\030)2429 229 y Fk(i)2483 267 y Fm(\024)2722 205 y Fo(1)p 2589 246 311 4 v 2589 332 a(\(1)h Fm(\000)2791 308 y Fo(\026)2781 332 y Fl(\030)t Fo(\))2860 306 y Fs(2)2925 205 y Fo(1)p 2920 246 55 4 v 2920 329 a Fl(n)2984 267 y(:)-20 560 y Fo(F)-8 b(or)36 b(\(39\))g(note)f(that)g(the)g(function)e Fl(x)g Fm(7!)f Fl(x)1541 536 y Fo(\026)1532 560 y Fl(\030)1576 527 y Fk(x)1620 560 y Fo(,)k Fl(x)c Fm(\025)g Fo(0)j(has)f(its)g(maxim) m(um)g(at)h Fl(x)d Fo(=)h(1)p Fl(=)15 b Fo(ln)o(\(1)p Fl(=)3292 536 y Fo(\026)3283 560 y Fl(\030)6 b Fo(\))34 b(whic)m(h)g(implies)-20 673 y(the)d(assertion.)p 3870 673 71 71 v -20 919 a Fp(Lemma)i(4.3)46 b Fc(L)-5 b(et)33 b Fo(\()p Fl(X)797 933 y Fk(n)845 919 y Fo(\))h Fc(b)-5 b(e)33 b(given)f(by)h(\(25\),\(26\))i(with)f Fl(A)2043 871 y Fs(\()p Fk(n)p Fs(\))2043 931 y Fk(r)2146 919 y Fl(;)15 b(b)2225 886 y Fs(\()p Fk(n)p Fs(\))2327 919 y Fl(;)g Fm(h)2426 896 y Fj(e)2402 919 y Fl(X)2477 933 y Fk(n)2525 919 y Fm(i)33 b Fc(satisfying)h(\(27\)-\(33\))h(with)f Fl(R)3665 933 y Fs(\001)3728 919 y Fc(,)f Fl(R)3858 933 y Fs(\006)3913 919 y Fc(,)-20 1048 y Fl(R)50 1000 y Fs(\(2\))49 1076 y(\006)144 1048 y Fc(,)g Fl(R)274 1063 y Fk(b)308 1048 y Fc(,)g Fl(R)438 1062 y Fk(X)537 1048 y Fc(satisfying)h(\(37\).) 43 b(Then,)33 b(for)g(al)5 b(l)33 b Fl(n)25 b Fm(\025)g Fo(3)1462 1305 y Fl(`)1500 1319 y Fs(2)1539 1305 y Fo(\()p Fl(X)1649 1319 y Fk(n)1697 1305 y Fl(;)15 b(X)7 b Fo(\))26 b Fm(\024)f Fl(C)2057 1244 y Fo(ln)o(\()p Fl(n)p Fo(\))p 2057 1285 202 4 v 2131 1368 a Fl(n)2268 1305 y(;)-20 1531 y Fc(wher)-5 b(e)34 b Fl(C)39 b Fc(is)33 b(given)f(by)1068 1798 y Fl(C)g Fo(:=)1296 1737 y Fm(k)p Fl(g)s Fm(k)1432 1704 y Fs(2)1432 1759 y Fg(1)1508 1737 y Fl(=)p Fo(\(1)22 b Fm(\000)e Fl(\030)t Fo(\))g(+)g Fm(k)p Fl(X)2056 1752 y Fk(`)2090 1737 y Fm(k)2135 1751 y Fg(1)p 1296 1777 914 4 v 1545 1864 a Fl(e)15 b Fo(ln\(1)p Fl(=)1813 1840 y Fo(\026)1804 1864 y Fl(\030)5 b Fo(\))1893 1840 y(\026)1884 1864 y Fl(\030)1928 1837 y Fk(`)2240 1798 y Fo(+)2479 1714 y Fj(e)2461 1737 y Fl(C)p 2341 1777 311 4 v 2341 1864 a Fo(\(1)21 b Fm(\000)2542 1840 y Fo(\026)2533 1864 y Fl(\030)t Fo(\))2612 1837 y Fs(2)2662 1798 y Fl(;)-20 2075 y Fc(with)196 2052 y Fj(e)178 2075 y Fl(C)6 b(;)15 b(`)33 b Fc(de\014ne)-5 b(d)34 b(in)e(\(40\),)i(\(41\),)1243 2051 y Fo(\026)1234 2075 y Fl(\030)29 b Fo(:=)c(\(1)c(+)f Fl(\030)t Fo(\))p Fl(=)p Fo(2)34 b Fc(and)f Fm(k)p Fl(X)2115 2090 y Fk(`)2149 2075 y Fm(k)2194 2089 y Fg(1)2302 2075 y Fc(estimate)-5 b(d)34 b(in)e(L)-5 b(emma)34 b(4.1.)-20 2287 y Fp(Pro)s(of:)77 b Fo(With)29 b(\(34\))j(and)e(\(37\))i(it)e(is) 1068 2596 y Fl(M)10 b Fo(\()p Fl(n)p Fo(\))84 b Fm(\024)1573 2482 y Fk(n)1528 2510 y Fj(X)1533 2705 y Fk(j)t Fs(=1)1675 2413 y Fj(0)1675 2577 y(@)1754 2482 y Fk(n)p Fg(\000)p Fs(1)1763 2510 y Fj(Y)1765 2705 y Fk(i)p Fs(=)p Fk(j)1991 2534 y Fl(i)p 1912 2575 188 4 v 1912 2658 a(i)21 b Fo(+)f(1)2110 2413 y Fj(1)2110 2577 y(A)2205 2596 y Fo(\()p Fl(R)2309 2611 y Fk(b)2344 2596 y Fo(\()p Fl(j)5 b Fo(\))21 b(+)f Fl(R)2637 2610 y Fk(X)2705 2596 y Fo(\()p Fl(j)5 b Fo(\)\))1375 2921 y(=)1543 2860 y(1)p 1538 2900 55 4 v 1538 2984 a Fl(n)1662 2808 y Fk(n)1618 2835 y Fj(X)1622 3030 y Fk(j)t Fs(=1)1764 2921 y Fl(j)g Fo(\()p Fl(R)1910 2936 y Fk(b)1946 2921 y Fo(\()p Fl(j)g Fo(\))21 b(+)f Fl(R)2239 2935 y Fk(X)2307 2921 y Fo(\()p Fl(j)5 b Fo(\)\))1375 3209 y Fm(\024)82 b Fo(\()p Fl(C)1628 3224 y Fk(b)1683 3209 y Fo(+)20 b Fl(C)1839 3223 y Fk(X)1906 3209 y Fo(\))1951 3148 y Fl(H)2027 3162 y Fk(n)p 1951 3188 123 4 v 1986 3272 a Fl(n)1375 3434 y Fm(\024)82 b Fo(\()p Fl(C)1628 3449 y Fk(b)1683 3434 y Fo(+)20 b Fl(C)1839 3448 y Fk(X)1906 3434 y Fo(\))1951 3373 y(1)h(+)f(ln)o(\()p Fl(n)p Fo(\))p 1951 3414 358 4 v 2103 3497 a Fl(n)2319 3434 y(;)107 b(n)24 b Fm(\025)h Fo(1)p Fl(:)-20 3660 y Fo(De\014ning)412 3858 y Fj(e)394 3881 y Fl(C)90 b Fo(:=)83 b Fl(C)793 3895 y Fk(X)880 3881 y Fo(+)20 b(max)1156 3781 y Fj(n)1216 3881 y Fo(2)p Fl(C)1326 3895 y Fk(A)1384 3881 y Fl(C)1456 3834 y Fs(\(2\))1449 3910 y(\006)1550 3881 y Fm(k)p Fl(X)7 b Fm(k)1722 3895 y Fs(2)1762 3881 y Fl(=\030)t(;)1903 b Fo(\(40\))1182 4078 y(\()p Fl(C)1289 4041 y Fs(2)1282 4101 y Fk(b)1349 4078 y Fo(+)20 b Fl(C)1512 4030 y Fs(\(2\))1505 4107 y(\006)1606 4078 y Fm(k)p Fl(X)7 b Fm(k)1778 4041 y Fs(2)1778 4101 y(2)1839 4078 y Fo(+)20 b(2)p Fl(C)2040 4092 y Fk(A)2097 4078 y Fl(C)2162 4093 y Fk(b)2196 4078 y Fo(\()p Fl(C)2296 4093 y Fk(b)2351 4078 y Fo(+)g Fl(C)2507 4092 y Fk(X)2574 4078 y Fo(\))h(+)f Fl(C)2793 4041 y Fg(0)2786 4101 y Fk(A)2843 4078 y Fo(\()p Fl(C)2943 4093 y Fk(b)2998 4078 y Fo(+)g Fl(C)3154 4092 y Fk(X)3221 4078 y Fo(\))3256 4041 y Fs(2)3296 4078 y Fo(\))3331 4041 y Fs(1)p Fk(=)p Fs(2)3441 3977 y Fj(o)3502 4078 y Fl(;)-20 4300 y Fo(w)m(e)31 b(ha)m(v)m(e)890 4529 y Fl(R)959 4543 y Fk(X)1026 4529 y Fo(\()p Fl(n)20 b Fm(\000)g Fl(i)p Fo(\))h(+)f Fl(d)1452 4543 y Fk(n)p Fg(\000)p Fk(i)1603 4529 y Fm(\024)1717 4506 y Fj(e)1699 4529 y Fl(C)1781 4468 y Fo(1)g Fm(_)g Fo(ln)o(\()p Fl(n)g Fm(\000)g Fl(i)p Fo(\))p 1781 4509 490 4 v 1927 4592 a Fl(n)g Fm(\000)g Fl(i)2306 4529 y Fm(\024)2420 4506 y Fj(e)2402 4529 y Fl(C)2483 4468 y Fo(1)h Fm(_)f Fo(ln)n(\()p Fl(n)p Fo(\))p 2483 4509 348 4 v 2558 4592 a Fl(n)g Fm(\000)g Fl(i)2841 4529 y(:)-20 4761 y Fo(Set)1494 4993 y Fl(`)25 b Fo(:=)1679 4864 y Fj(\030)1876 4929 y Fo(2)p Fl(C)1986 4944 y Fk(\030)p 1742 4972 417 4 v 1742 5055 a Fl(\030)f Fm(^)c Fo(\(1)h Fm(\000)e Fl(\030)t Fo(\))2168 4864 y Fj(\031)2236 4993 y Fl(;)1518 b Fo(\(41\))1915 5411 y(21)p eop %%Page: 22 22 22 21 bop -20 -21 a Fo(so)31 b(that)g Fl(\030)t Fo(\()p Fl(n)p Fo(\))25 b Fm(2)g Fo([)p Fl(\030)t(=)p Fo(2)p Fl(;)15 b Fo(\(1)23 b(+)d Fl(\030)t Fo(\))p Fl(=)p Fo(2])31 b(for)g Fl(n)24 b Fm(\025)h Fl(`)p Fo(,)31 b(and)f(w)m(e)g(obtain)g (with)f(\(36\))j(and)e(Lemma)h(4.2,)583 259 y Fl(`)621 273 y Fs(2)660 259 y Fo(\()p Fl(X)770 273 y Fk(n)818 259 y Fl(;)15 b(X)7 b Fo(\))84 b Fm(\024)1222 235 y Fo(\026)1213 259 y Fl(\030)1257 222 y Fk(n)1313 235 y Fo(\026)1304 259 y Fl(\030)1348 222 y Fg(\000)p Fk(`)1435 259 y Fo(\()p Fm(k)p Fl(X)7 b Fm(k)1642 273 y Fs(2)1704 259 y Fo(+)20 b Fm(k)p Fl(X)1915 274 y Fk(`)1948 259 y Fm(k)1993 273 y Fs(2)2033 259 y Fo(\))h(+)2198 236 y Fj(e)2180 259 y Fl(C)2266 146 y Fk(n)p Fg(\000)p Fs(1)2267 173 y Fj(X)2276 368 y Fk(i)p Fs(=0)2424 235 y Fo(\026)2415 259 y Fl(\030)2459 222 y Fk(i)2497 198 y Fo(1)f Fm(_)g Fo(ln)o(\()p Fl(n)p Fo(\))p 2497 238 348 4 v 2572 322 a Fl(n)g Fm(\000)g Fl(i)1059 561 y Fm(\024)1408 499 y Fo(1)p 1223 540 416 4 v 1223 626 a Fl(e)15 b Fo(ln)o(\(1)p Fl(=)1490 602 y Fo(\026)1481 626 y Fl(\030)5 b Fo(\))1570 602 y(\026)1561 626 y Fl(\030)1605 600 y Fk(`)1664 432 y Fj(\022)1740 499 y Fm(k)p Fl(g)s Fm(k)1876 466 y Fs(2)1876 522 y Fg(1)p 1740 540 213 4 v 1746 623 a Fo(1)21 b Fm(\000)f Fl(\030)1983 561 y Fo(+)g Fm(k)p Fl(X)2194 576 y Fk(`)2228 561 y Fm(k)2273 575 y Fg(1)2348 432 y Fj(\023)2444 499 y Fo(1)p 2440 540 55 4 v 2440 623 a Fl(n)2524 561 y Fo(+)2763 476 y Fj(e)2745 499 y Fl(C)p 2625 540 311 4 v 2625 626 a Fo(\(1)h Fm(\000)2827 602 y Fo(\026)2817 626 y Fl(\030)2861 600 y Fs(2)2901 626 y Fo(\))2956 499 y(1)g Fm(_)e Fo(ln)o(\()p Fl(n)p Fo(\))p 2956 540 348 4 v 3102 623 a Fl(n)3313 561 y(;)-20 812 y Fo(whic)m(h)29 b(implies)f(the)j(assertion)p 3870 812 71 71 v -20 1058 a(In)i(the)g(follo)m(wing)e(transp)s(osition) g(of)i(the)g Fl(`)1497 1072 y Fs(2)1569 1058 y Fo(rate)h(of)f(con)m(v)m (ergence)j(for)c(\()p Fl(X)2622 1072 y Fk(n)2670 1058 y Fo(\))h(in)m(to)g(a)h(rate)f(in)f(the)h(Kolmogoro)m(v)-20 1171 y(metric)c(w)m(e)g(use)g(an)g(estimate)h(of)f(Lemma)g(5.1)h(in)e (Fill)f(and)h(Janson)g([13)q(].)41 b(The)28 b(Kolmogoro)m(v)i(metric)f (is)f(denoted)-20 1283 y(b)m(y)1278 1480 y Fl(\045)p Fo(\()p Fl(\025;)15 b(\027)6 b Fo(\))26 b(:=)g(sup)1686 1558 y Fk(x)p Fg(2)p Fs(R)1840 1480 y Fm(j)p Fl(F)1923 1495 y Fk(\025)1969 1480 y Fo(\()p Fl(x)p Fo(\))21 b Fm(\000)f Fl(F)2261 1494 y Fk(\027)2304 1480 y Fo(\()p Fl(x)p Fo(\))p Fm(j)p Fl(;)-20 1727 y Fo(where)30 b Fl(F)301 1742 y Fk(\025)347 1727 y Fl(;)15 b(F)445 1741 y Fk(\027)519 1727 y Fo(denote)31 b(the)g(distribution)c(functions)h(of)j Fl(\025;)15 b(\027)31 b Fm(2)25 b(M)p Fo(.)-20 1932 y Fp(Lemma)33 b(4.4)46 b Fc(L)-5 b(et)33 b Fo(\()p Fl(X)797 1946 y Fk(n)845 1932 y Fo(\))f Fc(and)i Fl(C)39 b Fc(b)-5 b(e)33 b(as)g(in)f(L)-5 b(emma)34 b(4.3.)43 b(Then,)32 b(for)i(al)5 b(l)33 b Fl(n)25 b Fm(\025)g Fo(3)p Fc(:)1109 2229 y Fl(\045)p Fo(\()p Fl(X)1266 2243 y Fk(n)1313 2229 y Fl(;)15 b(X)7 b Fo(\))27 b Fm(\024)e Fo(2\()p Fl(C)7 b Fm(k)p Fl(w)1857 2192 y Fg(0)1881 2229 y Fm(k)1926 2243 y Fg(1)2001 2229 y Fo(\))2036 2192 y Fs(2)p Fk(=)p Fs(3)2147 2074 y Fj( )2229 2168 y Fo(ln)n(\()p Fl(n)p Fo(\))p 2229 2208 202 4 v 2302 2291 a Fl(n)2440 2074 y Fj(!)2512 2096 y Fs(2)p Fk(=)p Fs(3)2622 2229 y Fl(:)-20 2506 y Fp(Pro)s(of:)76 b Fo(F)-8 b(or)31 b(the)f(transp)s(osition)e(of) i(the)h Fl(`)1504 2520 y Fs(2)1573 2506 y Fo(rate)g(in)e(Lemma)h(4.3)h (in)m(to)f(a)h(rate)g(in)d(the)j(Kolmogoro)m(v)g(metric)f(w)m(e)-20 2619 y(note)k(that)f(the)g(b)s(ounded)d(deriv)-5 b(ativ)m(e)32 b(of)h(the)g(densit)m(y)f Fl(f)41 b Fo(implies)30 b(that)j(the)g(mo)s (dulus)d(of)j(con)m(tin)m(uit)m(y)f(\001)3653 2633 y Fk(X)3753 2619 y Fo(of)h Fl(X)-20 2732 y Fo(is)d(estimated)g(b)m(y)h (\001)689 2746 y Fk(X)756 2732 y Fo(\()p Fl(t)p Fo(\))26 b Fm(\024)f(k)p Fl(w)1093 2699 y Fg(0)1117 2732 y Fm(k)1162 2746 y Fg(1)1237 2732 y Fl(t)30 b Fo(for)g(all)g Fl(t)25 b(>)g Fo(0.)41 b(Using)29 b(the)i(inequalit)m(y)1184 2929 y Fl(\045)p Fo(\()p Fl(X)1341 2943 y Fk(n)1389 2929 y Fl(;)15 b(X)7 b Fo(\))26 b Fm(\024)f Fl(`)1706 2891 y Fs(2)1706 2951 y(2)1745 2929 y Fo(\()p Fl(X)1855 2943 y Fk(n)1903 2929 y Fl(;)15 b(X)7 b Fo(\))p Fl(t)2093 2891 y Fg(\000)p Fs(2)2209 2929 y Fo(+)19 b(\001)2375 2943 y Fk(X)2443 2929 y Fo(\()p Fl(t)p Fo(\))p Fl(;)-20 3125 y Fo(v)-5 b(alid)29 b(for)h(all)f Fl(t)d(>)e Fo(0)31 b(this)e(implies)1184 3376 y Fl(\045)p Fo(\()p Fl(X)1341 3390 y Fk(n)1388 3376 y Fl(;)15 b(X)7 b Fo(\))27 b Fm(\024)1678 3315 y Fl(C)1750 3282 y Fs(2)1804 3315 y Fo(ln)1879 3278 y Fs(2)1919 3315 y Fo(\()p Fl(n)p Fo(\))p 1678 3355 367 4 v 1814 3438 a Fl(n)1869 3412 y Fs(2)2077 3315 y Fo(1)p 2064 3355 73 4 v 2064 3438 a Fl(t)2097 3412 y Fs(2)2166 3376 y Fo(+)20 b Fm(k)p Fl(w)2369 3339 y Fg(0)2394 3376 y Fm(k)2439 3390 y Fg(1)2514 3376 y Fl(t;)-20 3594 y Fo(for)30 b(all)g Fl(t)25 b(>)g Fo(0.)41 b(W)-8 b(e)31 b(c)m(ho)s(ose)1385 3854 y Fl(t)25 b Fo(=)g Fl(t)1572 3868 y Fk(n)1644 3854 y Fo(=)1740 3726 y Fj(\022)1817 3792 y Fl(C)1889 3759 y Fs(2)1943 3792 y Fo(ln)2019 3756 y Fs(2)2058 3792 y Fo(\()p Fl(n)p Fo(\))p 1817 3833 367 4 v 1825 3916 a Fm(k)p Fl(w)1937 3890 y Fg(0)1961 3916 y Fm(k)2006 3930 y Fg(1)2081 3916 y Fl(n)2136 3890 y Fs(2)2193 3726 y Fj(\023)2260 3739 y Fs(1)p Fk(=)p Fs(3)-20 4103 y Fo(whic)m(h)k(leads)h(to)h(the)g(b)s(ound)d(stated.)p 3870 4103 71 71 v 121 4349 a(An)f(appro)m(ximation)f(of)i Fl(w)h Fo(can)f(no)m(w)f(b)s(e)f(constructed)i(as)f(in)f(Theorem)h(6.1) h(in)e(Fill)f(and)i(Janson)f([13)r(].)39 b(In)27 b(the)-20 4462 y(pro)s(of)32 b(w)m(e)i(use)e(a)i(T)-8 b(a)m(ylor)33 b(expansion)e(of)j(second)e(order)h(whic)m(h)e(impro)m(v)m(es)i(the)g (rate)h(of)f(con)m(v)m(ergence)i(compared)-20 4575 y(with)29 b(the)i(\014rst)f(order)f(expansion)h(used)f(b)m(y)i(Fill)d(and)i (Janson.)-20 4780 y Fp(Theorem)k(4.5)46 b Fc(L)-5 b(et)27 b Fo(\()p Fl(X)864 4794 y Fk(n)912 4780 y Fo(\))f Fc(and)i Fl(C)k Fc(b)-5 b(e)27 b(given)e(as)i(in)f(L)-5 b(emma)28 b(4.3)f(and)g(denote)h(by)e Fl(F)2880 4794 y Fk(n)2953 4780 y Fc(the)h(distribution)h(functions)-20 4893 y(of)33 b Fl(X)162 4907 y Fk(n)209 4893 y Fc(.)42 b(De\014ne)924 5132 y Fl(w)989 5146 y Fk(n)1036 5132 y Fo(\()p Fl(x)p Fo(\))26 b(:=)1316 5071 y Fl(F)1374 5085 y Fk(n)1421 5071 y Fo(\()p Fl(x)20 b Fo(+)g Fl(\016)1659 5085 y Fk(n)1707 5071 y Fl(=)p Fo(2\))h Fm(\000)f Fl(F)2002 5085 y Fk(n)2050 5071 y Fo(\()p Fl(a)g Fm(\000)g Fl(\016)2284 5085 y Fk(n)2331 5071 y Fl(=)p Fo(2\))p 1316 5111 1143 4 v 1843 5194 a Fl(\016)1883 5208 y Fk(n)2468 5132 y Fl(;)108 b(x)25 b Fm(2)g Fo(R)948 b(\(42\))1915 5411 y(22)p eop %%Page: 23 23 23 22 bop -20 -21 a Fc(with)1492 261 y Fl(\016)1532 275 y Fk(n)1605 261 y Fo(=)25 b Fl(L)1763 106 y Fj( )1845 200 y Fo(ln)o(\()p Fl(n)p Fo(\))p 1845 240 202 4 v 1918 323 a Fl(n)2056 106 y Fj(!)2128 128 y Fs(2)p Fk(=)p Fs(9)2238 261 y Fl(;)1516 b Fo(\(43\))-20 537 y Fc(with)34 b(an)f Fl(L)25 b(>)g Fo(0)p Fc(.)42 b(Then)435 796 y Fo(sup)434 874 y Fk(x)p Fg(2)p Fs(R)588 796 y Fm(j)p Fl(w)678 810 y Fk(n)726 796 y Fo(\()p Fl(x)p Fo(\))20 b Fm(\000)g Fl(w)r Fo(\()p Fl(x)p Fo(\))p Fm(j)27 b(\024)e Fl(r)1337 810 y Fk(n)1409 796 y Fo(:=)1530 668 y Fj(\022)1616 735 y Fo(4)p 1607 775 62 4 v 1607 858 a Fl(L)1679 695 y Fj(\020)1733 796 y Fl(C)7 b Fm(k)p Fl(w)1917 758 y Fg(0)1941 796 y Fm(k)1986 810 y Fg(1)2061 695 y Fj(\021)2116 718 y Fs(2)p Fk(=)p Fs(3)2246 796 y Fo(+)2347 735 y Fl(L)2409 702 y Fs(2)2448 735 y Fm(k)p Fl(w)2560 702 y Fg(0)q(0)2604 735 y Fm(k)2649 749 y Fg(1)p 2347 775 377 4 v 2490 858 a Fo(24)2734 668 y Fj(\023)15 b(\022)2893 735 y Fo(ln)n(\()p Fl(n)p Fo(\))p 2893 775 202 4 v 2966 858 a Fl(n)3104 668 y Fj(\023)3171 690 y Fs(4)p Fk(=)p Fs(9)3296 796 y Fl(:)-20 1064 y Fp(Pro)s(of:)77 b Fo(Let)31 b Fl(F)44 b Fo(denote)31 b(the)f(distribution)d(function)i(of)i Fl(X)7 b Fo(.)42 b(By)30 b(T)-8 b(a)m(ylor)31 b(expansion)e(w)m(e)i(ha) m(v)m(e)h Fl(w)r Fo(\()p Fl(y)s Fo(\))27 b(=)e Fl(w)r Fo(\()p Fl(x)p Fo(\))c(+)-20 1177 y Fl(w)47 1144 y Fg(0)71 1177 y Fo(\()p Fl(x)p Fo(\)\()p Fl(y)j Fm(\000)c Fl(x)p Fo(\))h(+)f(\()p Fl(w)689 1144 y Fg(0)q(0)732 1177 y Fo(\()p Fl(#)p Fo(\))p Fl(=)p Fo(2\)\()p Fl(y)25 b Fm(\000)20 b Fl(x)p Fo(\))1264 1144 y Fs(2)1334 1177 y Fo(with)29 b Fl(#)h Fo(b)s(et)m(w)m(een)h Fl(x)f Fo(and)g Fl(y)s Fo(.)41 b(This)28 b(yields)619 1381 y Fm(j)p Fl(F)13 b Fo(\()p Fl(x)21 b Fo(+)f Fl(\016)954 1395 y Fk(n)1002 1381 y Fl(=)p Fo(2\))h Fm(\000)f Fl(F)13 b Fo(\()p Fl(x)20 b Fm(\000)g Fl(\016)1548 1395 y Fk(n)1596 1381 y Fl(=)p Fo(2\))h Fm(\000)f Fl(\016)s(w)r Fo(\()p Fl(x)p Fo(\))p Fm(j)702 1596 y(\024)856 1437 y Fj(\014)856 1491 y(\014)856 1546 y(\014)856 1600 y(\014)856 1655 y(\014)902 1472 y(Z)992 1498 y Fk(x)p Fs(+)p Fk(\016)1118 1506 y Fe(n)1161 1498 y Fk(=)p Fs(2)952 1678 y Fk(x)p Fg(\000)p Fk(\016)1078 1686 y Fe(n)1120 1678 y Fk(=)p Fs(2)1251 1596 y Fl(w)r Fo(\()p Fl(x)p Fo(\))h(+)f Fl(w)1619 1558 y Fg(0)1643 1596 y Fo(\()p Fl(x)p Fo(\)\()p Fl(y)k Fm(\000)c Fl(x)p Fo(\))g(+)g(\()p Fl(w)2260 1558 y Fg(0)q(0)2304 1596 y Fo(\()p Fl(#)p Fo(\))p Fl(=)p Fo(2\)\()p Fl(y)25 b Fm(\000)20 b Fl(x)p Fo(\))2836 1558 y Fs(2)2895 1596 y Fm(\000)g Fl(w)r Fo(\()p Fl(x)p Fo(\))p Fl(dy)3270 1437 y Fj(\014)3270 1491 y(\014)3270 1546 y(\014)3270 1600 y(\014)3270 1655 y(\014)702 1890 y Fm(\024)866 1828 y(k)p Fl(w)978 1795 y Fg(0)q(0)1022 1828 y Fm(k)1067 1842 y Fg(1)p 866 1869 276 4 v 981 1952 a Fo(2)1167 1766 y Fj(Z)1258 1792 y Fk(\016)1289 1800 y Fe(n)1331 1792 y Fk(=)p Fs(2)1217 1972 y Fg(\000)p Fk(\016)1303 1980 y Fe(n)1346 1972 y Fk(=)p Fs(2)1436 1890 y Fl(y)1484 1852 y Fs(2)1523 1890 y Fl(dy)702 2142 y Fm(\024)889 2081 y Fo(1)p 866 2121 91 4 v 866 2205 a(24)967 2142 y Fm(k)p Fl(w)1079 2105 y Fg(0)q(0)1122 2142 y Fm(k)1167 2156 y Fg(1)1243 2142 y Fl(\016)1286 2105 y Fs(3)1283 2165 y Fk(n)1330 2142 y Fl(:)-20 2367 y Fo(Th)m(us)30 b(with)f Fl(\016)458 2381 y Fk(n)535 2367 y Fo(and)h Fl(w)777 2381 y Fk(n)854 2367 y Fo(as)h(giv)m(en)f(in)g(the)g(Lemma)h (w)m(e)f(obtain)512 2572 y(sup)511 2650 y Fk(x)p Fg(2)p Fs(R)665 2572 y Fm(j)p Fl(w)755 2586 y Fk(n)802 2572 y Fo(\()p Fl(x)p Fo(\))21 b Fm(\000)f Fl(w)r Fo(\()p Fl(x)p Fo(\))p Fm(j)594 2855 y(\024)84 b Fo(sup)748 2934 y Fk(x)p Fg(2)p Fs(R)902 2700 y Fj(\()975 2696 y(\014)975 2751 y(\014)975 2805 y(\014)975 2860 y(\014)975 2914 y(\014)1015 2794 y Fl(F)1073 2808 y Fk(n)1121 2794 y Fo(\()p Fl(x)20 b Fo(+)g Fl(\016)1359 2808 y Fk(n)1407 2794 y Fl(=)p Fo(2\))h Fm(\000)f Fl(F)1702 2808 y Fk(n)1749 2794 y Fo(\()p Fl(x)h Fm(\000)f Fl(\016)1988 2808 y Fk(n)2035 2794 y Fl(=)p Fo(2\))p 1015 2834 1147 4 v 1545 2918 a Fl(\016)1585 2932 y Fk(n)2192 2855 y Fm(\000)2292 2794 y Fl(F)13 b Fo(\()p Fl(x)21 b Fo(+)f Fl(\016)2602 2808 y Fk(n)2649 2794 y Fl(=)p Fo(2\))i Fm(\000)e Fl(F)13 b Fo(\()p Fl(x)20 b Fm(\000)g Fl(\016)3196 2808 y Fk(n)3244 2794 y Fl(=)p Fo(2\))p 2292 2834 1078 4 v 2787 2918 a Fl(\016)2827 2932 y Fk(n)3380 2696 y Fj(\014)3380 2751 y(\014)3380 2805 y(\014)3380 2860 y(\014)3380 2914 y(\014)980 3161 y Fo(+)1071 3002 y Fj(\014)1071 3057 y(\014)1071 3111 y(\014)1071 3166 y(\014)1071 3220 y(\014)1112 3100 y Fl(F)13 b Fo(\()p Fl(x)20 b Fo(+)g Fl(\016)1421 3114 y Fk(n)1468 3100 y Fl(=)p Fo(2\))i Fm(\000)e Fl(F)13 b Fo(\()p Fl(x)20 b Fm(\000)g Fl(\016)2015 3114 y Fk(n)2063 3100 y Fl(=)p Fo(2\))p 1112 3140 V 1607 3224 a Fl(\016)1647 3238 y Fk(n)2219 3161 y Fm(\000)g Fl(w)r Fo(\()p Fl(x)p Fo(\))2499 3002 y Fj(\014)2499 3057 y(\014)2499 3111 y(\014)2499 3166 y(\014)2499 3220 y(\014)2530 3006 y(\))594 3445 y Fm(\024)779 3384 y Fo(2)p 758 3425 88 4 v 758 3508 a Fl(\016)798 3522 y Fk(n)855 3445 y Fl(\045)p Fo(\()p Fl(X)1012 3459 y Fk(n)1060 3445 y Fl(;)15 b(X)7 b Fo(\))21 b(+)f Fm(k)p Fl(w)1441 3408 y Fg(0)q(0)1485 3445 y Fm(k)1530 3459 y Fg(1)1617 3384 y Fl(\016)1660 3351 y Fs(2)1657 3406 y Fk(n)p 1615 3425 91 4 v 1615 3508 a Fo(24)594 3710 y Fm(\024)766 3649 y Fo(2)p 758 3689 62 4 v 758 3772 a Fl(L)830 3710 y Fo(2)875 3609 y Fj(\020)930 3710 y Fl(C)7 b Fm(k)p Fl(w)1114 3672 y Fg(0)1137 3710 y Fm(k)1182 3724 y Fg(1)1258 3609 y Fj(\021)1312 3632 y Fs(2)p Fk(=)p Fs(3)1437 3582 y Fj(\022)1514 3649 y Fo(ln)o(\()p Fl(n)p Fo(\))p 1514 3689 202 4 v 1587 3772 a Fl(n)1725 3582 y Fj(\023)1792 3604 y Fs(2)p Fk(=)p Fs(3)1922 3710 y Fo(+)2023 3649 y Fm(k)p Fl(w)2135 3616 y Fg(0)q(0)2178 3649 y Fm(k)2223 3663 y Fg(1)p 2023 3689 276 4 v 2115 3772 a Fo(24)2308 3710 y Fl(L)2370 3672 y Fs(2)2425 3582 y Fj(\022)2502 3649 y Fo(ln)n(\()p Fl(n)p Fo(\))p 2502 3689 202 4 v 2575 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b(or)27 b(the)g(examples)f(of)g(the)h(sections)f(2.2.1-2.2.3)k(w)m(e)d (de\014ne)e(appropriate)h(discretizations)f(and)g(sho)m(w)i(\(37\).)41 b(The)-20 5015 y(algorithmic)29 b(computation)i(of)f(the)h (distributions)26 b(of)31 b(the)f(discretizations)g(is)f(done)h(in)f (the)i(next)f(section.)1915 5411 y(23)p eop %%Page: 24 24 24 23 bop 121 -21 a Fo(T)-8 b(o)23 b(de\014ne)e(the)i(discretized)e(v)m (ersions)g(of)h Fl(A)1591 -7 y Fk(r)1655 -21 y Fo(=)j Fl(f)1796 -7 y Fk(r)1833 -21 y Fo(\()p Fl(U)10 b Fo(\))23 b(and)e Fl(b)26 b Fo(=)e Fl(g)s Fo(\()p Fl(f)10 b Fo(\()p Fl(U)g Fo(\)\))24 b(w)m(e)e(denote,)j(for)d Fl(U)35 b Fo(=)25 b(\()p Fl(U)3521 -7 y Fs(1)3561 -21 y Fl(;)15 b(:)g(:)g(:)32 b(;)15 b(U)3840 -6 y Fk(d)3881 -21 y Fo(\),)1287 233 y([)p Fl(U)10 b Fo(])1409 247 y Fk(n)1482 233 y Fo(:=)1603 105 y Fj(\022)1680 172 y Fm(b)p Fl(nU)1837 186 y Fs(1)1876 172 y Fm(c)p 1680 212 237 4 v 1771 295 a Fl(n)1927 233 y(;)15 b(:)g(:)g(:)31 b(;)2154 172 y Fm(b)p Fl(nU)2311 187 y Fk(d)2351 172 y 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b(can)e(c)m(ho)s(ose)i Fl(C)639 2191 y Fs(\006)719 2177 y Fo(:=)f(2)p Fl(;)107 b(C)1089 2130 y Fs(\(2\))1082 2206 y(\006)1208 2177 y Fo(:=)25 b(2)g(and)f(b)m(y)g(Lemma)g(4.6)i(similarly)21 b(to)k(\(48\))g(w)m(e)g(obtain)f(the)g(c)m(hoice)i Fl(C)3690 2192 y Fk(b)3749 2177 y Fo(:=)f(6.)-20 2290 y(F)-8 b(urthermore)31 b Fl(C)573 2304 y Fk(X)665 2290 y Fo(:=)25 b(1)31 b(and)f(for)g Fl(C)1243 2305 y Fk(\030)1311 2290 y Fo(note)747 2544 y Fl(\030)791 2507 y Fs(2)856 2544 y Fo(=)25 b(2)15 b(E)g(med)1256 2507 y Fs(2)1296 2544 y Fo(\()p Fl(U)10 b Fo(\))26 b(=)f(2)1620 2421 y Fj(Z)1711 2447 y Fs(1)1671 2627 y(0)1766 2544 y Fl(x)1818 2507 y Fs(2)1958 2483 y Fl(x)2010 2450 y Fk(k)2052 2483 y Fo(\(1)c Fm(\000)f Fl(x)p Fo(\))2331 2450 y Fk(k)p 1867 2523 599 4 v 1867 2607 a Fl(B)5 b Fo(\()p Fl(k)23 b Fo(+)d(1)p Fl(;)15 b(k)24 b Fo(+)c(1\))2490 2544 y Fl(dx)26 b Fo(=)2743 2483 y Fl(k)e Fo(+)c(2)p 2721 2523 253 4 v 2721 2607 a(2)p Fl(k)k Fo(+)19 b(3)2983 2544 y Fl(:)771 b Fo(\(50\))-20 2800 y(This)36 b(yields)g Fm(j)p Fl(\030)t Fo(\()p Fl(n)p Fo(\))26 b Fm(\000)f Fl(\030)t Fm(j)37 b(\024)h Fo(4)p Fl(=)p Fo(\()p Fl(\030)t(n)p Fo(\))g(=)f Fl(C)1459 2815 y Fk(\030)1497 2800 y Fl(=n)h Fo(with)e Fl(C)1914 2815 y Fk(\030)1990 2800 y Fo(:=)2123 2722 y Fj(p)p 2214 2722 692 4 v 78 x Fo(8\(2)p Fl(k)25 b Fo(+)20 b(3\))p Fl(=)p Fo(\()p Fl(k)25 b Fo(+)19 b(2\))q(.)63 b(Finally)36 b Fl(C)3375 2814 y Fk(A)3469 2800 y Fo(:=)i Fl(C)3675 2767 y Fg(0)3668 2827 y Fk(A)3762 2800 y Fo(:=)g(2)-20 2913 y(completes)31 b(the)g(c)m(hoices.)-20 3264 y Fp(4.2.3)106 b(Quadtree)-20 3435 y Fo(F)-8 b(or)31 b(quadtrees)g(note)g(th)m(tn)f(b) m(y)h(induction)d(w)m(e)i(ha)m(v)m(e)i(for)e(all)f Fl(a)2144 3449 y Fs(1)2184 3435 y Fl(;)15 b(:)g(:)g(:)32 b(;)15 b(a)2449 3449 y Fk(n)2496 3435 y Fl(;)g(b)2575 3449 y Fs(1)2615 3435 y Fl(;)g(:)g(:)g(:)32 b(;)15 b(b)2871 3449 y Fk(n)2943 3435 y Fm(2)25 b Fo([)p Fm(\000)p Fo(1)p Fl(;)15 b Fo(1]:)1290 3553 y Fj(\014)1290 3607 y(\014)1290 3662 y(\014)1290 3717 y(\014)1290 3771 y(\014)1346 3598 y Fk(K)1320 3626 y Fj(Y)1321 3821 y Fk(i)p Fs(=1)1451 3712 y Fl(a)1499 3726 y Fk(i)1548 3712 y Fm(\000)1665 3598 y Fk(K)1639 3626 y Fj(Y)1640 3821 y Fk(i)p Fs(=1)1770 3712 y Fl(b)1809 3726 y Fk(i)1837 3553 y Fj(\014)1837 3607 y(\014)1837 3662 y(\014)1837 3717 y(\014)1837 3771 y(\014)1893 3712 y Fm(\024)2022 3598 y Fk(K)1989 3626 y Fj(X)1997 3821 y Fk(i)p Fs(=1)2135 3712 y Fm(j)p Fl(a)2208 3726 y Fk(i)2257 3712 y Fm(\000)20 b Fl(b)2387 3726 y Fk(i)2415 3712 y Fm(j)p Fl(;)-20 3979 y Fo(th)m(us)1090 4189 y Fm(j)p Fl(f)1160 4203 y Fk(r)1198 4189 y Fo(\([)p Fl(U)10 b Fo(])1355 4203 y Fk(n)1403 4189 y Fo(\))20 b Fm(\000)g Fl(f)1594 4203 y Fk(r)1632 4189 y Fo(\()p Fl(U)10 b Fo(\))p Fm(j)26 b(\024)1934 4128 y Fl(d)p 1931 4168 55 4 v 1931 4252 a(n)1995 4189 y(;)107 b(r)27 b Fo(=)e(1)p Fl(;)15 b(:)g(:)g(:)33 b(;)15 b Fo(2)2599 4152 y Fk(d)2640 4189 y Fl(;)-20 4416 y Fo(the)31 b(case)g Fl(r)d Fo(=)d(2)539 4383 y Fk(d)610 4416 y Fo(b)s(eing)k(also)i (trivial.)38 b(This)29 b(yields)1370 4617 y Fl(C)1435 4631 y Fs(\006)1515 4617 y Fo(=)c Fl(d)p Fo(2)1703 4580 y Fk(d)1744 4617 y Fl(;)107 b(C)1948 4569 y Fs(\(2\))1941 4646 y(\006)2067 4617 y Fo(:=)25 b Fl(d)2235 4580 y Fs(2)2275 4617 y Fo(2)2320 4580 y Fk(d)2360 4617 y Fl(;)-20 4808 y Fo(and)30 b(b)m(y)g(Lemma)h(4.6)g Fl(C)819 4823 y Fk(b)879 4808 y Fo(:=)25 b(3)p Fl(d)p Fo(2)1137 4775 y Fk(d)1179 4808 y Fo(.)40 b(F)-8 b(urthermore)31 b Fl(C)1837 4822 y Fk(X)1929 4808 y Fo(=)25 b(1)31 b(and)f(for)g Fl(C)2482 4823 y Fk(\030)2550 4808 y Fo(note)1357 5090 y Fl(\030)1401 5052 y Fs(2)1465 5090 y Fo(=)25 b(2)1606 5052 y Fk(d)1662 5090 y Fo(E)1795 4976 y Fk(d)1755 5003 y Fj(Y)1756 5199 y Fk(i)p Fs(=1)1886 5090 y Fl(U)1958 5052 y Fs(2)1948 5112 y Fk(i)2023 5090 y Fo(=)2119 4962 y Fj(\022)2195 5028 y Fo(2)p 2195 5069 46 4 v 2195 5152 a(3)2251 4962 y Fj(\023)2318 4984 y Fk(d)2373 5090 y Fl(;)1381 b Fo(\(51\))1915 5411 y(25)p eop %%Page: 26 26 26 25 bop -20 -21 a Fo(so)31 b(that)1310 219 y Fm(j)p Fl(\030)t Fo(\()p Fl(n)p Fo(\))20 b Fm(\000)g Fl(\030)t Fm(j)26 b(\024)1815 158 y Fo(2)p Fl(d)21 b Fo(+)f Fl(d)2066 125 y Fs(2)2106 158 y Fo(2)2151 125 y Fk(d)p 1815 198 377 4 v 1954 281 a Fl(\030)t(n)2227 219 y Fo(=)2333 156 y Fl(C)2398 171 y Fk(\030)p 2333 198 103 4 v 2357 281 a Fl(n)-20 482 y Fo(with)29 b Fl(C)252 497 y Fk(\030)315 482 y Fo(:=)d(\(2)p Fl(d)21 b Fo(+)f Fl(d)723 449 y Fs(2)763 482 y Fo(2)808 449 y Fk(d)848 482 y Fo(\)\(3)p Fl(=)p Fo(2\))1088 449 y Fk(d=)p Fs(2)1202 482 y Fo(.)41 b(Finally)28 b Fl(C)1642 496 y Fk(A)1724 482 y Fo(:=)d(2)1890 449 y Fk(d)1931 482 y Fl(;)15 b(C)2043 449 y Fg(0)2036 508 y Fk(A)2119 482 y Fo(:=)25 b(2)2285 449 y Fk(d)2326 482 y Fo(\(2)2406 449 y Fk(d)2468 482 y Fm(\000)19 b Fo(1\))32 b(completes)e(the)h(c)m(hoices.)-20 768 y Fn(5)135 b(The)45 b(rejection)h(algorithm)-20 971 y Fo(The)f(dominan)m(t)e(curv)m(e)i (and)f(the)h(asso)s(ciate)h(sample)e(needed)g(for)h(the)g(rejection)f (metho)s(d)g(w)m(ere)i(deriv)m(ed)d(in)-20 1084 y(section)32 b(3.2.)47 b(It)32 b(remains)e(the)i(problem)f(of)h(appro)m(ximating)e (the)i(densit)m(y)g Fl(w)i Fo(in)c(order)i(to)g(decide)g(the)g(outcome) -20 1197 y(of)f(a)g(rejection)f(test.)121 1310 y(Let)j Fl(K)363 1324 y Fs(2)403 1310 y Fo(,)g Fl(K)538 1324 y Fs(4)611 1310 y Fo(b)s(e)e(upp)s(er)g(b)s(ounds)f(for)j Fm(k)p Fl(w)1571 1277 y Fg(0)1595 1310 y Fm(k)1640 1324 y Fg(1)1715 1310 y Fl(;)15 b Fm(k)p Fl(w)1867 1277 y Fg(0)r(0)1911 1310 y Fm(k)1956 1324 y Fg(1)2031 1310 y Fo(,)33 b(e.g.,)i(the)e(c)m(hoices)g(giv)m(en)f(in)g(Lemma)g(3.3.)49 b(Then)31 b Fl(r)3894 1324 y Fk(n)-20 1423 y Fo(giv)m(en)g(in)e (Theorem)h(4.5)h(is)f(estimated)g(with)f(the)i(c)m(hoice)1479 1684 y Fl(L)25 b Fo(:=)1697 1623 y(96)1787 1590 y Fs(1)p Fk(=)p Fs(3)1898 1623 y Fo(\()p Fl(C)7 b(K)2082 1637 y Fs(2)2121 1623 y Fo(\))2156 1590 y Fs(2)p Fk(=)p Fs(9)p 1697 1663 570 4 v 1885 1772 a Fl(K)1969 1724 y Fs(1)p Fk(=)p Fs(3)1962 1798 y(4)3779 1684 y Fo(\(52\))-20 1961 y(b)m(y)897 2233 y Fl(r)938 2247 y Fk(n)1010 2233 y Fm(\024)25 b Fl(R)1175 2247 y Fk(n)1247 2233 y Fo(:=)1368 2105 y Fj(\022)1445 2172 y Fo(16)p 1445 2212 91 4 v 1468 2296 a(3)1546 2105 y Fj(\023)1613 2128 y Fs(1)p Fk(=)p Fs(3)1738 2132 y Fj(\020)1792 2233 y Fl(C)7 b(K)1941 2247 y Fs(2)1980 2132 y Fj(\021)2035 2155 y Fs(4)p Fk(=)p Fs(9)2145 2233 y Fl(K)2229 2185 y Fs(1)p Fk(=)p Fs(3)2222 2259 y(4)2354 2105 y Fj(\022)2431 2172 y Fo(ln)n(\()p Fl(n)p Fo(\))p 2431 2212 202 4 v 2504 2296 a Fl(n)2642 2105 y Fj(\023)2708 2128 y Fs(4)p Fk(=)p Fs(9)2834 2233 y Fl(:)920 b Fo(\(53\))-20 2482 y(Th)m(us)1377 2687 y(sup)1376 2765 y Fk(x)p Fg(2)p Fs(R)1530 2687 y Fm(j)p Fl(w)1620 2701 y Fk(n)1667 2687 y Fo(\()p Fl(x)p Fo(\))21 b Fm(\000)f Fl(w)r Fo(\()p Fl(x)p Fo(\))p Fm(j)27 b(\024)e Fl(R)2307 2701 y Fk(n)2354 2687 y Fl(;)-20 2946 y Fo(with)k Fl(w)252 2960 y Fk(n)330 2946 y Fo(giv)m(en)h(in)f(\(42\))j(and)e Fl(L)g Fo(in)f(the)i (de\014nition)d(of)i Fl(\016)1944 2960 y Fk(n)2022 2946 y Fo(there)g(giv)m(en)h(b)m(y)f(\(52\).)-20 3189 y Ff(5.1)113 b(Algorithmic)33 b(appro)m(ximation)j(of)i(the)f(densit)m(y)-20 3361 y Fo(F)-8 b(or)31 b(the)g(computation)f(of)h(the)f(appro)m (ximations)g Fl(w)1798 3375 y Fk(n)1875 3361 y Fo(of)g Fl(w)j Fo(w)m(e)e(k)m(eep)g(and)f(up)s(date)g(arra)m(ys)g Fm(A)3246 3375 y Fk(n)3323 3361 y Fo(de\014ned)f(b)m(y)1236 3610 y Fm(A)1309 3624 y Fk(n)1355 3610 y Fo([)p Fl(k)s Fo(])d(:=)f(P)1679 3482 y Fj(\022)1746 3610 y Fl(X)1821 3624 y Fk(n)1894 3610 y Fo(=)2002 3549 y Fl(k)p 2000 3589 55 4 v 2000 3673 a(n)2064 3482 y Fj(\023)2146 3610 y Fl(;)107 b(k)28 b Fm(2)d Fo(Z)p Fl(;)-20 3864 y Fo(so)30 b(that)h Fm(A)361 3878 y Fk(n)407 3864 y Fo([)p Fl(k)s Fo(])26 b Fm(6)p Fo(=)f(0)31 b(at)f(most)g(for)g Fm(\000)p Fl(Q)1319 3878 y Fk(n)1391 3864 y Fm(\024)25 b Fl(k)j Fm(\024)d Fl(Q)1730 3878 y Fk(n)1807 3864 y Fo(with)j Fl(Q)2085 3878 y Fk(n)2162 3864 y Fo(giv)m(en)i(in)e(Lemma)i(4.1.)42 b(According)30 b(to)g(the)g(recur-)-20 3977 y(siv)m(e)d(de\014nition)e (of)i Fl(X)730 3991 y Fk(n)804 3977 y Fo(in)e(\(25\),)k(\(26\))g(and)d (the)h(c)m(hoice)h(of)f(discretizations)e(in)h(\(44\)-\(47\))k(and)c (with)g(the)h(notation)-20 4106 y Fl(f)35 4058 y Fs(\()p Fk(n)p Fs(\))25 4118 y Fk(r)162 4106 y Fo(:=)e Fl(f)328 4120 y Fk(r)396 4106 y Fo(for)31 b Fl(r)d Fo(=)d(1)p Fl(;)15 b(:)g(:)g(:)32 b(;)15 b(K)28 b Fm(\000)20 b Fo(1)30 b(and)g Fl(f)1466 4058 y Fs(\()p Fk(n)p Fs(\))1456 4135 y Fk(K)1593 4106 y Fo(:=)25 b(1)c Fm(\000)f Fo(1)p Fl(=n)h Fm(\000)2128 4038 y Fj(P)2224 4064 y Fk(K)5 b Fg(\000)p Fs(1)2224 4133 y Fk(r)r Fs(=1)2398 4106 y Fl(f)2443 4120 y Fk(r)2511 4106 y Fo(w)m(e)31 b(de\014ne)e(\014rst)h Fm(A)3168 4120 y Fs(0)3207 4106 y Fo([0])d(:=)e(1)p Fl(;)15 b Fm(A)3608 4120 y Fs(0)3648 4106 y Fo([)p Fl(k)s Fo(])26 b(:=)g(0)-20 4219 y(for)k Fl(k)e Fm(6)p Fo(=)d(0)31 b(\(whic)m(h)e(w)m (e)h(call)g Fa(initialize)45 b Fm(A)1559 4233 y Fs(0)1598 4219 y Fo(\))30 b(and)f(for)h(the)g(up)s(date)f(w)m(e)i(assume)e(that)i Fm(A)3156 4233 y Fk(n)p Fg(\000)p Fs(1)3322 4219 y Fo(is)e(already)h (giv)m(en)-20 4332 y(and)g Fm(A)230 4346 y Fk(n)277 4332 y Fo([)p Fl(k)s Fo(])c(:=)f(0)31 b(is)e(initialized)e(for)j(all)g Fl(k)e Fm(2)d Fo(Z.)40 b(Then)29 b(w)m(e)i(obtain)f Fm(A)2385 4346 y Fk(n)2462 4332 y Fo(algorithmically)e(b)m(y)i(the)h(pro)s (cedure)171 4558 y Fa(for)47 b Fl(i)393 4572 y Fs(1)433 4558 y Fl(;)15 b(:)g(:)g(:)32 b(;)15 b(i)681 4573 y Fk(d)747 4558 y Fo(=)25 b(0)48 b Fa(to)f Fl(n)1134 4525 y Fs(2)1173 4558 y Fm(d)p Fo(ln\()p Fl(n)p Fo(\))p Fm(e)20 b(\000)g Fo(1)49 b Fa(do)1915 5411 y Fo(26)p eop %%Page: 27 27 27 26 bop 267 -21 a Fa(for)47 b Fl(j)495 -7 y Fs(1)550 -21 y Fl(:)15 b(:)g(:)31 b(;)15 b(j)763 -7 y Fk(K)857 -21 y Fo(=)25 b Fm(\000)p Fl(R)1093 -7 y Fk(n)p Fg(\000)p Fs(1)1278 -21 y Fa(to)47 b Fl(R)1490 -7 y Fk(n)p Fg(\000)p Fs(1)1675 -21 y Fa(do)1187 207 y Fl(u)25 b Fo(:=)1400 145 y(1)p 1395 186 55 4 v 1395 269 a Fl(n)1475 78 y Fj(\022\026)1738 145 y Fl(i)1769 159 y Fs(1)p 1605 186 337 4 v 1605 269 a Fl(n)p Fm(d)p Fo(ln)o(\()p Fl(n)p Fo(\))p Fm(e)1951 78 y Fj(\027)2020 207 y Fl(;)15 b(:)g(:)g(:)31 b(;)2236 78 y Fj(\026)2432 145 y Fl(i)2463 160 y Fk(d)p 2300 186 V 2300 269 a Fl(n)p Fm(d)p Fo(ln)n(\()p Fl(n)p Fo(\))p Fm(e)2646 78 y Fj(\027\023)1187 446 y Fl(v)e Fo(:=)1556 385 y(1)p 1391 425 376 4 v 1391 509 a Fl(n)1446 482 y Fs(2)1485 509 y Fm(d)p Fo(ln)o(\()p Fl(n)p Fo(\))p Fm(e)1777 446 y Fo(\()p Fl(i)1843 460 y Fs(1)1883 446 y Fl(;)15 b(:)g(:)g(:)31 b(;)15 b(i)2130 461 y Fk(d)2172 446 y Fo(\))1187 733 y Fl(k)28 b Fo(:=)1418 671 y(1)p 1394 712 95 4 v 1394 795 a Fl(n)1449 769 y Fs(2)1513 577 y Fj($)1571 733 y Fl(n)1626 695 y Fs(2)1680 577 y Fj( )1785 619 y Fk(K)1752 646 y Fj(X)1756 841 y Fk(r)r Fs(=1)1898 733 y Fl(f)1953 695 y Fs(\()p Fk(n)p Fs(\))1943 755 y Fk(r)2055 733 y Fo(\()p Fl(u)p Fo(\))p Fl(j)2214 747 y Fk(r)2273 733 y Fo(+)23 b(~)-48 b Fl(g)s Fo(\()p Fl(f)2500 695 y Fs(\()p Fk(n)p Fs(\))2602 733 y Fo(\()p Fl(v)s Fo(\)\))2754 577 y Fj(!)q(\045)1187 1051 y Fm(A)1260 1065 y Fk(n)1307 1051 y Fo([)p Fl(k)s Fo(])25 b(:=)h Fm(A)1627 1065 y Fk(n)1673 1051 y Fo([)p Fl(k)s Fo(])21 b(+)1885 950 y Fj(\020)1939 1051 y Fl(n)1994 1014 y Fs(2)2033 1051 y Fm(d)p Fo(ln\()p Fl(n)p Fo(\))p Fm(e)2315 950 y Fj(\021)2369 973 y Fk(d)2455 937 y(K)2429 965 y Fj(Y)2425 1159 y Fk(r)r Fs(=1)2564 1051 y Fm(A)2637 1065 y Fk(n)p Fg(\000)p Fs(1)2774 1051 y Fo([)p Fl(j)2836 1065 y Fk(r)2875 1051 y Fo(])267 1299 y Fa(enddo)171 1412 y(enddo)-20 1751 y Fo(W)-8 b(e)35 b(call)e(this)f(pro)s(cedure)h Fa(update)p Fo(\()p Fm(A)1312 1765 y Fk(n)p Fg(\000)p Fs(1)1447 1751 y Fl(;)15 b Fm(A)1560 1765 y Fk(n)1607 1751 y Fo(\).)51 b(Then)32 b(with)h(the)g(arra)m(y)h Fm(A)2642 1765 y Fk(n)2723 1751 y Fo(the)f(discrete)h(appro)m(ximation) e Fl(w)3894 1765 y Fk(n)-20 1864 y Fo(of)f Fl(w)i Fo(as)d(in)f(Theorem) h(4.5)i(is)d(obtained)h(b)m(y)1085 2077 y Fl(w)1150 2091 y Fk(n)1197 2077 y Fo(\()p Fl(x)p Fo(\))c(:=)1497 2015 y(1)p 1476 2056 88 4 v 1476 2139 a Fl(\016)1516 2153 y Fk(n)1934 1990 y Fj(X)1588 2192 y Fk(n)p Fs(\()p Fk(x)p Fg(\000)p Fk(\016)1784 2200 y Fe(n)1827 2192 y 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b(for)f(sim)m(ulation)e(algorithms)i(is)f(still)f(lac)m(king.)-20 2811 y Fn(Ac)l(kno)l(wledgmen)l(t)-20 3014 y Fo(W)-8 b(e)32 b(thank)e(Jim)f(Fill)f(for)j(making)e(the)i(tec)m(hnical)f(rep)s (orts)g([12)q(,)g(13)q(])h(a)m(v)-5 b(ailable)30 b(to)h(us.)-20 3298 y Fn(References)26 3501 y Fo([1])46 b(Bic)m(k)m(el,)e(P)-8 b(.)41 b(J.)f(and)g(P)-8 b(.)40 b(A.)h(F)-8 b(reedman)41 b(\(1981\).)73 b(Some)40 b(asymptotic)h(theory)f(for)g(the)h(b)s(o)s (otstrap.)70 b Fc(A)n(nn.)167 3614 y(Statist.)32 b(9)p Fo(,)f(1196{1217.)26 3797 y([2])46 b(Bougerol,)37 b(P)-8 b(.)35 b(and)f(N.)g(Picard)g(\(1992\).)56 b(Strict)34 b(stationarit)m(y)h(of)f(generalized)g(autoregressiv)m(e)i(pro)s (cesses.)167 3910 y Fc(A)n(nn.)c(Pr)-5 b(ob)g(ab.)32 b(20)p Fo(,)f(1714{1798.)26 4093 y([3])46 b(Chen,)31 b(R.,)g(R.)h(Go)s(o)s(dman,)f(and)f(A.)i(Zame)f(\(1984\).)45 b(Limiting)29 b(distributions)e(of)32 b(t)m(w)m(o)g(random)f (sequences.)167 4205 y Fc(J.)h(Multivariate)h(A)n(nal.)e(14)p Fo(,)g(221{230.)26 4388 y([4])46 b(Devro)m(y)m(e,)33 b(L.)d(\(1986\).)43 b Fc(Nonuniform)34 b(R)-5 b(andom)35 b(V)-7 b(ariate)33 b(Gener)-5 b(ation)7 b Fo(.)43 b(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork.)26 4571 y([5])46 b(Devro)m(y)m(e,)33 b(L.)d(\(2001\).)43 b(Sim)m(ulating)28 b(p)s(erp)s(etuities.)39 b Fc(Metho)-5 b(dol.)34 b(Comput.)g(Appl.)f(Pr)-5 b(ob)g(ab.)31 b(3)p Fo(,)g(97{115.)26 4754 y([6])46 b(Devro)m(y)m(e,)39 b(L.,)f(J.)d(A.)h(Fill,)f(and)g(R.)h(Neininger)e(\(2000\).)59 b(P)m(erfect)37 b(sim)m(ulation)c(from)i(the)h(Quic)m(ksort)f(limit)167 4867 y(distribution.)i Fc(Ele)-5 b(ctr)g(on.)34 b(Comm.)g(Pr)-5 b(ob)g(ab.)31 b(5)p Fo(,)g(95{99)i(\(electronic\).)26 5049 y([7])46 b(de)38 b(Bruijn,)h(N.)g(G.)g(\(1951\).)67 b(The)38 b(asymptotic)h(b)s(eha)m(viour)e(of)h(a)h(function)e(o)s (ccurring)g(in)h(the)g(theory)h(of)167 5162 y(primes.)g Fc(J.)32 b(Indian)i(Math.)f(So)-5 b(c.)33 b(\(N.S.\))h(15)p Fo(,)d(25{32.)1915 5411 y(28)p eop %%Page: 29 29 29 28 bop 26 -21 a Fo([8])46 b(Dobro)m(w,)37 b(R.)f(P)-8 b(.)36 b(and)e(J.)h(A.)h(Fill,)f(\(1999\).)57 b(T)-8 b(otal)36 b(path)f(length)g(for)g(random)f(recursiv)m(e)h(trees.)56 b Fc(Combin.)167 92 y(Pr)-5 b(ob)g(ab.)34 b(Comput.)d(8)p Fo(,)g(317{333.)26 279 y([9])46 b(Em)m(brec)m(h)m(ts,)j(P)-8 b(.)45 b(and)e(C.)h(M.)h(Goldie,)i(\(1994\).)85 b(P)m(erp)s(etuities)43 b(and)h(random)f(equations.)82 b Fc(Asymptotic)167 392 y(statistics)34 b(\(Pr)-5 b(ague,)33 b(1993\))s Fo(,)g(75{86.)-20 580 y([10])47 b(Em)m(brec)m(h)m(ts,)39 b(P)-8 b(.,)38 b(C.)e(Kl)s(\177)-48 b(upp)s(elb)s(erg)32 b(and)j(T.)h(Mik)m(osc)m(h,)j (\(1997\).)60 b Fc(Mo)-5 b(del)5 b(ling)38 b(extr)-5 b(emal)40 b(events)7 b Fo(,)38 b(Springer,)167 693 y(Berlin.)-20 880 y([11])47 b(Fill,)38 b(J.)f(A.)g(and)g(S.)g(Janson)g(\(2000\).)64 b(Smo)s(othness)36 b(and)h(deca)m(y)h(prop)s(erties)e(of)i(the)f (limiting)e(Quic)m(ksort)167 993 y(densit)m(y)29 b(function.)37 b Fc(Mathematics)c(and)g(Computer)g(Scienc)-5 b(e)31 b(\(V)-7 b(ersail)5 b(les,)32 b(2000\))p Fo(,)g(53{64.)41 b(T)-8 b(rends)28 b(Math.,)167 1106 y(Birkh\177)-45 b(auser,)29 b(Basel.)-20 1294 y([12])47 b(Fill,)29 b(J.)i(A.)g(and)f(S.)g(Janson)g (\(2001\).)44 b(Appro)m(ximating)30 b(the)g(Limiting)f(Quic)m(ksort)h (Distribution.)39 b Fc(R)-5 b(andom)167 1407 y(Structur)g(es)34 b(A)n(lgorithms)39 b(19)p Fo(,)31 b(376{406.)-20 1594 y([13])47 b(Fill,)30 b(J.)i(A.)g(and)f(S.)g(Janson)g(\(2001\).)47 b(Quic)m(ksort)31 b(Asymptotics.)44 b(T)-8 b(ec)m(hnical)31 b(Rep)s(ort)h(#597,)h(Departmen)m(t)167 1707 y(of)e(Mathematical)g (Sciences,)f(The)g(Johns)g(Hopkins)f(Univ)m(ersit)m(y)-8 b(.)30 b(Av)-5 b(ailable)29 b(from)167 1820 y Fa (http://www.mts.jhu.edu/)p Fm(\030)-6 b Fa(fill/papers/quick)p 2158 1820 29 4 v 29 w(asy.ps)p Fo(.)-20 2008 y([14])47 b(Fla)5 b(jolet,)24 b(P)-8 b(.,)24 b(G.)e(Lab)s(elle,)g(L.)f(Laforest,) j(and)d(B.)h(Salvy)f(\(1995\).)28 b(Hyp)s(ergeometrics)22 b(and)e(the)i(cost)h(structure)167 2121 y(of)31 b(quadtrees.)40 b Fc(R)-5 b(andom)35 b(Structur)-5 b(es)34 b(A)n(lgorithms)39 b(10)p Fo(,)31 b(117-144.)-20 2308 y([15])47 b(Goldie,)38 b(C.)f(M.)h(and)e(R.)h(Gr)s(\177)-48 b(ub)s(el)35 b(\(1996\).)63 b(P)m(erp)s(etuities)36 b(with)g(thin)f(tails.)60 b Fc(A)-5 b(dv.)39 b(in)g(Appl.)g(Pr)-5 b(ob)g(ab.)38 b(28)p Fo(,)167 2421 y(463{480.)-20 2609 y([16])47 b(Goldie,)30 b(C.)g(M.)h(and)f(R.)g (A.)h(Maller)f(\(2000\).)43 b(Stabilit)m(y)28 b(of)j(p)s(erp)s (etuities.)38 b Fc(A)n(nn.)32 b(Pr)-5 b(ob)g(ab.)31 b(28)p Fo(,)h(1195{1218.)-20 2797 y([17])47 b(Gr)s(\177)-48 b(ub)s(el,)23 b(R.)g(and)f(U.)h(R\177)-45 b(osler)22 b(\(1996\).)31 b(Asymptotic)23 b(distribution)c(theory)k(for)f(Hoare's) i(selection)f(algorithm.)167 2910 y Fc(A)-5 b(dv.)33 b(in)f(Appl.)h(Pr)-5 b(ob)g(ab.)32 b(28)p Fo(,)f(252{269.)-20 3097 y([18])47 b(Hoare,)32 b(C.)e(A.)h(R.)f(\(1961\).)43 b(Algorithm)29 b(64,)j(Quic)m(ksort.)40 b Fc(Comm.)34 b(A)n(CM)44 b(4)p Fo(,)31 b(321-322.)-20 3285 y([19])47 b(Hennequin,)d(P)-8 b(.)43 b(\(1989\))79 b(Com)m(binatorial)41 b(analysis)h(of)g(quic)m(ksort)h(algorithm.)76 b Fc(RAIR)n(O)43 b(Informatique)167 3398 y(th)n(\023)-44 b(eorique)33 b(et)g(Applic)-5 b(ations)40 b(23)p Fo(,)31 b(317{333.)-20 3585 y([20])47 b(Hennequin,)g(P)-8 b(.)45 b(\(1991\))86 b Fc(A)n(nalyse)46 b(en)g(moyenne)g(d'algorithme,)51 b(tri)c(r)-5 b(apide)47 b(et)f(arbr)-5 b(es)47 b(de)f(r)-5 b(e)g(cher)g(che)p Fo(,)167 3698 y(Ph.D.)31 b(Thesis,)e(Ecole)i(P)m (olytec)m(hnique.)-20 3886 y([21])47 b(Hoare,)32 b(C.)e(A.)h(R.)f (\(1962\).)43 b(Quic)m(ksort.)e Fc(Comput.)33 b(J.)p Fo(,)d(10{15.)-20 4073 y([22])47 b(Hw)m(ang,)33 b(H.-K.)f(and)e(R.)i (Neininger)e(\(2001\).)46 b(Phase)31 b(c)m(hange)i(of)e(limit)e(la)m (ws)i(in)g(the)g(quic)m(ksort)g(recurrence)167 4186 y(under)e(v)-5 b(arying)30 b(toll)f(functions.)39 b(Preprin)m(t)29 b(a)m(v)-5 b(ailable)30 b(from)167 4299 y Fa(http://neyman.mathematik.)o(uni-)o (fre)o(ibur)o(g.de)o(/ho)o(mepa)o(ges/)o(nei)o(ning)o(er/q)o(fin)o (al.p)o(s)-20 4487 y Fo([23])47 b(Hw)m(ang,)31 b(H.-K.)62 b(and)30 b(T.-H.)g(Tsai)g(\(2000\).)43 b(Quic)m(kselect)30 b(and)g(Dic)m(kman)h(function.)39 b(Preprin)m(t.)-20 4675 y([24])47 b(Kesten,)40 b(H.)f(\(1973\).)66 b(Random)37 b(di\013erence)g(equations)h(and)f(renew)m(al)h(theory)g(for)g(pro)s (ducts)f(of)h(random)167 4787 y(matrices.)j Fc(A)-5 b(cta)33 b(Math.)e(131)p Fo(,)h(207{248.)-20 4975 y([25])47 b(Kn)m(uth,)h(D.)e (E.)f(\(1997\).)86 b Fc(The)46 b(A)n(rt)g(of)g(Computer)i(Pr)-5 b(o)g(gr)g(amming.)48 b(V)-7 b(ol.)46 b(3:)69 b(Sorting)47 b(and)g(Se)-5 b(ar)g(ching)p Fo(.)167 5088 y(Addison-W)d(esley)g(.)1915 5411 y(29)p eop %%Page: 30 30 30 29 bop -20 -21 a Fo([26])47 b(Mahmoud,)41 b(H.)e(M.)g(\(1991\).)68 b(Limiting)36 b(distributions)f(for)k(path)f(lengths)g(in)g(recursiv)m (e)g(trees.)66 b Fc(Pr)-5 b(ob)g(ab.)167 92 y(Engr)g(g.)33 b(Inform.)g(Sci.)d(5)p Fo(,)h(53-59.)-20 279 y([27])47 b(Mahmoud,)34 b(H.)g(M.)f(\(1992\).)52 b Fc(Evolution)36 b(of)g(R)-5 b(andom)37 b(Se)-5 b(ar)g(ch)37 b(T)-7 b(r)i(e)g(es)p Fo(.)50 b(John)33 b(Wiley)f(&)h(Sons,)h(New)f(Y)-8 b(ork,)167 392 y(1992.)-20 580 y([28])47 b(Mahmoud,)25 b(H.)g(M.,)h(R.)e(Mo)s (darres,)h(and)f(R.)g(T.)g(Sm)m(ythe)g(\(1995\).)33 b(Analysis)22 b(of)j(quic)m(kselect:)37 b(An)24 b(algorithm)167 693 y(for)30 b(order)g(statistics.)41 b Fc(RAIR)n(O)32 b(Informatique)i(Th) n(\023)-44 b(eorique)33 b(et)f(Applic)-5 b(ations)40 b Fp(29)p Fo(,)31 b(255-276.)-20 880 y([29])47 b(Neininger,)39 b(R.)f(\(2001\).)66 b(On)38 b(a)g(m)m(ultiv)-5 b(ariate)37 b(con)m(traction)i(metho)s(d)f(for)g(random)f(recursiv)m(e)h (structures)167 993 y(with)29 b(applications)g(to)i(quic)m(ksort.)40 b Fc(R)-5 b(andom)35 b(Structur)-5 b(es)34 b(A)n(lgorithms)39 b(19)p Fo(,)31 b(498{524.)-20 1181 y([30])47 b(Neininger,)35 b(R.)h(and)e(L.)i(R)s(\177)-48 b(usc)m(hendorf)34 b(\(1999\).)58 b(On)34 b(the)h(in)m(ternal)f(path)i(length)e(of)i Fl(d)p Fo(-dimensional)d(quad)167 1294 y(trees.)41 b Fc(R)-5 b(andom)35 b(Structur)-5 b(es)34 b(A)n(lgorithms)39 b(15)p Fo(,)31 b(25{41.)-20 1482 y([31])47 b(Neininger,)32 b(R.)h(and)g(L.)f (R)s(\177)-48 b(usc)m(hendorf)32 b(\(2001\).)51 b(A)33 b(general)g(con)m(traction)h(theorem)f(and)f(asymptotic)h(nor-)167 1594 y(malit)m(y)d(in)f(com)m(binatorial)h(structures.)40 b(Preprin)m(t)28 b(a)m(v)-5 b(ailable)30 b(from)167 1707 y Fa(http://neyman.mathematik.)o(uni-)o(fre)o(ibur)o(g.de)o(/ho)o(mepa) o(ges/)o(nei)o(ning)o(er/a)o(syn)o(orm.)o(ps)-20 1895 y Fo([32])47 b(Rac)m(hev,)36 b(S.)e(T.)f(and)h(L.)f(R)s(\177)-48 b(usc)m(hendorf)33 b(\(1995\).)54 b(Probabilit)m(y)31 b(metrics)j(and)f(recursiv)m(e)g(algorithms.)50 b Fc(A)-5 b(dv.)167 2008 y(in)33 b(Appl.)g(Pr)-5 b(ob)g(ab.)31 b(27)p Fo(,)h(770{799.)-20 2196 y([33])47 b(R)m(\023)-43 b(egnier,)38 b(M.)e(\(1989\).)58 b(A)36 b(limiting)c(distribution)g (for)j(quic)m(ksort.)56 b Fc(RAIR)n(O)37 b(Informatique)h(Th)n(\023)-44 b(eorique)38 b(et)167 2308 y(Applic)-5 b(ation)32 b(23)p Fo(,)g(335{343.)-20 2496 y([34])47 b(R\177)-45 b(osler,)30 b(U.)h(\(1991\).)43 b(A)31 b(limit)d(theorem)j(for)f(\\Quic)m(ksort".) 41 b Fc(RAIR)n(O)32 b(Inform.)h(Th)n(\023)-44 b(eor.)34 b(Appl.)d(25)p Fo(,)g(85{100.)-20 2684 y([35])47 b(R\177)-45 b(osler,)24 b(U.)g(\(1992\).)31 b(A)23 b(\014xed)f(p)s(oin)m(t)g (theorem)i(for)f(distributions.)i Fc(Sto)-5 b(chastic)27 b(Pr)-5 b(o)g(c)g(ess.)27 b(Appl.)d(42)p Fo(,)i(195{214.)-20 2871 y([36])47 b(R\177)-45 b(osler,)26 b(U.)f(\(2001\).)35 b(On)24 b(the)h(analysis)f(of)h(sto)s(c)m(hastic)h(divide)d(and)i (conquer)g(algorithms.)30 b Fc(A)n(lgorithmic)-5 b(a)27 b(29)p Fo(,)167 2984 y(238{261.)-20 3172 y([37])47 b(R\177)-45 b(osler,)31 b(U.)g(and)f(L.)h(R)s(\177)-48 b(usc)m(hendorf)30 b(\(2001\).)44 b(The)30 b(con)m(traction)i(metho)s(d)e(for)h(recursiv)m (e)f(algorithms.)41 b Fc(A)n(lgo-)167 3285 y(rithmic)-5 b(a)32 b(29)p Fo(,)g(3{33.)-20 3472 y([38])47 b(Sedgewic)m(k,)31 b(R.)f(\(1977\).)43 b(The)30 b(analysis)f(of)i(Quic)m(ksort)f (programs.)40 b Fc(A)-5 b(cta)33 b(Informat.)f(7)p Fo(,)f(327{355.)-20 3660 y([39])47 b(Sedgewic)m(k,)31 b(R.)40 b Fc(Quicksort)p Fo(.)h(Gurland)28 b(Publishing,)f(NY,)k(1980.)-20 3848 y([40])47 b(Sedgewic)m(k,)32 b(R.)f(and)g(P)-8 b(.)31 b(Fla)5 b(jolet)32 b(\(1996\).)45 b Fc(A)n(n)33 b(intr)-5 b(o)g(duction)35 b(to)f(the)g(analysis)h(of)e(algorithms.)p Fo(.)46 b(Addison-)167 3961 y(W)-8 b(esley)g(,)32 b(Amsterdam.)-20 4148 y([41])47 b(T)-8 b(ak\023)-45 b(acs,)38 b(L.)d(\(1955\).)57 b(On)35 b(sto)s(c)m(hastic)h(pro)s(cesses)f(connected)h(with)e(certain) h(ph)m(ysical)f(recording)g(appara-)167 4261 y(tuses.)41 b Fc(A)-5 b(cta)33 b(Math.)g(A)-5 b(c)g(ad.)33 b(Sci.)f(Hungar.)f(6)p Fo(,)g(363{380.)-20 4449 y([42])47 b(T)-8 b(enen)m(baum,)32 b(G.)g(\(1995\).)47 b Fc(Intr)-5 b(o)g(duction)37 b(to)d(A)n(nalytic)g (and)h(Pr)-5 b(ob)g(abilistic)35 b(Numb)-5 b(er)33 b(The)-5 b(ory)8 b Fo(.)47 b(T)-8 b(ranslated)167 4562 y(from)36 b(the)h(second)f(F)-8 b(renc)m(h)37 b(edition)e(\(1995\))k(b)m(y)e(C.)f (B.)h(Thomas.)58 b(Cam)m(bridge)36 b(Univ)m(ersit)m(y)f(Press,)j(Cam-) 167 4675 y(bridge.)-20 4862 y([43])47 b(V)-8 b(erv)j(aat,)38 b(W.)e(\(1979\).)56 b(On)34 b(a)i(sto)s(c)m(hastic)f(di\013erence)g (equation)f(and)h(a)g(represen)m(tation)g(of)g(non-negativ)m(e)167 4975 y(in\014nitely)28 b(divisible)e(random)k(v)-5 b(ariables.)39 b Fc(A)-5 b(dv.)33 b(in)f(Appl.)h(Pr)-5 b(ob)g(ab.)32 b(11,)f Fo(750{783.)1915 5411 y(30)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF