%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: paper.dvi %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -o universallbd.ps paper %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1999.08.24:1023 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B 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(y)m(e)d(and)0 359 y(Gy\177)-46 b(or\014,)30 b(1985\):)0 587 y Fm(Lemma)f(1.)50 b Fd(Let)26 b Fl(X)701 601 y Ff(1)737 587 y Fl(;)15 b(:)g(:)g(:)j(;)d(X)1016 601 y Fk(n)1087 587 y Fd(b)s(e)25 b(iid)h(zero)f(mean)f(random)h(v)-5 b(ariables)25 b(with)h(\014nite)g(\014rst)g(absolute)f(momen)m(t.)0 695 y(Then)1217 873 y Fj(E)1301 717 y Fi(\()1375 713 y(\014)1375 768 y(\014)1375 823 y(\014)1375 877 y(\014)1375 932 y(\014)1458 811 y Fn(1)p 1415 852 131 4 v 1415 870 a Fh(p)p 1491 870 55 4 v 66 x Fl(n)1617 759 y Fk(n)1571 786 y Fi(X)1583 978 y Fk(i)p Ff(=1)1718 873 y Fl(X)1794 887 y Fk(i)1822 713 y Fi(\014)1822 768 y(\014)1822 823 y(\014)1822 877 y(\014)1822 932 y(\014)1852 717 y(\))1951 873 y Fh(\025)2095 811 y Fn(1)p 2057 852 122 4 v 2057 870 a Fh(p)p 2133 870 46 4 v 76 x Fn(8)2189 873 y Fj(E)15 b Fh(fj)p Fl(X)2420 887 y Ff(1)2457 873 y Fh(jg)h Fl(:)299 1203 y Fn(The)32 b(pro)s(of)g(of)g(Theorem)g(1)h(is)f(based)g(up)s(on)g (F)-8 b(ourier)31 b(transforms.)47 b(Let)33 b Fl(')h 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2623 y Fh(\000)g Fl(')p Fh(jj)p Fl( )s Fh(j)2627 2494 y Fi(\033)1721 2864 y Fh(\025)1828 2802 y Fn(1)p 1828 2843 V 1828 2926 a(2)1898 2864 y(sup)1955 2936 y Fk(t)2051 2864 y Fj(E)15 b Fh(fj)p Fl(')2265 2878 y Fk(n)2332 2864 y Fh(\000)k Fl(')p Fh(jj)p Fl( )s Fh(jg)0 3087 y Fn(where)38 b Fl(')332 3101 y Fk(n)418 3087 y Fn(=)543 3051 y Ff(1)p 539 3066 42 4 v 539 3119 a Fk(n)606 3019 y Fi(P)702 3114 y Fk(j)777 3087 y Fn(=)i(1)934 3054 y Fk(n)979 3087 y Fl(e)1022 3054 y Fk(itX)1124 3062 y Fb(j)1199 3087 y Fn(is)f(the)f(empirical)e(c)m(haracteristic)f (function.)65 b(Note)38 b(that)g(w)m(e)i(used)e(the)0 3195 y(fact)k(that)g(the)h(F)-8 b(ourier)41 b(transform)h(of)h Fl(f)1499 3209 y Fk(n)1587 3195 y Fn(is)g Fl(')1751 3209 y Fk(n)1797 3195 y Fl( )s Fn(.)81 b(No)m(w,)48 b(w)m(e)c(can)e (\014nally)g(use)h(Lemma)g(1.)80 b(Indeed,)0 3303 y Fh(j)p Fl(')84 3317 y Fk(n)151 3303 y Fh(\000)19 b Fl(')p Fh(j)26 b(\025)f(j<)p Fn(\()p Fl(')635 3317 y Fk(n)681 3303 y Fn(\))19 b Fh(\000)h(<)p Fn(\()p Fl(')p 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Fh(\000)g(=)p Fn(\()p Fl(')p Fn(\))p Fh(jg)25 b(\025)1793 4951 y Fn(1)p 1732 4991 168 4 v 1732 5010 a Fh(p)p 1808 5010 92 4 v 75 x Fn(32)1925 4884 y Fi(\022)2002 4951 y Fn(1)p 2002 4991 46 4 v 2002 5075 a(2)2078 5012 y Fh(\000)2179 4951 y(<)p Fn(\()p Fl(')p Fn(\(2)p Fl(t)p Fn(\)\))p 2179 4991 347 4 v 2329 5075 a(2)2556 5012 y Fh(\000)19 b(=)2713 4975 y Ff(2)2750 5012 y Fn(\()p Fl(')p Fn(\()p Fl(t)p Fn(\)\))2986 4884 y Fi(\023)3068 5012 y Fl(:)1872 5280 y Fc(3)p eop %%Page: 4 4 4 3 bop 0 83 a Fn(Av)m(eraging)29 b(the)g(t)m(w)m(o)i(b)s(ounds)d (yields)1198 220 y Fh(p)p 1274 220 55 4 v 70 x Fl(n)p Fj(E)15 b Fh(fj)p Fl(')1543 304 y Fk(n)1609 290 y Fh(\000)20 b Fl(')p Fh(jg)26 b(\025)2047 228 y Fn(1)p 1963 269 214 4 v 1963 287 a Fh(p)p 2039 287 137 4 v 76 x Fn(128)2186 290 y(\(1)20 b Fh(\000)f(j)p Fl(')p Fh(j)2488 252 y Ff(2)2527 290 y Fn(\))p Fl(:)0 507 y Fn(Collecting)26 b(all)j(the)g(b)s(ounds)g (sho)m(ws)h(that)494 726 y Fj(E)578 597 y Fi(\032)647 602 y(Z)753 726 y Fh(j)p Fl(f)823 740 y Fk(n)889 726 y Fh(\000)19 b Fl(f)10 b Fh(j)1060 597 y Fi(\033)1154 726 y Fh(\025)24 b Fn(max)1436 597 y Fi(\022)1503 726 y Fn(sup)1559 798 y Fk(t)1655 726 y Fh(j)p Fn(1)d Fh(\000)e Fl( )s Fh(jj)p Fl(')p Fh(j)33 b Fl(;)46 b Fn(sup)2194 798 y Fk(t)2291 726 y Fh(j)p Fl( )s Fh(j)p Fl(n)2458 688 y Fg(\000)p Ff(1)p Fk(=)p Ff(2)2709 664 y Fn(1)p 2625 705 214 4 v 2625 723 a Fh(p)p 2702 723 137 4 v 2702 799 a Fn(128)2849 726 y(\(1)19 b Fh(\000)g(j)p Fl(')p Fh(j)3150 688 y Ff(2)3189 726 y Fn(\))3225 597 y Fi(\023)1154 978 y Fh(\025)24 b Fn(sup)1307 1051 y Fk(t)1403 978 y Fn(max)1588 849 y Fi(\022)1655 978 y Fh(j)p Fn(1)d Fh(\000)e Fl( )s Fh(jj)p Fl(')p Fh(j)33 b Fl(;)46 b Fh(j)p Fl( )s Fh(j)p Fl(n)2305 941 y Fg(\000)p Ff(1)p Fk(=)p Ff(2)2556 916 y Fn(1)p 2473 957 214 4 v 2473 975 a Fh(p)p 2549 975 137 4 v 76 x Fn(128)2696 978 y(\(1)19 b Fh(\000)h(j)p Fl(')p Fh(j)2998 941 y Ff(2)3036 978 y Fn(\))3072 849 y Fi(\023)1154 1231 y Fh(\025)1260 1169 y Fn(1)p 1260 1210 46 4 v 1260 1293 a(2)1331 1231 y(sup)1387 1303 y Fk(t)1484 1231 y Fn(min)1651 1102 y Fi(\022)1719 1231 y Fh(j)p Fl(')p Fh(j)31 b Fl(;)46 b(n)1985 1193 y Fg(\000)p Ff(1)p Fk(=)p Ff(2)2235 1169 y Fn(1)p 2152 1210 214 4 v 2152 1228 a Fh(p)p 2228 1228 137 4 v 75 x Fn(128)2375 1231 y(\(1)19 b Fh(\000)g(j)p Fl(')p Fh(j)2676 1193 y Ff(2)2715 1231 y Fn(\))2751 1102 y Fi(\023)2833 1231 y Fl(:)0 1449 y Fn(The)30 b(minim)m(um)f(is)h(maximal)f(when)g(b)s(oth) g(op)s(erands)f(are)h(equal,)h(whic)m(h)f(o)s(ccurs)f(for)1019 1656 y Fh(j)p Fl(')p Fh(j)f Fn(=)1262 1595 y(1)p 1262 1635 46 4 v 1262 1719 a(2)1332 1555 y Fi(\020)1387 1656 y Fh(\000)1459 1576 y(p)p 1534 1576 192 4 v 1534 1656 a Fn(128)p Fl(n)19 b Fn(+)1838 1580 y Fh(p)p 1914 1580 350 4 v 76 x Fn(128)p Fl(n)g Fn(+)h(4)2263 1555 y Fi(\021)1155 1912 y Fn(=)1262 1850 y(1)p 1262 1891 46 4 v 1262 1975 a(2)1317 1832 y Fh(p)p 1393 1832 192 4 v 80 x Fn(128)p Fl(n)1600 1756 y Fi( )1673 1912 y Fh(\000)p Fn(1)f(+)1901 1769 y Fi(r)p 1992 1769 370 4 v 143 x Fn(1)h(+)2233 1850 y(4)p 2160 1891 192 4 v 2160 1975 a(128)p Fl(n)2361 1756 y Fi(!)1155 2247 y Fh(\025)1262 2185 y Fn(1)p 1262 2226 46 4 v 1262 2309 a(2)1317 2167 y Fh(p)p 1393 2167 192 4 v 80 x Fn(128)p Fl(n)1600 2063 y Fi(0)1600 2228 y(@)1680 2247 y Fh(\000)p Fn(1)f(+)i(1)f(+)2353 2185 y(4)p 2076 2226 601 4 v 2076 2348 a(256)p Fl(n)2269 2244 y Fi(q)p 2359 2244 318 4 v 2359 2348 a Fn(1)g(+)2580 2312 y Ff(4)p 2526 2327 141 4 v 2526 2379 a(128)p Fk(n)2686 2063 y Fi(1)2686 2228 y(A)1313 2518 y Fn(\(use)1506 2449 y Fh(p)p 1582 2449 202 4 v 69 x Fn(1)g(+)g Fl(\030)30 b Fh(\025)24 b Fn(1)c(+)2173 2478 y Fk(\030)p 2072 2497 233 4 v 2072 2551 a Ff(2)2105 2506 y Fg(p)p 2160 2506 146 3 v 45 x Ff(1+)p Fk(xi)2315 2518 y Fn(\))1155 2715 y(=)1561 2654 y(1)p 1262 2694 644 4 v 1262 2741 a Fh(p)p 1338 2741 192 4 v 75 x Fn(128)p Fl(n)1530 2713 y Fi(q)p 1621 2713 285 4 v 103 x Fn(1)g(+)1825 2780 y Ff(1)p 1788 2795 108 4 v 1788 2848 a(32)p Fk(n)1915 2715 y Fl(:)0 3010 y Fn(The)27 b(v)-5 b(alue)26 b(for)h Fh(j)p Fl(')p Fh(j)j Fn(is)d(b)s(et)m(w)m(een) f(zero)g(and)h(one,)g(so)h(that)e(b)m(y)i(the)f(con)m(tin)m(uit)m(y)e (of)j Fl(')p Fn(,)h(it)e(is)g(attained)d(for)j(some)0 3118 y Fl(t)p Fn(.)41 b(The)30 b(sough)m(t)f(lo)m(w)m(er)h(b)s(ound)e (is)i(1)p Fl(=)p Fn(2)f(times)g(the)h(giv)m(en)f(v)-5 b(alue)29 b(of)h Fh(j)p Fl(')p Fh(j)p Fn(.)p 2515 3066 65 4 v 2515 3125 4 59 v 2576 3125 V 2515 3128 65 4 v 299 3258 a(A)h(table)d(of)i(constan)m(ts)e(is)i(pro)m(vided)f(b)s(elo)m (w:)1313 3452 y(Sample)f(size)h Fl(n)92 b Fn(Lo)m(w)m(er)30 b(b)s(ound)1815 3610 y(1)91 b(0.043519)1769 3718 y(10)g(0.013953)1723 3825 y(100)g(0.004418)1678 3933 y(1000)f(0.001397)1632 4041 y(10000)g(0.000441)1586 4149 y(100000)g(0.000139)1541 4257 y(1000000)f(0.000044)1495 4365 y(10000000)g(0.000013)1872 5280 y Fc(4)p eop %%Page: 5 5 5 4 bop 0 83 a Fj(An)36 b(asymptotic)d(b)s(ound.)299 247 y Fn(W)-8 b(e)33 b(can)g(giv)m(e)g(another)e(inequalit)m(y)g(whic)m (h)h(is)h(sligh)m(tly)f(b)s(etter)f(\(b)m(y)i(a)h(factor)d(of)i(1)p Fl(=)3252 171 y Fh(p)p 3328 171 46 4 v 76 x Fn(2\))g(than)f(the)0 355 y(b)s(ound)c(of)h(Theorem)e(1)j(for)e(v)m(ery)i(large)e Fl(n)p Fn(.)40 b(It)29 b(is)g(not)f(as)h(go)s(o)s(d)f(for)g(small)g (and)h(mo)s(derate)e(v)-5 b(alues)27 b(of)j Fl(n)p Fn(.)40 b(W)-8 b(e)0 463 y(will)29 b(only)g(indicate)e(where)i(the)g(pro)s(ofs) g(require)f(mo)s(di\014cation.)0 682 y Fm(Theorem)34 b(2.)50 b Fd(W)-8 b(e)30 b(ha)m(v)m(e)1023 906 y Fn(inf)992 962 y Fk(K;h;f)1181 906 y Fj(E)1265 777 y Fi(\032)1333 782 y(Z)1440 906 y Fh(j)p Fl(f)1510 920 y Fk(n)1575 906 y Fh(\000)20 b Fl(f)10 b Fh(j)1747 777 y Fi(\033)1841 906 y Fh(\025)2035 844 y Fn(1)p 1947 885 223 4 v 1947 969 a(16)2039 903 y Fh(p)p 2114 903 55 4 v 2114 969 a Fl(n)2194 750 y Fi( )2277 824 y Fn(1)20 b Fh(\000)2536 788 y Ff(8)p 2444 803 218 4 v 2444 856 a(\(4)p Fk(n)p Ff(\))2570 839 y Fa(1)p Fb(=)p Fa(4)p 2277 885 395 4 v 2321 969 a Fn(1)g(+)2536 933 y Ff(1)p 2488 948 129 4 v 2488 1001 a(4)2521 959 y Fg(p)p 2576 959 42 3 v 42 x Fk(n)2681 750 y Fi(!)2769 906 y 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2108 y Fk(n)1161 2136 y Fi(X)1173 2328 y Fk(i)p Ff(=1)1308 2223 y Fl(X)1384 2237 y Fk(i)1411 2223 y Fh(j)1436 2066 y Fi(\))1536 2223 y Fh(\025)1642 2161 y Fn(1)p 1642 2202 46 4 v 1642 2285 a(2)1698 2140 y Fi(p)p 1789 2140 296 4 v 83 x Fj(E)15 b Fh(f)p Fl(X)2002 2191 y Ff(2)1995 2243 y(1)2039 2223 y Fh(g)20 b(\000)2196 2094 y Fi(\022)2301 2161 y Fn(1)p 2273 2202 101 4 v 2273 2285 a(4)p Fl(n)2384 2094 y Fi(\023)2461 2088 y Fa(1)p 2461 2097 29 3 v 2461 2131 a(4)2518 2223 y Fj(E)2587 2185 y Ff(1)p Fk(=)p Ff(4)2707 2149 y Fi(\010)2760 2223 y Fl(X)2843 2185 y Ff(4)2836 2245 y(1)2880 2149 y Fi(\011)2949 2223 y Fl(:)0 2654 y Fm(Pr)n(oof.)50 b Fn(W)-8 b(e)31 b(pro)s(ceed)c(as)j(in)g(the)f(pro)s(of)g(of)h(Theorem)f (1.)40 b(By)31 b(Lemma)f(2,)915 2748 y Fh(p)p 991 2748 55 4 v 70 x Fl(n)p Fj(E)15 b Fh(fj)q(<)p Fn(\()p Fl(')1363 2832 y Fk(n)1408 2818 y Fn(\))20 b Fh(\000)f(<)p Fn(\()p Fl(')p Fn(\))p Fh(jg)1072 3035 y(\025)1178 2973 y Fn(1)p 1178 3014 46 4 v 1178 3098 a(2)1234 2898 y Fi(r)p 1325 2898 1066 4 v 137 x Fj(E)1409 2934 y Fi(n)1470 3035 y Fn(\(cos)o(\()p Fl(tX)1773 3049 y Ff(1)1809 3035 y Fn(\))h Fh(\000)f(<)p Fn(\()p Fl(')p Fn(\()p Fl(t)p Fn(\)\)\))2293 2987 y Ff(2)2330 2934 y Fi(o)2411 3035 y Fh(\000)g Fn(2)2563 2906 y Fi(\022)2668 2973 y Fn(1)p 2640 3014 101 4 v 2640 3098 a(4)p Fl(n)2751 2906 y Fi(\023)2828 2901 y Fa(1)p 2828 2910 29 3 v 2828 2943 a(4)1072 3316 y Fh(\025)1178 3254 y Fn(1)p 1178 3295 46 4 v 1178 3379 a(2)1234 3234 y Fi(p)p 1325 3234 1002 4 v 82 x Fj(E)c Fh(f)p Fn(cos)1577 3290 y Ff(2)1614 3316 y Fn(\()p Fl(tX)1759 3330 y Ff(1)1796 3316 y Fn(\))20 b Fh(\000)f(<)2009 3290 y Ff(2)2046 3316 y Fn(\()p Fl(')p Fn(\()p Fl(t)p Fn(\)\))p Fh(g)g(\000)g Fn(2)2499 3187 y Fi(\022)2604 3254 y Fn(1)p 2576 3295 101 4 v 2576 3379 a(4)p Fl(n)2687 3187 y Fi(\023)2764 3182 y Fa(1)p 2764 3191 29 3 v 2764 3224 a(4)1072 3597 y Fn(=)1178 3535 y(1)p 1178 3576 46 4 v 1178 3660 a(2)1234 3449 y Fi(r)p 1325 3449 994 4 v 1335 3535 a Fn(1)p 1335 3576 46 4 v 1335 3660 a(2)1411 3597 y(+)1512 3535 y Fh(<)p Fn(\()p Fl(')p Fn(\(2)p Fl(t)p Fn(\)\))p 1512 3576 347 4 v 1662 3660 a(2)1889 3597 y Fh(\000)g(<)2046 3571 y Ff(2)2083 3597 y Fn(\()p Fl(')p Fn(\()p Fl(t)p Fn(\)\))g Fh(\000)h Fn(2)2491 3468 y Fi(\022)2595 3535 y Fn(1)p 2568 3576 101 4 v 2568 3660 a(4)p Fl(n)2678 3468 y Fi(\023)2755 3463 y Fa(1)p 2755 3472 29 3 v 2755 3505 a(4)2843 3597 y Fl(:)0 3811 y Fn(A)31 b(similar)d(argumen)m(t)h(on)h(the)f(imaginary) f(part)h(can)g(b)s(e)h(used)f(to)h(sho)m(w)g(that)528 3984 y Fh(p)p 604 3984 55 4 v 70 x Fl(n)p Fj(E)15 b Fh(fj=)p Fn(\()p Fl(')975 4068 y Fk(n)1020 4054 y Fn(\))20 b Fh(\000)g(=)p Fn(\()p Fl(')p Fn(\))p Fh(jg)25 b(\025)1567 3992 y Fn(1)p 1567 4033 46 4 v 1567 4117 a(2)1623 3906 y Fi(r)p 1714 3906 994 4 v 1724 3992 a Fn(1)p 1724 4033 46 4 v 1724 4117 a(2)1800 4054 y Fh(\000)1901 3992 y(<)p Fn(\()p Fl(')p Fn(\(2)p Fl(t)p Fn(\)\))p 1901 4033 347 4 v 2052 4117 a(2)2278 4054 y Fh(\000)19 b(=)2435 4028 y Ff(2)2472 4054 y Fn(\()p Fl(')p Fn(\()p Fl(t)p Fn(\)\))g 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Fh(j)p Fl(')p Fh(j)p 2475 4507 314 4 v 2609 4591 a Fn(4)2819 4528 y Fh(\000)2910 4399 y Fi(\022)2992 4466 y Fn(4)p 2988 4507 55 4 v 2988 4591 a Fl(n)3052 4399 y Fi(\023)3130 4394 y Fa(1)p 3130 4403 29 3 v 3130 4436 a(4)3218 4528 y Fl(:)0 4742 y Fn(Collecting)26 b(all)j(the)g(b)s(ounds) g(sho)m(ws)h(that)338 4985 y Fj(E)422 4856 y Fi(\032)490 4861 y(Z)597 4985 y Fh(j)p Fl(f)667 4999 y Fk(n)732 4985 y Fh(\000)20 b Fl(f)10 b Fh(j)904 4856 y Fi(\033)997 4985 y Fh(\025)25 b Fn(max)1279 4829 y Fi( )1351 4985 y Fn(sup)1408 5058 y Fk(t)1504 4985 y Fh(j)p Fn(1)20 b Fh(\000)g Fl( )s Fh(jj)p Fl(')p Fh(j)33 b Fl(;)46 b Fn(sup)2043 5058 y Fk(t)2139 4985 y Fh(j)p Fl( )s Fh(j)p Fl(n)2306 4947 y Fg(\000)p Ff(1)p Fk(=)p Ff(2)2479 4829 y Fi( )2562 4923 y Fn(1)19 b Fh(\000)h Fn(2)p Fh(j)p Fl(')p Fh(j)p 2562 4964 314 4 v 2695 5048 a Fn(8)2905 4985 y Fh(\000)2997 4856 y Fi(\022)3101 4923 y Fn(1)p 3074 4964 101 4 v 3074 5048 a(4)p Fl(n)3184 4856 y Fi(\023)3261 4851 y Fa(1)p 3261 4860 29 3 v 3261 4893 a(4)3304 4829 y Fi(!)o(!)1872 5280 y Fc(5)p eop %%Page: 6 6 6 5 bop 997 161 a Fh(\025)25 b Fn(sup)1150 234 y Fk(t)1247 161 y Fn(max)1432 5 y Fi( )1504 161 y Fh(j)p Fn(1)20 b Fh(\000)g Fl( )s Fh(jj)p Fl(')p Fh(j)33 b Fl(;)46 b Fh(j)p Fl( )s Fh(j)p Fl(n)2154 123 y Fg(\000)p Ff(1)p Fk(=)p Ff(2)2327 5 y Fi( )2409 99 y Fn(1)20 b Fh(\000)f Fn(2)p Fh(j)p Fl(')p Fh(j)p 2409 140 314 4 v 2543 224 a Fn(8)2752 161 y Fh(\000)2844 32 y Fi(\022)2948 99 y Fn(1)p 2921 140 101 4 v 2921 224 a(4)p Fl(n)3031 32 y Fi(\023)3109 27 y Fa(1)p 3109 36 29 3 v 3109 69 a(4)3151 5 y Fi(!)o(!)997 469 y Fh(\025)1104 408 y Fn(1)p 1104 448 46 4 v 1104 532 a(2)1175 469 y(sup)1231 542 y Fk(t)1327 469 y Fn(min)1495 313 y Fi( )1567 469 y Fh(j)p Fl(')p Fh(j)32 b Fl(;)46 b(n)1834 432 y Fg(\000)p Ff(1)p Fk(=)p Ff(2)2005 313 y Fi( )2088 408 y Fn(1)20 b Fh(\000)f Fn(2)p Fh(j)p Fl(')p Fh(j)p 2088 448 314 4 v 2221 532 a Fn(8)2431 469 y Fh(\000)2523 341 y Fi(\022)2627 408 y Fn(1)p 2600 448 101 4 v 2600 532 a(4)p Fl(n)2710 341 y Fi(\023)2787 335 y Fa(1)p 2787 344 29 3 v 2787 377 a(4)2830 313 y Fi(!)o(!)2990 469 y Fl(:)0 705 y Fn(The)30 b(minim)m(um)f(is)h(maximal) f(when)g(b)s(oth)g(op)s(erands)f(are)h(equal,)h(whic)m(h)f(o)s(ccurs)f (for)1489 936 y Fh(j)p Fl(')p Fh(j)f Fn(=)1789 818 y Ff(1)p 1741 833 129 4 v 1741 886 a(8)1774 844 y Fg(p)p 1829 844 42 3 v 42 x Fk(n)1900 854 y Fh(\000)2110 818 y Ff(1)p 2002 833 250 4 v 2002 886 a(\(4)p Fk(n)2102 869 y Fa(3)2134 886 y Ff(\))2160 869 y Fa(1)p Fb(=)p Fa(4)p 1731 915 530 4 v 1843 999 a Fn(1)20 b(+)2058 963 y Ff(1)p 2011 978 129 4 v 2011 1031 a(4)2044 989 y Fg(p)p 2098 989 42 3 v 2098 1031 a Fk(n)2271 936 y Fl(:)0 1171 y Fn(The)30 b(lo)m(w)m(er)f(b)s(ound)g(in)h(question)e(is)i(half)f (this)g(v)-5 b(alue)29 b(of)h Fh(j)p Fl(')p Fh(j)p Fn(.)p 2144 1119 65 4 v 2144 1177 4 59 v 2206 1177 V 2144 1180 65 4 v 0 1453 a Fj(References.)0 1711 y Fn(K.)35 b(B.)f(Da)m(vis,)g (\\Mean)f(square)g(error)g(prop)s(erties)d(of)k(densit)m(y)f (estimates,")e 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