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b(1\))j(can)g(b)r(e)h(used)f(instead)g(to)g(estimate)g(the)g (directional)g(distribution)g(of)g(\010)2756 1427 y Fm(d)2756 1480 y(k)2797 1457 y Fo(.)48 b(F)-7 b(or)30 b(this)i(purp)r(ose,)f(the) h(rose)84 1556 y(of)f(neigh)n(b)r(orho)r(o)r(d)e(is)i(in)n(tro)r(duced) f(and)g(its)h(prop)r(erties)f(are)f(studied.)46 b(Namely)-7 b(,)32 b(w)n(e)e(\\blo)n(w")f(the)i(test)g(line)f(\(or)g(the)84 1656 y(lo)n(w)n(er)j(dimensional)h(\015at\))g Fn(\021)j Fo(up)e(to)f(a)g(test)g(cylinder)g(whose)f(in)n(tersection)h(with)g (\015ats)g(of)g(the)h(pro)r(cess)e(\010)3531 1626 y Fm(d)3531 1680 y(k)3606 1656 y Fo(is)h(not)84 1758 y(empt)n(y)e(an)n(ymore.)45 b(Then,)32 b(one)f(can)f(coun)n(t)h(the)h(\015ats)e(of)h(\010)1996 1728 y Fm(d)1996 1782 y(k)2068 1758 y Fo(that)h(in)n(tersect)e(the)h (test)h(cylinder.)46 b(Collecting)31 b(suc)n(h)84 1858 y(information)f(for)f(the)i(test)f(\015ats)g Fn(\021)j Fo(with)e(v)-5 b(arious)28 b(directions,)i(a)g(conclusion)f(ab)r(out)h (the)g(directional)f(distribution)84 1958 y(can)f(b)r(e)g(made.)184 2061 y(Notice)i(that)f(the)h(idea)g(of)f(\\blo)n(wing")f(test)i (\015ats)f(up)h(to)f(their)h(neigh)n(b)r(orho)r(o)r(ds)e(w)n(as)g (already)h(men)n(tioned)g(in)h([6)o(])84 2160 y(and)e([21)o(].)184 2263 y(In)g(Section)g(4.2,)f(an)h(asymptotically)e(un)n(biased)i (estimator)f(for)g(the)h(rose)f(of)h(neigh)n(b)r(orho)r(o)r(d)e(is)i (prop)r(osed.)37 b(This)84 2363 y(enables)28 b(us)f(to)h(estimate)f (the)h(densit)n(y)f(of)h(the)g(directional)f(distribution)g(of)h(the)g (pro)r(cess)e(\010)3033 2333 y Fm(d)3033 2387 y(k)3074 2363 y Fo(,)i(see)f(Section)h(4.3.)84 2623 y Fp(2)113 b(Stationary)37 b Ff(k)r Fp({\015at)g(pro)s(cesses)84 2793 y Fo(In)29 b(this)g(section,)f(w)n(e)g(follo)n(w)f(the)i(framew)n (ork)d(of)i([16)o(])h(in)f(in)n(tro)r(ducing)g(the)h(basic)f(notions)f (of)i Fn(k)s Fo({\015at)e(pro)r(cesses)g(\(cf.)84 2893 y([15],)g([22)o(],)h([25)o(])g(for)f(other)g(constructions\).)184 2996 y(Let)34 b Fn(F)12 b Fo(\()p Fn(k)s(;)i(d)p Fo(\))35 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y(k)1926 5145 y Fo(that)g(b)r(elong)f(to)h Fn(D)g Fj(2)e(F)8 b Fo(.)47 b(Then)31 b(the)h(measure)e(\003)e(:)h Fj(F)36 b(!)84 5245 y Fo([0)p Fn(;)14 b Fj(1)p Fo(\))28 b(with)511 5438 y(\003\()p Fn(D)r Fo(\))51 b(=)f Fn(E)19 b Fo(\010)1010 5404 y Fm(d)1010 5459 y(k)1051 5438 y Fo(\()p Fn(D)r Fo(\))14 b Fn(;)97 b(D)25 b Fj(2)f(F)p eop %%Page: 3 3 3 2 bop 84 30 a FB(Math.)35 b(Nac)n(hr.)26 b FA(251)g FB(\(2003\))2838 b(3)p 84 68 3732 4 v 84 258 a Fo(is)28 b(called)f(the)h Fk(intensity)i(me)l(asur)l(e)d Fo(of)h(\010)1362 228 y Fm(d)1362 281 y(k)1403 258 y Fo(.)184 360 y(F)-7 b(or)27 b(stationary)f(\010)788 330 y Fm(d)788 384 y(k)829 360 y Fo(,)i(the)g(follo)n(wing)e(factorization)g(of)i(its)g(in)n (tensit)n(y)f(measure)g(holds)g(\(cf.)h([22)o(]\):)511 585 y(\003\()p Fn(D)r Fo(\))51 b(=)f Fn(\025)988 472 y Fd(Z)913 714 y Fm(G)p Fw(\()p Fm(k)q(;d)p Fw(\))1141 472 y Fd(Z)1123 716 y Fm(\030)1155 700 y Fb(?)1238 585 y Fn(I)1274 597 y Fm(D)1334 585 y Fo(\()p Fn(y)22 b Fo(+)c Fn(\030)t Fo(\))c 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1247 y Fw(\010)951 1227 y Fe(d)951 1267 y(k)988 1247 y Fx(\\)p Fm(\021)1102 1227 y Fo(b)r(e)h(the)f(in)n(tensit)n(y)g(of)g (\010)1858 1197 y Fm(d)1858 1250 y(k)1918 1227 y Fj(\\)20 b Fn(\021)s Fo(.)41 b(Due)30 b(to)e(the)i(stationarit)n(y)d(of)i(\010) 3129 1197 y Fm(d)3129 1250 y(k)3170 1227 y Fo(,)g(it)h(is)f(su\016cien) n(t)g(to)84 1346 y(consider)g(only)h(those)g(a\016ne)g(\015ats)f Fn(\021)34 b Fo(that)c(con)n(tain)f(the)i(origin,)e(i.e.,)i Fn(\021)f Fj(2)e Fn(G)2559 1279 y Fd(\000)2597 1346 y Fn(d)21 b Fj(\000)e Fn(k)k Fo(+)d Fn(j;)14 b(d)3010 1279 y Fd(\001)3048 1346 y Fo(.)44 b(Then,)31 b(the)g(in)n(tensit)n(y)84 1446 y Fn(\025)132 1466 y Fw(\010)179 1446 y Fe(d)179 1486 y(k)217 1466 y Fx(\\)p Fm(\021)329 1446 y Fo(as)c(a)h(function)g (of)f(the)h(directional)f(distribution)h Fn(\022)i Fo(and)d(the)h(test) g(\015at)g Fn(\021)i Fo(rewrites)511 1621 y Fd(\000)549 1688 y Fn(T)598 1700 y Fm(k)q(;d)p Fx(\000)p Fm(k)q Fw(+)p Fm(j)863 1688 y Fn(\022)904 1621 y Fd(\001)942 1688 y Fo(\()p Fn(\021)s Fo(\))52 b(=)e Fn(\025)1335 1575 y Fd(Z)1260 1817 y Fm(G)p Fw(\()p Fm(k)q(;d)p Fw(\))1446 1688 y Fo([)p Fn(\030)t(;)14 b(\021)s Fo(])g Fn(\022)r Fo(\()p Fn(d\030)t Fo(\))1830 b(\(2.3\))84 1999 y(where)34 b([)p Fn(\030)t(;)14 b(\021)s Fo(])34 b(is)f(the)770 1932 y Fd(\000)808 1999 y Fn(d)23 b Fj(\000)f Fn(j)1000 1932 y Fd(\001)1038 1999 y Fo({v)n(olume)32 b(of)i(the)g(parallelepip)r (ed)e(spanned)i(b)n(y)f(orthonormal)f(bases)g(in)i Fn(\030)3458 1969 y Fx(?)3548 1999 y Fo(and)f Fn(\021)3759 1969 y Fx(?)84 2107 y Fo(\(cf.)c([3)o(],)f([9]\).)37 b(The)28 b(function)1044 2040 y Fd(\000)1082 2107 y Fn(T)1131 2119 y Fm(k)q(;d)p Fx(\000)p Fm(k)q Fw(+)p Fm(j)1396 2107 y Fn(\022)1437 2040 y Fd(\001)1475 2107 y Fo(\()p Fn(\021)s Fo(\),)h Fn(\021)d Fj(2)d Fn(G)1845 2040 y Fd(\000)1884 2107 y Fn(d)18 b Fj(\000)g Fn(k)k Fo(+)c Fn(j;)c(d)2290 2040 y Fd(\001)2328 2107 y Fo(,)28 b(is)f(called)h(the)g Fk(r)l(ose)i(of)g(interse)l(ctions)e Fo(of)g(\010)3751 2077 y Fm(d)3751 2130 y(k)3791 2107 y Fo(.)84 2348 y Fp(3)113 b(Isop)s(erimetric)34 b(problems)84 2512 y Fy(3.1)95 b(Connection)31 b(to)h(con)m(v)m(ex)h(geometry)84 2647 y Fo(Supp)r(ose)c(the)f(stationary)f Fn(k)s Fo({\015at)h(pro)r(cess)f (\010)1536 2617 y Fm(d)1536 2671 y(k)1605 2647 y Fo(to)h(b)r(e)h(P)n (oisson)d(\(cf.)j([15)o(],)g([25)o(])f(for)g(the)g(de\014nition\).)40 b(F)-7 b(or)28 b(in)n(tegers)f Fn(k)84 2747 y Fo(and)h Fn(d)g Fo(with)g(2)p Fn(k)d Fj(\025)e Fn(d)p Fo(,)28 b(in)n(tro)r(duce)f(the)h(\(2)p Fn(k)21 b Fj(\000)d Fn(d)p Fo(\){\015at)28 b(pro)r(cess)511 2931 y Fn(X)580 2943 y Fw(2)617 2864 y Fd(\000)655 2931 y Fo(\010)715 2897 y Fm(d)715 2951 y(k)756 2864 y Fd(\001)845 2931 y Fo(=)960 2864 y Fd(\010)1008 2931 y Fn(\030)1044 2943 y Fw(1)1100 2931 y Fj(\\)19 b Fn(\030)1210 2943 y Fw(2)1271 2931 y Fo(:)k Fn(\030)1353 2943 y Fw(1)1390 2931 y Fn(;)14 b(\030)1463 2943 y Fw(2)1524 2931 y Fj(2)23 b Fo(\010)1662 2897 y Fm(d)1662 2951 y(k)1703 2931 y Fn(;)42 b Fo(dim\()p Fn(\030)1974 2943 y Fw(1)2030 2931 y 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y(b)r(e)j(c)n(hosen)e(from)h(the)g(smaller)f(class)g(of)h(pro)5 b(jection)31 b(generating)g(measures)f(of)i(zonoids)f(in)i Fl(R)3134 1429 y Fm(d)3210 1459 y Fo(for)f Fn(d)f Fo(=)e(2)p Fn(k)s Fo(,)k(then)84 1559 y(problem)28 b(\(3.2\))g(rewrites)f(\(up)h (to)g(a)f(constan)n(t)h(factor\))f(as)g(follo)n(ws:)37 b(\014nd)28 b(zonoids)f Fn(K)34 b Fo(of)28 b(maxim)n(um)f(v)n(olume)h Fn(\027)3636 1571 y Fm(d)3674 1559 y Fo(\()p Fn(K)6 b Fo(\))84 1658 y(pro)n(vided)27 b(that)h(the)g(total)f(mass)g(of)h (their)f(pro)5 b(jection)27 b(generating)f(measure)g(is)i Fn(\025)p Fo(,)g(i.e.,)g Fn(\032)2951 1670 y Fm(k)2992 1658 y Fo(\()p Fn(K)q(;)14 b(G)p Fo(\()p Fn(k)s(;)g(d)p Fo(\)\))24 b(=)e Fn(\025)p Fo(.)184 1764 y(Denote)h(b)n(y)f Fn(\022)613 1776 y Fm(K)700 1764 y Fo(the)h Fk(gener)l(ating)j(me)l (asur)l(e)c Fo(for)h(a)f(zonoid)g Fn(K)6 b Fo(.)35 b(By)22 b(de\014nition,)i(this)f(is)g(the)g(measure)f(on)g(the)h(sphere)84 1864 y(suc)n(h)28 b(that)g(the)g(supp)r(ort)f(function)h Fn(h)1275 1876 y 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b(to)f(the)h(stationarit)n(y)e(of)h(\010)3659 4157 y Fm(d)3659 4210 y(k)3700 4187 y Fo(,)i(it)84 4286 y(su\016ces)28 b(to)f(consider)g(only)g(those)g(\015ats)h Fn(\021)e Fj(2)d Fn(F)12 b Fo(\()p Fn(r)n(;)i(d)p Fo(\))29 b(that)f(con)n(tain)e(the)i(origin,)f(i.e.)37 b Fn(\021)26 b Fj(2)e Fn(G)p Fo(\()p Fn(r)n(;)14 b(d)p Fo(\).)184 4386 y(F)-7 b(or)32 b(an)n(y)g(stationary)f Fn(k)s Fo({\015at)h(pro)r (cess)f(\010)1494 4356 y Fm(d)1494 4410 y(k)1568 4386 y Fo(with)i(in)n(tensit)n(y)g Fn(\025)g Fo(and)f(directional)g (distribution)h Fn(\022)r Fo(,)h(in)n(tro)r(duce)e(the)84 4488 y(family)23 b(of)f Fk(dual)j(pr)l(o)l(c)l(esses)e Fn(D)r Fo(\()p Fn(\025;)14 b(\022)r Fo(\):)36 b(a)21 b(stationary)g(\()p Fn(d)8 b Fj(\000)g Fn(k)s Fo(\){\015at)21 b(pro)r(cess)g(\010)2477 4458 y Fm(d)2477 4512 y(d)p Fx(\000)p Fm(k)2627 4488 y Fo(b)r(elongs)g(to)h Fn(D)r Fo(\()p Fn(\025;)14 b(\022)r Fo(\))24 b(i\013)e(its)h(in)n(tensit)n(y) 84 4598 y(is)h(equal)e(to)i Fn(\025)f Fo(and)h(its)f(directional)f (distribution)i(is)f Fn(\022)1791 4568 y Fx(?)1847 4598 y Fo(\()p Fn(d\020)6 b Fo(\))25 b(=)d Fn(\022)2149 4530 y Fd(\000)2188 4598 y Fn(d\020)2273 4568 y Fx(?)2329 4530 y Fd(\001)2391 4598 y Fo(for)h Fn(\020)29 b Fj(2)23 b Fn(G)p Fo(\()p Fn(d)10 b Fj(\000)g Fn(k)s(;)k(d)p Fo(\).)36 b(The)24 b(follo)n(wing)e(result)84 4697 y(connects)28 b(the)g(rose)e(of)i(neigh)n(b)r(orho)r(o)r(d)e(with)i(the)g(rose)e(of)i (in)n(tersections)e(of)i(a)f(dual)h(pro)r(cess.)184 4822 y Fy(Theorem)33 b(4.1)41 b Fk(F)-6 b(or)32 b Fn(k)22 b Fo(+)d Fn(r)29 b(<)d(d)p Fk(,)32 b(the)g(fol)t(lowing)i(r)l (elationship)f(holds)g(for)f(the)g Fn(a)p Fk({r)l(ose)g(of)g(neighb)l (orho)l(o)l(d)i(of)e(the)84 4921 y(stationary)f Fn(k)s Fk({\015at)f(pr)l(o)l(c)l(ess)g Fo(\010)1056 4891 y Fm(d)1056 4945 y(k)1097 4921 y Fk(:)511 5109 y Fn(N)578 5121 y Fm(k)q(r)651 5109 y Fo(\()p Fn(a;)14 b(\021)s Fo(\))53 b(=)g Fn(\024)1059 5121 y Fm(d)p Fx(\000)p Fm(k)q Fx(\000)p Fm(r)1270 5109 y Fn(a)1314 5075 y Fm(d)p Fx(\000)p Fm(k)q Fx(\000)p Fm(r)1526 5042 y Fd(\000)1564 5109 y Fn(T)1613 5121 y Fm(d)p Fx(\000)p Fm(k)q(;d)p Fx(\000)p Fm(r)1879 5109 y Fn(\022)1920 5075 y Fx(?)1976 5042 y Fd(\001)14 b(\000)2066 5109 y Fn(\021)2110 5075 y Fx(?)2166 5042 y Fd(\001)2218 5109 y Fn(;)99 b(\021)26 b Fj(2)e Fn(G)p Fo(\()p Fn(r)n(;)14 b(d)p Fo(\))915 b(\(4.1\))84 5299 y Fk(wher)l(e)31 b Fn(T)368 5311 y Fm(d)p Fx(\000)p Fm(k)q(;d)p Fx(\000)p Fm(r)634 5299 y Fn(\022)675 5269 y Fx(?)761 5299 y Fk(is)f(the)h(r)l(ose)f(of)h(interse)l(ctions)f(of)h(the)f(dual) h(pr)l(o)l(c)l(ess)f Fo(\010)2505 5269 y Fm(d)2505 5322 y(d)p Fx(\000)p Fm(k)2656 5299 y Fj(2)24 b Fn(D)r Fo(\()p Fn(\025;)14 b(\022)r Fo(\))31 b Fk(with)g Fo(\()p Fn(d)19 b Fj(\000)f Fn(r)r Fo(\))p Fk({\015ats.)41 b(By)84 5408 y Fo(\(2.3\))p Fk(,)30 b Fn(T)359 5420 y Fm(d)p Fx(\000)p Fm(k)q(;d)p Fx(\000)p Fm(r)625 5408 y Fn(\022)666 5378 y Fx(?)752 5408 y Fk(is)g(the)g(same)g(for)h(al)t(l)g(pr)l(o)l(c)l (esses)f Fo(\010)1856 5378 y Fm(d)1856 5431 y(d)p Fx(\000)p Fm(k)2006 5408 y Fj(2)24 b Fn(D)r Fo(\()p Fn(\025;)14 b(\022)r Fo(\))p Fk(.)p eop %%Page: 9 9 9 8 bop 84 30 a FB(Math.)35 b(Nac)n(hr.)26 b FA(251)g FB(\(2003\))2838 b(9)p 84 68 3732 4 v 184 258 a Fo(P)13 b(r)h(o)f(o)h(f.)41 b(F)-7 b(or)31 b(an)n(y)g(nonparallel)e(\()p Fn(d)22 b Fj(\000)e Fn(k)s Fo(\){\015at)31 b Fn(\020)38 b Fo(and)31 b(\()p Fn(d)21 b Fj(\000)g Fn(r)r Fo(\){\015at)32 b Fn(\014)t Fo(,)g(their)g(in)n(tersection)e(is)h(not)g(empt)n(y)h (since)84 357 y Fn(d)23 b Fj(\000)f Fn(k)j Fo(+)d Fn(d)h Fj(\000)f Fn(r)35 b(>)e(d)p Fo(.)54 b(Therefore,)34 b(the)g(usual)f (rose)f(of)h(in)n(tersections)g(of)g(\010)2536 327 y Fm(d)2536 381 y(d)p Fx(\000)p Fm(k)2696 357 y Fj(2)g Fn(D)r Fo(\()p Fn(\025;)14 b(\022)r Fo(\))35 b(with)f(\()p Fn(d)23 b Fj(\000)f Fn(r)r Fo(\){\015ats)34 b(is)84 467 y(w)n(ell{de\014ned.)j(The)28 b(in)n(tensit)n(y)f(of)h(\010)1250 437 y Fm(d)1250 490 y(k)1309 467 y Fj(\014)18 b Fn(\021)31 b Fo(is)c(giv)n(en)g(b)n(y)511 739 y Fn(N)578 751 y Fm(k)q(r)651 739 y Fo(\()p 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Fm(\026)p Fw(=0)3382 3168 y Fn(!)3434 3180 y Fw(3)3471 3168 y Fo(\()p Fn(du)p Fo(\))84 3492 y(for)28 b(an)n(y)e(ev)n(en)h (four)h(times)g(con)n(tin)n(uously)e(di\013eren)n(tiable)h(function)i Fn(g)c Fj(2)f Fn(C)2468 3462 y Fw(4)2505 3425 y Fd(\000)2543 3492 y Fy(S)2596 3462 y Fw(2)2633 3425 y Fd(\001)2671 3492 y Fo(,)k(where)511 3812 y Fn(W)12 b Fo(\()p Fn(u)p Fo(\))51 b(=)879 3694 y Fd(\022)1012 3755 y Fn(d)p 936 3792 196 4 v 936 3869 a(d)p Fo(\()p Fn(\026)1061 3845 y Fw(2)1099 3869 y Fo(\))1127 3694 y Fd(\023)1185 3620 y(2)1185 3766 y(6)1185 3819 y(4)1370 3699 y(Z)1240 3947 y Fx(h)p Fm(u;t)p Fx(i)1378 3924 y Fc(2)1422 3947 y Fm(>\026)1514 3930 y Fc(2)1561 3755 y Fn(N)1628 3767 y Fw(11)1698 3755 y Fo(\()p Fn(t)p Fo(\))p Fj(j)14 b(h)q Fn(u;)g(t)p Fj(i)f(j)h Fn(!)2111 3767 y Fw(3)2148 3755 y Fo(\()p Fn(dt)p Fo(\))p 1561 3792 725 4 v 1679 3809 a Fd(q)p 1762 3809 405 4 v 104 x Fj(h)p Fn(u;)g(t)p Fj(i)1941 3872 y Fw(2)1997 3913 y Fj(\000)k Fn(\026)2130 3889 y Fw(2)2296 3620 y Fd(3)2296 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(end)f(of)g(this)h(section\).)35 b(In)22 b(practice,)h(it)g(is)f(imp)r (ossible)g(to)g(coun)n(t)g(the)g(in)n(tersections)f(of)i(a)e(line)i (pro)r(cess)84 4910 y(with)33 b(an)e(in\014nite)i(test)f(cylinder.)49 b(Th)n(us,)33 b(w)n(e)e(prop)r(ose)g(an)g(estimator)g(for)h Fn(N)2579 4922 y Fw(11)2649 4910 y Fo(\()p Fn(a;)14 b(\021)s Fo(\))32 b(b)n(y)g(supp)r(osing)f(the)h(cylinder)84 5010 y(with)26 b(the)g(axis)e(at)i Fn(\021)i Fo(to)d(ha)n(v)n(e)f(\014nite)i (length)g(2)p Fn(b)p Fo(.)35 b(This)25 b(estimator)g(app)r(ears)f(to)h (b)r(e)h(asymptotically)e(un)n(biased)h(as)f(the)84 5110 y(length)i(2)p Fn(b)f Fo(tends)h(to)f(in\014nit)n(y)i(or)d(the)i (radius)f(of)h(the)g(cylinder)f Fn(a)h Fo(b)r(ecomes)f(arbitrarily)f (small.)35 b(Due)27 b(to)e(stationarit)n(y)84 5209 y(of)j(\010)239 5179 y Fw(3)239 5230 y(1)276 5209 y Fo(,)g(it)g(su\016ces)f(to)h (consider)e(only)h(test)h(lines)g(going)e(through)h(the)h(origin.)184 5324 y(Coun)n(t)18 b(all)g(in)n(tersections)g(of)g(\010)1147 5294 y Fw(3)1147 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y(b)1308 1913 y Fo(\(0\))19 b Fj(\002)f Fn(B)1583 1878 y Fm(\021)1619 1853 y Fb(?)1579 1933 y Fm(a)1672 1913 y Fo(\(0\))1778 1845 y Fd(\001)1839 1913 y Fj(6)p Fo(=)23 b Fj(;)1969 1821 y Fd(o)619 2087 y Fo(=)50 b Fn(E)847 2008 y Fd(X)828 2193 y Fm(\020)t Fx(2)p Fw(\010)954 2173 y Fc(3)954 2211 y(1)1000 2087 y Fn(I)1036 2062 y Fd(\010)1085 2129 y Fm(\020)t Fx(\\)1164 2062 y Fd(\000)1202 2129 y Fm(B)1255 2101 y Fe(\021)1252 2149 y(b)1292 2129 y Fx(\002)p Fm(B)1397 2101 y Fe(\021)1429 2089 y Fb(?)1394 2138 y Fe(a)1482 2129 y Fw(\(0\))1567 2062 y Fd(\001)1605 2129 y Fx(6)p Fw(=)p Fx(;)1690 2062 y Fd(\011)1743 2087 y Fo(\()p Fn(\020)6 b Fo(\))619 2366 y(=)50 b Fn(\025)850 2253 y Fd(Z)777 2495 y Fm(G)p Fw(\(1)p Fm(;)p Fw(3\))1000 2253 y Fd(Z)982 2497 y Fm(\030)1014 2480 y Fb(?)1097 2366 y Fn(I)1133 2341 y Fd(\010)1182 2408 y Fw(\()p Fm(y)r Fw(+)p Fm(\030)r Fw(\))p Fx(\\)1398 2341 y Fd(\000)1435 2408 y Fm(B)1488 2380 y Fe(\021)1485 2428 y(b)1525 2408 y Fx(\002)p Fm(B)1630 2380 y Fe(\021)1662 2368 y Fb(?)1627 2417 y Fe(a)1715 2408 y Fw(\(0\))1800 2341 y Fd(\001)1838 2408 y Fx(6)p Fw(=)p Fx(;)1923 2341 y Fd(\011)1976 2366 y Fo(\()p Fn(y)s(;)14 b(\030)t Fo(\))g Fn(\027)2221 2326 y Fm(\030)2253 2301 y Fb(?)2216 2388 y Fw(2)2307 2366 y Fo(\()p Fn(dy)s Fo(\))g Fn(\022)r Fo(\()p Fn(d\030)t Fo(\))g Fn(:)3645 2188 y Fo(\(4.5\))84 2697 y(With)30 b(the)g(substitution)f Fn(y)f Fo(=)d Fn(P)12 b(r)1169 2714 y Fm(\030)1201 2697 y Fb(?)d Fo(\()p Fn(x)p Fo(\),)31 b Fn(x)25 b Fj(2)h Fn(H)36 b Fo(\(cf.)30 b(the)f(pro)r(of)f(of)h(Theorem)f(4.1\))h(where)f Fn(P)12 b(r)3146 2709 y Fm(')3194 2697 y Fo(\()p Fj(\001)p Fo(\))30 b(stands)f(for)f(the)84 2797 y(orthogonal)e(pro)5 b(jection)26 b(op)r(erator)g(on)n(to)h(the)h(linear)f(subspace)g Fn(')p Fo(,)h(the)g(inner)f(in)n(tegral)g(in)g(\(4.5\))h(rewrites)511 2980 y Fd(\002)545 3048 y Fn(\030)585 3013 y Fx(?)642 3048 y Fn(;)14 b(\021)723 3013 y Fx(?)779 2980 y Fd(\003)827 2935 y(Z)821 3173 y Fm(H)924 3048 y Fn(I)960 3022 y Fd(\010)q(\000)1047 3090 y Fm(P)9 b(r)1129 3111 y Fe(\030)1158 3099 y Fb(?)1211 3090 y Fm(x)p Fw(+)p Fm(\030)1332 3022 y Fd(\001)1370 3090 y Fx(\\)1415 3022 y Fd(\000)1452 3090 y Fm(B)1505 3061 y Fe(\021)1502 3110 y(b)1542 3090 y Fx(\002)p Fm(B)1647 3061 y Fe(\021)1679 3049 y Fb(?)1644 3099 y Fe(a)1732 3090 y Fw(\(0\))1817 3022 y Fd(\001)1856 3090 y Fx(6)p Fw(=)p Fx(;)1941 3022 y Fd(\011)1993 3048 y Fo(\()p Fn(y)s(;)14 b(\030)t Fo(\))g Fn(\027)2238 3013 y Fm(H)2233 3068 y Fw(2)2301 3048 y Fo(\()p Fn(dx)p Fo(\))g Fn(:)1153 b Fo(\(4.6\))84 3369 y(W)-7 b(e)29 b(compute)f(this)g(in)n(tegral)e(for)i (lines)f Fn(\021)32 b Fo(suc)n(h)27 b(that)1786 3301 y Fd(\002)1821 3369 y Fn(\030)1861 3339 y Fx(?)1917 3369 y Fn(;)14 b(\021)1998 3339 y Fx(?)2054 3301 y Fd(\003)2112 3369 y Fn(>)23 b Fo(0.)37 b(Since)28 b Fn(x)19 b Fj(\000)g Fn(P)12 b(r)2771 3385 y Fm(\030)2803 3369 y Fb(?)c Fo(\()p Fn(x)p Fo(\))29 b(is)f(parallel)f(to)g Fn(\030)t Fo(,)i(w)n(e)e(can)84 3468 y(write)h Fn(P)12 b(r)399 3485 y Fm(\030)431 3469 y Fb(?)485 3468 y Fo(\()p Fn(x)p Fo(\))19 b(+)f Fn(\030)27 b Fo(=)c Fn(x)c Fo(+)f Fn(\030)t Fo(,)28 b(and)f(the)h(in)n(tegral)f (in)g(\(4.6\))h(tak)n(es)e(the)i(form)511 3614 y Fd(Z)505 3852 y Fm(H)608 3727 y Fn(I)644 3701 y Fd(\010)693 3769 y Fw(\()p Fm(x)p Fw(+)p Fm(\030)r Fw(\))p Fx(\\)911 3701 y Fd(\000)948 3769 y Fm(B)1001 3740 y Fe(\021)998 3789 y(b)1038 3769 y Fx(\002)p Fm(B)1143 3740 y Fe(\021)1175 3728 y Fb(?)1140 3778 y Fe(a)1228 3769 y Fw(\(0\))1313 3701 y Fd(\001)1351 3769 y Fx(6)p Fw(=)p Fx(;)1436 3701 y Fd(\011)1489 3727 y Fo(\()p Fn(y)s(;)14 b(\030)t Fo(\))g Fn(\027)1734 3692 y Fm(H)1729 3747 y Fw(2)1797 3727 y Fo(\()p Fn(dx)p Fo(\))g Fn(:)1657 b Fo(\(4.7\))84 4035 y(Due)29 b(to)e(the)h(ob)n(vious)e(relationship)511 4243 y Fn(I)547 4218 y Fd(\010)596 4285 y Fw(\()p Fm(x)p Fw(+)p Fm(\030)r Fw(\))p Fx(\\)814 4218 y Fd(\000)851 4285 y Fm(B)904 4257 y Fe(\021)901 4305 y(b)941 4285 y Fx(\002)p Fm(B)1046 4257 y Fe(\021)1078 4245 y Fb(?)1043 4294 y Fe(a)1131 4285 y Fw(\(0\))1216 4218 y Fd(\001)1254 4285 y Fx(6)p Fw(=)p Fx(;)1339 4218 y Fd(\011)1443 4243 y Fo(=)50 b Fn(I)1594 4218 y Fd(\010)1643 4285 y Fm(x)p Fx(2)p Fm(B)1779 4257 y Fe(\021)1776 4305 y(b)1815 4285 y Fx(\002)p Fm(B)1920 4257 y Fe(\021)1952 4245 y Fb(?)1917 4294 y Fe(a)2005 4285 y Fw(\(0\))2090 4218 y Fd(\011)2161 4243 y Fo(+)18 b Fn(I)2280 4218 y Fd(\010)2329 4285 y Fm(x)7 b(=)-41 b Fx(2)p Fm(B)2465 4257 y Fe(\021)2462 4305 y(b)2501 4285 y Fx(\002)p Fm(B)2606 4257 y Fe(\021)2638 4245 y Fb(?)2603 4294 y Fe(a)2692 4285 y Fw(\(0\))p Fm(;)18 b Fw(\()p Fm(x)p Fw(+)p Fm(\030)r Fw(\))p Fx(\\)3033 4218 y Fd(\000)3071 4285 y Fm(B)3124 4257 y Fe(\021)3121 4305 y(b)3160 4285 y Fx(\002)p Fm(B)3265 4257 y Fe(\021)3297 4245 y Fb(?)3262 4294 y Fe(a)3351 4285 y Fw(\(0\))3436 4218 y Fd(\001)3474 4285 y Fx(6)p Fw(=)p Fx(;)3559 4218 y Fd(\011)3625 4243 y Fn(;)84 4494 y Fo(the)28 b(in)n(tegral)f(in)h (\(4.7\))f(is)g(equal)h(to)511 4727 y Fn(\027)557 4693 y Fm(H)552 4748 y Fw(2)620 4635 y Fd(\020)670 4727 y Fn(H)d Fj(\\)838 4660 y Fd(\000)876 4727 y Fn(B)943 4687 y Fm(\021)939 4752 y(b)983 4727 y Fo(\(0\))19 b Fj(\002)f Fn(B)1258 4693 y Fm(\021)1294 4668 y Fb(?)1254 4748 y Fm(a)1348 4727 y Fo(\(0\))1454 4660 y Fd(\001)1492 4635 y(\021)695 4910 y Fo(+)41 b Fn(\027)847 4875 y Fm(H)842 4930 y Fw(2)911 4818 y Fd(\020)960 4910 y Fn(x)24 b Fj(2)f Fn(H)30 b Fo(:)23 b Fn(x)33 b(=)-51 b Fj(2)23 b Fn(B)1470 4870 y Fm(\021)1466 4935 y(b)1511 4910 y Fo(\(0\))18 b Fj(\002)g Fn(B)1785 4875 y Fm(\021)1821 4850 y Fb(?)1781 4930 y Fm(a)1875 4910 y Fo(\(0\))p Fn(;)37 b Fo(\()p Fn(x)19 b Fo(+)f Fn(\030)t Fo(\))h Fj(\\)2387 4842 y Fd(\000)2425 4910 y Fn(B)2492 4870 y Fm(\021)2488 4935 y(b)2551 4910 y Fj(\002)f Fn(B)2701 4875 y Fm(\021)2737 4850 y Fb(?)2697 4930 y Fm(a)2791 4910 y Fo(\(0\))2897 4842 y Fd(\001)2958 4910 y Fj(6)p Fo(=)k Fj(;)3087 4818 y Fd(\021)617 5077 y Fo(=)46 b(4)p Fn(ab)17 b Fo(+)h Fn(\027)996 5043 y Fm(H)991 5098 y Fw(2)1059 5010 y Fd(\000)1097 5077 y Fn(P)12 b(r)1201 5037 y Fm(\030)1199 5102 y(H)1263 5077 y Fo(\()p Fn(B)1358 5089 y Fm(a)1398 5077 y Fo(\()p Fn(o)p Fo(\)\))1534 5010 y Fd(\001)617 5307 y Fo(=)46 b(4)p Fn(ab)17 b Fo(+)h(4)c(cot)f Fn(\013)1239 5178 y Fm(a)1197 5194 y Fd(Z)1204 5430 y Fw(0)1294 5226 y Fd(p)p 1377 5226 250 4 v 81 x Fn(a)1421 5283 y Fw(2)1477 5307 y Fj(\000)18 b Fn(t)1590 5283 y Fw(2)1641 5307 y Fn(dt)p eop %%Page: 12 12 12 11 bop 84 34 a FB(12)2535 b(Sp)r(o)r(darev:)26 b(Roses)g(of)h(neigh) n(b)r(orho)r(o)r(d)p 84 68 3732 4 v 84 258 a Fo(where)h Fn(\013)g Fo(is)g(the)g(angle)f(b)r(et)n(w)n(een)h(the)g(lines)g Fn(\030)k Fo(and)27 b Fn(\021)s Fo(,)h(and)g Fn(P)12 b(r)2094 218 y Fm(\030)2092 282 y(H)2155 258 y Fo(\()p Fj(\001)p Fo(\))29 b(denotes)e(the)i(pro)5 b(jection)26 b(on)n(to)h Fn(H)35 b Fo(in)28 b(direction)84 357 y Fn(\030)t Fo(.)38 b(Then,)27 b(\(4.5\))h(rewrites)511 556 y Fn(E)19 b Fo(#)674 464 y Fd(n)729 556 y Fn(\020)30 b Fj(2)23 b Fo(\010)933 521 y Fw(3)933 576 y(1)993 556 y Fo(:)g Fn(\020)i Fj(\\)1174 489 y Fd(\000)1212 556 y Fn(B)1279 516 y Fm(\021)1275 581 y(b)1320 556 y Fo(\(0\))18 b Fj(\002)g Fn(B)1594 521 y Fm(\021)1630 496 y Fb(?)1590 576 y Fm(a)1684 556 y Fo(\(0\))1790 489 y Fd(\001)1851 556 y Fj(6)p Fo(=)23 b Fj(;)1981 464 y Fd(o)617 756 y Fo(=)50 b(4)p Fn(ab\025)969 643 y Fd(Z)897 885 y Fm(G)p Fw(\(1)p Fm(;)p Fw(3\))1119 689 y Fd(\002)1153 756 y Fn(\030)1193 722 y Fx(?)1250 756 y Fn(;)14 b(\021)1331 722 y Fx(?)1387 689 y Fd(\003)1435 756 y Fn(\022)r Fo(\()p Fn(d\030)t Fo(\))20 b(+)e Fn(\025\031)s(a)1868 722 y Fw(2)1973 643 y Fd(Z)1901 885 y Fm(G)p Fw(\(1)p Fm(;)p Fw(3\))2123 689 y Fd(\002)2158 756 y Fn(\030)t(;)c(\021)2279 722 y Fx(?)2335 689 y Fd(\003)2383 756 y Fn(\022)r Fo(\()p Fn(d\030)t Fo(\))617 1016 y(=)50 b(4)p Fn(ab)868 948 y Fd(\000)905 1016 y Fn(T)954 1028 y Fw(22)1024 1016 y Fn(\022)1065 981 y Fx(?)1121 948 y Fd(\001\000)1197 1016 y Fn(\021)1241 981 y Fx(?)1298 948 y Fd(\001)1354 1016 y Fo(+)18 b Fn(\031)s(a)1531 981 y Fw(2)1583 1016 y Fo(\()p Fn(T)1664 1028 y Fw(12)1734 1016 y Fn(\022)r Fo(\))1807 948 y Fd(\000)1845 1016 y Fn(\021)1889 981 y Fx(?)1946 948 y Fd(\001)617 1157 y Fo(=)46 b(2)p Fn(bN)873 1169 y Fw(11)942 1157 y Fo(\()p Fn(a;)14 b(\021)s Fo(\))19 b(+)f Fn(\031)s(a)1327 1123 y Fw(2)1378 1157 y Fo(\()p Fn(T)1459 1169 y Fw(12)1529 1157 y Fn(\022)r Fo(\))1617 1090 y Fd(\000)1655 1157 y Fn(\021)1699 1123 y Fx(?)1755 1090 y Fd(\001)1807 1157 y Fn(:)84 1336 y Fo(Since)28 b Fn(\022)i Fo(is)e(a)f(probabilit)n(y)f(measure,)h(the)h(upp)r(er)g(b) r(ound)g(in)g(\(4.4\))f(is)g(ob)n(vious.)p 3755 1336 4 57 v 3759 1283 50 4 v 3759 1336 V 3808 1336 4 57 v 184 1500 a(The)h(follo)n(wing)e(result)i(is)f(an)g(immediate)h (consequence)f(of)g(Theorem)g(4.5.)184 1628 y Fy(Corollary)35 b(4.6)41 b Fk(F)-6 b(or)31 b(e)l(ach)h Fn(a)25 b(>)f Fo(0)p Fk(,)32 b(the)f(estimator)1909 1607 y Fd(b)1887 1628 y Fn(N)1954 1640 y Fw(11)2024 1628 y Fo(\()p Fn(a;)14 b(\021)s Fo(\))31 b Fk(given)h(in)f Fo(\(4.3\))f Fk(is)h(asymptotic)l (al)t(ly)j(unbiase)l(d)d(as)84 1728 y Fn(b)23 b Fj(!)g(1)p Fk(:)533 1881 y Fd(b)511 1902 y Fn(N)578 1914 y Fw(11)648 1902 y Fo(\()p Fn(a;)14 b(\021)s Fo(\))19 b Fj(\000)f Fn(N)1006 1914 y Fw(11)1076 1902 y Fo(\()p Fn(a;)c(\021)s Fo(\))53 b(=)g Fn(O)r Fo(\(1)p Fn(=b)p Fo(\))14 b Fn(:)84 2085 y Fk(Mor)l(e)l(over,)497 2064 y Fd(b)475 2085 y Fn(N)542 2097 y Fw(11)612 2085 y Fo(\()p Fn(a;)g(\021)s Fo(\))p Fn(=a)30 b Fk(is)g(an)g(asymptotic)l(al)t(ly)i(unbiase)l(d)e (estimator)g(for)h Fn(N)2575 2097 y Fw(11)2645 2085 y Fo(\()p Fn(\021)s Fo(\))g Fk(as)f Fn(a)23 b Fj(!)g Fo(0)29 b Fk(or)h Fn(b)23 b Fj(!)g(1)p Fk(,)30 b(wher)l(e)511 2325 y Fn(E)622 2248 y Fd(b)601 2269 y Fn(N)668 2281 y Fw(11)738 2269 y Fo(\()p Fn(a;)14 b(\021)s Fo(\))p 601 2306 327 4 v 742 2382 a Fn(a)956 2325 y Fj(\000)k Fn(N)1106 2337 y Fw(11)1176 2325 y Fo(\()p Fn(\021)s Fo(\))53 b(=)g Fn(O)r Fo(\()p Fn(a=b)p Fo(\))14 b Fn(:)184 2527 y Fy(Remark)33 b(4.7)41 b Fo(It)30 b(is)g(clear)e(ho)n(w)h(the)i (estimator)d(\(4.3\))i(can)f(b)r(e)h(generalized)f(in)h(order)e(to)i (estimate)f(the)h Fn(a)p Fo({rose)84 2626 y(of)d(neigh)n(b)r(orho)r(o)r (d)e Fn(N)765 2638 y Fm(k)q(r)838 2626 y Fo(\()p Fn(a;)14 b(\021)s Fo(\))27 b(for)e(the)i(pro)r(cess)e(\010)1669 2596 y Fm(d)1669 2650 y(k)1736 2626 y Fo(of)h Fn(k)s Fo({dimensional)f(\015ats)h(in)h(the)f(higher)g(dimensional)f(space)h Fl(R)3771 2596 y Fm(d)84 2726 y Fo(with)j Fn(d)23 b(>)f Fo(3,)28 b Fn(k)21 b Fo(+)d Fn(r)26 b(<)d(d)p Fo(:)533 2985 y Fd(b)511 3006 y Fn(N)578 3018 y Fm(k)q(r)651 3006 y Fo(\()p Fn(a;)14 b(\021)s Fo(\))51 b(=)1017 2920 y(#)1100 2827 y Fd(n)1155 2920 y Fn(\030)27 b Fj(2)d Fo(\010)1357 2889 y Fm(d)1357 2943 y(k)1420 2920 y Fo(:)g Fn(\030)e Fj(\\)1599 2852 y Fd(\000)1637 2920 y Fn(B)1704 2880 y Fm(\021)1700 2945 y(b)1745 2920 y Fo(\(0\))c Fj(\002)g Fn(B)2019 2889 y Fm(\021)2055 2864 y Fb(?)2015 2940 y Fm(a)2109 2920 y Fo(\(0\))2215 2852 y Fd(\001)2276 2920 y Fj(6)p Fo(=)23 b Fj(;)2406 2827 y Fd(o)p 1017 2987 1444 4 v 1660 3063 a Fn(\024)1708 3075 y Fm(r)1745 3063 y Fn(b)1781 3039 y Fm(r)2484 3006 y Fn(:)84 3215 y Fo(Ho)n(w)n(ev)n (er,)31 b(in)g(this)h(section,)f(w)n(e)g(considered)f(just)i(the)g (three{dimensional)e(case)g(in)i(order)d(to)i(a)n(v)n(oid)f (unnecessary)84 3315 y(tec)n(hnical)e(di\016culties)g(in)g(the)g(pro)r (of)f(of)g(Theorem)g(4.5.)84 3529 y Fy(4.3)95 b(Estimating)29 b(the)j(directional)f(distribution)g(densit)m(y)84 3664 y Fo(Iden)n(tifying)21 b(an)n(y)f(line)i(through)e(the)h(origin)f(with) h(the)g(pair)f(of)h(its)g(unit)h(direction)e(v)n(ectors,)h(eac)n(h)f (function)h(\(measure\))84 3764 y(on)29 b Fn(G)p Fo(\(1)p Fn(;)14 b Fo(3\))29 b(can)g(b)r(e)g(though)n(t)g(of)g(as)g(an)g(ev)n (en)f(function)i(\(measure\))e(on)h(the)h(sphere)e Fy(S)2859 3734 y Fw(2)2896 3764 y Fo(.)42 b(Supp)r(ose)29 b(the)g(directional)84 3863 y(distribution)38 b Fn(\022)i Fo(has)c(a)h(densit)n(y)g Fn( )k Fo(with)c(resp)r(ect)h(to)f(the)g(uniform)h(directional)e (distribution)h Fn(\015)42 b Fo(on)37 b Fy(S)3476 3833 y Fw(2)3514 3863 y Fo(,)j Fn(\015)5 b Fo(\()p Fj(\001)p Fo(\))39 b(=)84 3963 y Fn(!)136 3975 y Fw(3)173 3963 y Fo(\()p Fj(\001)p Fo(\))p Fn(=!)354 3975 y Fw(3)391 3896 y Fd(\000)430 3963 y Fy(S)483 3933 y Fw(2)520 3896 y Fd(\001)558 3963 y Fo(.)76 b(Supp)r(ose)40 b Fn( )k Fo(to)c(b)r(e)h(t)n(wice)g(con)n(tin)n(uously)e(di\013eren)n(tiable)h (on)h Fy(S)2749 3933 y Fw(2)2786 3963 y Fo(:)63 b Fn( )47 b Fj(2)e Fn(C)3138 3933 y Fw(2)3132 3984 y Fm(e)3176 3896 y Fd(\000)3214 3963 y Fy(S)3267 3933 y Fw(2)3304 3896 y Fd(\001)3383 3963 y Fo(\(subindex)c Fn(e)84 4068 y Fo(means)29 b(\\ev)n(en"\).)42 b(By)29 b(the)h(example)f(considered)f (in)i(Section)f(4.1,)g(the)h(rose)e(of)h(neigh)n(b)r(orho)r(o)r(d)g Fn(N)3250 4080 y Fw(11)3320 4068 y Fo(\()p Fn(u)p Fo(\),)h Fn(u)25 b Fj(2)i Fy(S)3693 4038 y Fw(2)3760 4068 y Fo(is)84 4168 y(prop)r(ortional)e(to)h(the)h(rose)e(of)h(in)n(tersections)1543 4100 y Fd(\000)1581 4168 y Fn(T)1630 4180 y Fw(22)1700 4168 y Fn( )1757 4138 y Fx(?)1813 4100 y Fd(\001\000)1889 4168 y Fn(u)1937 4138 y Fx(?)1993 4100 y Fd(\001)2031 4168 y Fo(.)36 b(By)26 b(Prop)r(osition)f(8.1)g(of)h([23],)g(the)h (densit)n(y)f Fn( )j Fo(can)d(b)r(e)84 4276 y(restored)31 b(from)g Fn(T)657 4288 y Fw(22)727 4276 y Fn( )784 4245 y Fx(?)871 4276 y Fo(using)h(its)f(expansion)g(in)h(spherical)e (harmonics)h(\(cf.)h([14)o(]\).)49 b(Com)n(bining)31 b(b)r(oth)h(assertions)84 4375 y(with)d(that)e(of)h(Corollary)d(4.3,)i (one)g(can)g(write)511 4664 y Fn( )s Fo(\()p Fn(u)p Fo(\))51 b(=)f Fn( )903 4629 y Fx(?)959 4596 y Fd(\000)997 4664 y Fn(u)1045 4629 y Fx(?)1101 4596 y Fd(\001)1190 4664 y Fo(=)1332 4560 y Fx(1)1305 4585 y Fd(X)1305 4763 y Fm(k)q Fw(=0)1439 4560 y(4)p Fm(k)q Fw(+1)1456 4585 y Fd(X)1459 4761 y Fm(j)s Fw(=1)1627 4473 y Fd(R)1609 4623 y Fa(S)1651 4606 y Fc(2)1705 4540 y Fn(N)1772 4552 y Fw(11)1842 4540 y Fo(\()p Fn(a;)14 b(v)s Fo(\))p Fn(S)2081 4552 y Fw(2)p Fm(k)q(;j)2206 4540 y Fo(\()p Fn(v)s Fo(\))g Fn(!)2379 4552 y Fw(3)2417 4540 y Fo(\()p Fn(dv)s Fo(\))p 1617 4644 951 4 v 2032 4720 a Fn(ac)2112 4732 y Fm(k)2592 4664 y Fn(S)2643 4676 y Fw(2)p Fm(k)q(;j)2767 4664 y Fo(\()p Fn(u)p Fo(\))g Fn(;)97 b(u)22 b Fj(2)i Fy(S)3215 4629 y Fw(2)3645 4664 y Fo(\(4.8\))84 4921 y(where)511 5121 y Fn(c)547 5133 y Fm(k)638 5121 y Fo(=)51 b Fj(\000)p Fn(\031)892 5065 y Fo(\000)14 b(\()p Fn(k)22 b Fo(+)c(1)p Fn(=)p Fo(2\))12 b(\000)i(\()p Fn(k)22 b Fj(\000)c Fo(1)p Fn(=)p Fo(2\))p 892 5102 819 4 v 1122 5178 a(\()p Fn(k)j Fo(+)d(1\)!)c Fn(k)s Fo(!)1734 5121 y Fn(;)97 b(k)26 b Fj(2)e Fl(N)84 5339 y Fo(and)30 b Fn(S)299 5351 y Fm(nj)406 5339 y Fo(is)g(a)g(spherical)f(harmonic)g(of)h(order)f Fn(n)p Fo(.)44 b(Note)31 b(that)f(in)h(accordance)d(with)i(Corollary)e (4.3,)i(b)r(oth)h(sides)f(of)84 5438 y(equation)d(\(4.8\))h(do)f(not)h (dep)r(end)g(on)f Fn(a)p Fo(.)37 b(F)-7 b(or)27 b(instance,)h(w)n(e)f (can)g(set)h Fn(a)22 b Fo(=)h(1.)p eop %%Page: 13 13 13 12 bop 84 30 a FB(Math.)35 b(Nac)n(hr.)26 b FA(251)g FB(\(2003\))2800 b(13)p 84 68 3732 4 v 184 258 a Fo(T)-7 b(o)27 b(estimate)h Fn( )s Fo(,)g(tak)n(e)f(a)g(\014nite)h(sum)g(in)g (\(4.8\))f(and)g(substitute)i(the)f(v)-5 b(alue)27 b Fn(N)2653 270 y Fw(11)2723 258 y Fo(\()p Fn(a;)14 b(v)s Fo(\))28 b(b)n(y)3076 237 y Fd(b)3055 258 y Fn(N)3122 270 y Fw(11)3192 258 y Fo(\()p Fn(a;)14 b(v)s Fo(\):)526 553 y Fd(b)511 574 y Fn( )565 586 y Fm(N)s(;b)673 574 y Fo(\()p Fn(u)p Fo(\))51 b(=)982 471 y Fm(N)952 496 y Fd(X)951 674 y Fm(k)q Fw(=0)1086 471 y(4)p Fm(k)q Fw(+1)1103 496 y Fd(X)1105 672 y Fm(j)s Fw(=1)1287 384 y Fd(R)1269 534 y Fa(S)1311 517 y Fc(2)1387 430 y Fd(b)1365 451 y Fn(N)1432 463 y Fw(11)1503 451 y Fo(\()p Fn(a;)14 b(v)s Fo(\))p Fn(S)1742 463 y Fw(2)p Fm(k)q(;j)1866 451 y Fo(\()p Fn(v)s Fo(\))g Fn(!)2039 463 y Fw(3)2077 451 y Fo(\()p Fn(dv)s Fo(\))p 1277 555 951 4 v 1692 631 a Fn(ac)1772 643 y Fm(k)2252 574 y Fn(S)2303 586 y Fw(2)p Fm(k)q(;j)2427 574 y Fo(\()p Fn(u)p Fo(\))g Fn(;)97 b(u)23 b Fj(2)g Fy(S)2875 540 y Fw(2)2912 574 y Fn(:)710 b Fo(\(4.9\))84 865 y(T)-7 b(o)21 b(ev)-5 b(aluate)20 b(the)i(in)n(tegral)960 798 y Fd(R)944 947 y Fa(S)986 931 y Fc(2)1060 844 y Fd(b)1038 865 y Fn(N)1105 877 y Fw(11)1175 865 y Fo(\()p Fn(a;)14 b(v)s Fo(\))p Fn(S)1414 877 y Fw(2)p Fm(k)q(;j)1539 865 y Fo(\()p Fn(v)s Fo(\))g Fn(!)1712 877 y Fw(3)1750 865 y Fo(\()p Fn(dv)s Fo(\))22 b(in)f(\(4.9\))g(n)n(umerically)-7 b(,)21 b(v)-5 b(alues)21 b(of)g(the)g(estimator)3512 844 y Fd(b)3490 865 y Fn(N)3557 877 y Fw(11)3627 865 y Fo(\()p Fn(a;)14 b(v)s Fo(\))84 1021 y(in)30 b(a)g(\014nite)g(n)n(um) n(b)r(er)f(of)h(directions)f Fn(v)k Fo(are)c(required.)42 b(These)29 b(directions)g(can)h(b)r(e)g(c)n(hosen)f(at)g(random,)g(in)h (order)f(to)84 1121 y(use)f(Mon)n(te{Carlo)d(metho)r(ds)j(of)g(n)n (umerical)e(in)n(tegration.)184 1246 y Fy(Theorem)35 b(4.8)41 b Fk(L)l(et)33 b(the)g(density)h Fn( )i Fk(of)e(the)g(dir)l(e) l(ctional)g(distribution)g(of)g(the)g(stationary)g(line)f(pr)l(o)l(c)l (ess)h Fo(\010)3673 1216 y Fw(3)3673 1266 y(1)3743 1246 y Fk(b)l(e)84 1357 y(twic)l(e)39 b(c)l(ontinuously)e(di\013er)l (entiable)j(on)e Fy(S)1474 1327 y Fw(2)1511 1357 y Fk(.)64 b(Then)38 b(the)h(estimator)2364 1335 y Fd(b)2349 1357 y Fn( )2403 1369 y Fm(N)s(;b)2511 1357 y Fo(\()p Fn(u)p Fo(\))f Fk(given)h(in)f Fo(\(4.9\))g Fk(is)g(asymptotic)l(al)t(ly)84 1456 y(unbiase)l(d)31 b(as)f Fn(b)23 b Fj(!)g(1)p Fk(,)30 b Fn(N)i Fj(!)23 b(1)29 b Fo(\()p Fk(or)i Fn(a)23 b Fj(!)g Fo(0)p Fk(,)30 b Fn(N)h Fj(!)24 b(1)p Fo(\))p Fk(:)38 b(for)31 b(al)t(l)g Fn(u)22 b Fj(2)i Fy(S)2395 1426 y Fw(2)549 1651 y Fo(lim)511 1704 y Fm(N)6 b Fx(!1)739 1651 y Fo(lim)716 1705 y Fm(b)p Fx(!1)891 1651 y Fn(E)986 1629 y Fd(b)971 1651 y Fn( )1025 1663 y Fm(N)s(;b)1134 1651 y Fo(\()p Fn(u)p Fo(\))52 b(=)91 b(lim)1416 1704 y Fm(N)6 b Fx(!1)1631 1651 y Fo(lim)1621 1702 y Fm(a)p Fx(!)p Fw(0)1770 1651 y Fn(E)1865 1629 y Fd(b)1850 1651 y Fn( )1904 1663 y Fm(N)s(;b)2012 1651 y Fo(\()p Fn(u)p Fo(\))53 b(=)f Fn( )s Fo(\()p Fn(u)p Fo(\))14 b Fn(:)184 1887 y Fo(P)f(r)h(o)f(o)h(f.)41 b(By)28 b(F)-7 b(ubini's)28 b(Theorem,)f(the)h(mean)f(v)-5 b(alue)28 b(of)2009 1865 y Fd(b)1994 1887 y Fn( )2048 1899 y Fm(N)s(;b)2157 1887 y Fo(\()p Fn(u)p Fo(\))f(is)h(giv)n(en)f(b)n(y)511 2206 y Fn(E)605 2184 y Fd(b)591 2206 y Fn( )645 2218 y Fm(N)s(;b)753 2206 y Fo(\()p Fn(u)p Fo(\))51 b(=)1062 2103 y Fm(N)1032 2127 y Fd(X)1031 2306 y Fm(k)q Fw(=0)1166 2103 y(4)p Fm(k)q Fw(+1)1183 2127 y Fd(X)1185 2304 y Fm(j)s Fw(=1)1353 2016 y Fd(R)1335 2166 y Fa(S)1377 2149 y Fc(2)1432 2083 y Fn(E)1533 2062 y Fd(b)1511 2083 y Fn(N)1578 2095 y Fw(11)1649 2083 y Fo(\()p Fn(a;)14 b(v)s Fo(\))p Fn(S)1888 2095 y Fw(2)p Fm(k)q(;j)2012 2083 y Fo(\()p Fn(v)s Fo(\))g Fn(!)2185 2095 y Fw(3)2223 2083 y Fo(\()p Fn(dv)s Fo(\))p 1344 2187 1031 4 v 1798 2263 a Fn(ac)1878 2275 y Fm(k)2398 2206 y Fn(S)2449 2218 y Fw(2)p Fm(k)q(;j)2573 2206 y Fo(\()p Fn(u)p Fo(\))g Fn(:)84 2473 y Fo(By)28 b(Theorem)f(4.5,)f(one)i (can)f(write)513 2739 y Fn(E)608 2717 y Fd(b)593 2739 y Fn( )647 2751 y Fm(N)s(;b)755 2739 y Fo(\()p Fn(u)p Fo(\))51 b(=)1083 2635 y Fm(N)1052 2660 y Fd(X)1052 2839 y Fm(k)q Fw(=0)1186 2635 y(4)p Fm(k)q Fw(+1)1203 2660 y Fd(X)1206 2837 y Fm(j)s Fw(=1)1373 2549 y Fd(R)1356 2698 y Fa(S)1398 2681 y Fc(2)1452 2615 y Fn(N)1519 2627 y Fw(11)1589 2615 y Fo(\()p Fn(a;)14 b(v)s Fo(\))p Fn(S)1828 2627 y Fw(2)p Fm(k)q(;j)1953 2615 y Fo(\()p Fn(v)s Fo(\))g Fn(!)2126 2627 y Fw(3)2164 2615 y Fo(\()p Fn(dv)s Fo(\))p 1364 2720 951 4 v 1779 2796 a Fn(ac)1859 2808 y Fm(k)2338 2739 y Fn(S)2389 2751 y Fw(2)p Fm(k)q(;j)2514 2739 y Fo(\()p Fn(u)p Fo(\))1056 3083 y(+)1172 3027 y Fn(\031)s(a)p 1172 3064 95 4 v 1181 3140 a Fo(2)p Fn(b)1321 2979 y Fm(N)1291 3004 y Fd(X)1290 3183 y Fm(k)q Fw(=0)1425 2979 y(4)p Fm(k)q Fw(+1)1442 3004 y Fd(X)1445 3181 y Fm(j)s Fw(=1)1612 2893 y Fd(R)1595 3043 y Fa(S)1637 3026 y Fc(2)1691 2960 y Fo(\()p Fn(T)1772 2972 y Fw(12)1842 2960 y Fn( )s Fo(\))1931 2893 y Fd(\000)1969 2960 y Fn(v)2012 2930 y Fx(?)2068 2893 y Fd(\001)2107 2960 y Fn(S)2158 2972 y Fw(2)p Fm(k)q(;j)2282 2960 y Fo(\()p Fn(v)s Fo(\))g Fn(!)2455 2972 y Fw(3)2492 2960 y Fo(\()p 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Fn(u)p Fo(\))g Fn(:)84 4006 y Fo(T)-7 b(aking)27 b(the)h(limit)h(as)d Fn(N)32 b Fj(!)23 b(1)28 b Fo(completes)f(the)h (pro)r(of.)p 3755 4006 4 57 v 3759 3954 50 4 v 3759 4006 V 3808 4006 4 57 v 84 4211 a FA(Ac)n(kno)n(wledgemen)n(ts)75 b FB(The)37 b(author)f(w)n(ould)h(lik)n(e)f(to)h(thank)e(Prof.)j(J.)f (Mec)n(k)n(e)g(for)g(dra)n(wing)g(his)f(atten)n(tion)h(to)f(pap)r(ers) 84 4311 y([17)r(])f(and)f([21)q(],)k(for)e(suggesting)g(the)e(problem)g (and)h(for)g(helpful)g(discussions.)63 b(He)35 b(is)g(also)h(indebted)e (to)h(Prof.)h(G.)g(Last,)84 4410 y(Dr.)26 b(W.)g(Nagel,)h(Prof.)g(V.)e (Sc)n(hmidt)f(and)h(the)h(referees)h(for)f(their)g(commen)n(ts)e(on)i (preliminary)f(v)n(ersions)h(of)g(this)g(pap)r(er.)84 4653 y Fp(References)123 4809 y FB([1])42 b(R.)22 b(Da)n(vidson,)g (Line{pro)r(cesses,)i(roads,)f(and)e(\014bres,)i(in:)32 b(Sto)r(c)n(hastic)22 b(Geometry,)g(edited)f(b)n(y)f(E.)i(F.)f(Harding) h(and)f(D.)g(G.)245 4900 y(Kendall)26 b(\(Wiley)-6 b(,)26 b(London,)g(1974\),)h(pp.)e(248)14 b({)g(251.)123 4989 y([2])42 b(W.)27 b(F)-6 b(a)n(vis)27 b(and)f(V.)g(W)-6 b(ei\031,)28 b(Mean)f(v)l(alues)f(of)h(w)n(eigh)n(ted)g(cells)h(of)f (stationary)g(P)n(oisson)h(h)n(yp)r(erplane)e(tessellations)j(of)e Ft(R)3755 4958 y Fs(d)3794 4989 y FB(,)245 5081 y(Math.)g(Nac)n(hr.)e FA(193)p FB(,)i(37)13 b({)h(48)26 b(\(1998\).)123 5168 y([3])42 b(P)-6 b(.)33 b(Go)r(o)r(dey)g(and)f(R.)g(Ho)n(w)n(ard,)j(Pro) r(cesses)f(of)f(\015ats)g(induced)e(b)n(y)g(higher{dimensional)j(pro)r (cesses,)h(Adv.)d(Math.)h FA(80)p FB(,)245 5260 y(92)14 b({)f(109)27 b(\(1990\).)123 5347 y([4])42 b(P)-6 b(.)22 b(Go)r(o)r(dey)f(and)g(W.)g(W)-6 b(eil,)23 b(In)n(tegral)e(geometric)h (form)n(ulae)f(for)h(pro)t(jection)g(functions,)h(Geom.)e(Dedicata)g FA(41)p FB(,)i(117)14 b({)f(126)245 5438 y(\(1992\).)p eop %%Page: 14 14 14 13 bop 84 34 a FB(14)2535 b(Sp)r(o)r(darev:)26 b(Roses)g(of)h(neigh) n(b)r(orho)r(o)r(d)p 84 68 3732 4 v 123 258 a([5])42 b(P)-6 b(.)26 b(Go)r(o)r(dey)f(and)g(W.)h(W)-6 b(eil,)25 b(Zonoids)h(and)f(generalizations,)j(in:)34 b(Handb)r(o)r(ok)25 b(of)h(Con)n(v)n(ex)f(Geometry,)f(edited)h(b)n(y)g(P)-6 b(.)25 b(M.)245 349 y(Grub)r(er)h(and)f(J.)i(M.)f(Wills)h (\(North{Holland,)f(Amsterdam,)e(1993\),)j(pp.)f(1297)14 b({)f(1326.)123 436 y([6])42 b(S.)31 b(Janson)h(and)e(O.)h(Kallen)n(b)r (erg,)i(Maximizing)e(the)g(in)n(tersection)g(densit)n(y)f(of)i(\014bre) e(pro)r(cess,)k(J.)d(Appl.)f(Probab.)i FA(18)p FB(,)245 528 y(820)14 b({)g(828)26 b(\(1981\).)123 615 y([7])42 b(J.)h(Keutel,)k(Ein)c(Extremalproblem)e(f)r(\177)-41 b(ur)43 b(zuf\177)-38 b(allige)45 b(Eb)r(enen)d(und)f(f)r(\177)-41 b(ur)43 b(Eb)r(enenprozesse)g(in)g(h\177)-38 b(oherdimensionalen)245 706 y(R\177)g(aumen,)25 b(PhD)g(thesis,)h(F)-6 b(riedric)n(h{Sc)n (hiller)27 b(Univ)n(ersit)n(y)e(\(Jena,)h(1992\).)123 793 y([8])42 b(A.)19 b(N.)g(Kolmogoro)n(v)h(and)e(S.)h(V.)g(F)-6 b(omin,)20 b(Reelle)f(F)-6 b(unktionen)18 b(und)g(F)-6 b(unktionalanalysis,)21 b(Dt.)e(V)-6 b(erlag)20 b(der)e(Wiss.)i (\(Berlin,)245 885 y(1975\).)123 972 y([9])42 b(G.)27 b(Matheron,)f(Random)e(Sets)i(and)f(In)n(tegral)h(Geometry)f(\(Wiley)-6 b(,)26 b(New)g(Y)-6 b(ork,)25 b(1975\).)84 1059 y([10])43 b(J.)27 b(Mec)n(k)n(e,)34 b(On)25 b(some)h(inequalities)g(for)h (Poisson)g(net)n(w)n(orks,)35 b(Math.)26 b(Nac)n(hr.)g FA(128)p FB(,)g(81)14 b({)f(86)26 b(\(1986\).)84 1146 y([11])43 b(J.)27 b(Mec)n(k)n(e,)34 b(An)25 b(extremal)g(prop)r(ert)n (y)g(of)i(random)d(\015ats,)35 b(J.)26 b(Microscop)n(y)h FA(151)p FB(,)g(205)14 b({)f(209)27 b(\(1988\).)84 1233 y([12])43 b(J.)27 b(Mec)n(k)n(e,)34 b(Random)25 b Fu(r)r FB({\015ats)h(meeting)f(a)h(ball,)35 b(Arc)n(h.)25 b(Math.)h FA(51)p FB(,)h(378)14 b({)f(384)27 b(\(1988\).)84 1321 y([13])43 b(J.)27 b(Mec)n(k)n(e,)34 b(On)25 b(the)h(in)n(tersection)g (densit)n(y)f(of)h(\015at)g(pro)r(cesses,)36 b(Math.)26 b(Nac)n(hr.)g FA(151)p FB(,)g(69)14 b({)f(74)26 b(\(1991\).)84 1408 y([14])43 b(C.)27 b(M)r(\177)-41 b(uller,)36 b(Spherical)26 b(Harmonics)g(\(Springer{V)-6 b(erlag,)26 b(Berlin,)h(1966\).)84 1495 y([15])43 b(J.)27 b(Mec)n(k)n(e,)f(R.)f(Sc)n(hneider,)h(D.)f(Sto)n (y)n(an,)g(and)h(W.)g(W)-6 b(eil,)34 b(Sto)r(c)n(hastisc)n(he)27 b(Geometrie)f(\(Birkh\177)-38 b(auser,)26 b(Basel,)h(1990\).)84 1582 y([16])43 b(J.)36 b(Mec)n(k)n(e)f(and)f(C.)i(Thomas,)i(On)c(an)h (extreme)f(v)l(alue)g(problem)g(for)i(\015at)f(pro)r(cesses,)65 b(Comm.)34 b(Statist.)i(Sto)r(c)n(hastic)245 1673 y(Mo)r(dels)27 b FA(2)p FB(\(2\),)g(273)14 b({)f(280)27 b(\(1986\).)84 1761 y([17])43 b(I.)24 b(S.)g(Molc)n(hano)n(v)g(and)f(S.)h(A.)f(Zuy)n (ev,)g(V)-6 b(ariational)25 b(analysis)g(of)f(functionals)h(of)f(a)g(P) n(oisson)h(pro)r(cess,)g(T)-6 b(ec)n(hnical)25 b(Rep)r(ort)245 1852 y(RR{3302,)i(INRIA,)d(Sophia{An)n(tip)r(olis)j(\(No)n(v)n(em)n(b)r (er)d(1997\).)84 1939 y([18])43 b(I.)31 b(S.)g(Molc)n(hano)n(v)h(and)e (S.)h(A.)g(Zuy)n(ev,)50 b(V)-6 b(ariational)32 b(analysis)g(of)g (functionals)g(of)g(a)f(P)n(oisson)h(pro)r(cess,)53 b(Math.)31 b(Op)r(er.)245 2030 y(Researc)n(h.)c FA(25)p FB(,)f(458)14 b({)f(508)27 b(\(2000\).)84 2118 y([19])43 b(I.)29 b(S.)f(Molc)n(hano)n (v)h(and)f(S.)g(A.)h(Zuy)n(ev,)f(T)-6 b(angen)n(t)28 b(sets)h(in)f(the)g(space)h(of)g(measures:)40 b(with)29 b(applications)h(to)e(v)l(ariational)245 2209 y(analysis,)36 b(J.)26 b(Math.)h(Anal.)f(Appl.)f FA(249)p FB(,)h(539)14 b({)g(552)26 b(\(2000\).)84 2296 y([20])43 b(R.)25 b(Sc)n(hneider,)f (Isop)r(erimetric)g(inequalities)i(for)f(in\014nite)f(h)n(yp)r(erplane) f(systems,)i(Discrete)g(Comput.)e(Geom.)i FA(13)p FB(,)g(609)14 b({)245 2387 y(627)27 b(\(1995\).)84 2475 y([21])43 b(R.)26 b(Sc)n(hneider,)f(A)g(dualit)n(y)h(for)g(Poisson)h(\015ats,)f(Adv.)f (Appl.)g(Probab.)i(\(SGSA\))d FA(31)p FB(,)j(63)13 b({)g(69)27 b(\(1999\).)84 2562 y([22])43 b(R.)36 b(Sc)n(hneider)f(and)g(W.)h(W)-6 b(eil,)38 b(Sto)r(c)n(hastisc)n(he)f(Geometrie,)h(T)-6 b(eubner)35 b(Skripten)g(zur)g(Mathematisc)n(hen)g(Sto)r(c)n(hastik)245 2653 y(\(T)-6 b(eubner,)26 b(Stuttgart,)f(2000\).)84 2740 y([23])43 b(E.)33 b(Sp)r(o)r(darev,)h(On)d(the)h(rose)h(of)g(in)n (tersections)g(for)g(stationary)g(\015at)f(pro)r(cesses,)j(Adv.)c (Appl.)h(Probab.)h(\(SGSA\))e FA(33)p FB(,)245 2832 y(584)14 b({)g(599)26 b(\(2001\).)84 2919 y([24])43 b(E.)22 b(Sp)r(o)r(darev,)g (Selected)f(topics)h(in)f(the)g(theory)f(of)i(spatial)h(stationary)e (\015at)g(pro)r(cesses,)j(PhD)d(thesis,)i(F)-6 b(riedric)n(h{Sc)n (hiller)245 3010 y(Univ)n(ersit)n(y)25 b(\(Jena,)i(2001\).)84 3097 y([25])43 b(D.)25 b(Sto)n(y)n(an,)f(W.)g(S.)h(Kendall,)g(and)f(J.) h(Mec)n(k)n(e,)g(Sto)r(c)n(hastic)g(Geometry)f(and)g(its)h (Applications,)g(second)g(edition)f(\(Wiley)-6 b(,)245 3189 y(Chic)n(hester,)27 b(1995\).)84 3276 y([26])43 b(C.)28 b(Thomas,)f(Extrem)n(um)d(prop)r(erties)j(of)g(the)f(in)n (tersection)h(densities)g(of)h(stationary)f(Poisson)h(h)n(yp)r(erplane) e(pro)r(cesses,)245 3367 y(Math.)h(Op)r(erationsforsc)n(h.)g(Statist.)f (Ser.)g(Statistik)g FA(15)p FB(\(3\),)g(443)14 b({)g(449)26 b(\(1984\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF