Content-Type: multipart/mixed; boundary="-------------0009140949464" This is a multi-part message in MIME format. ---------------0009140949464 Content-Type: text/plain; name="00-360.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-360.comments" number of pages : 12 ---------------0009140949464 Content-Type: text/plain; name="00-360.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-360.keywords" Stationary, nonstationary processes, Spectral representation, Cumulants, Consistency, Spectral analysis. ---------------0009140949464 Content-Type: application/postscript; name="mstat.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mstat.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: mstat.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips mstat -o %DVIPSParameters: dpi=600, compressed, comments 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1917 y Fm(A.M.S.)k(classi\014cation:)36 b Fr(Primary)26 b(62M15;)g(secondary)g(62G20.)166 2116 y Fm(Key-w)m(ords:)42 b Fr(Stationary)-7 b(,)30 b(nonstationary)f(pro)r(cesses,)h(Sp)r (ectral)g(represen)n(tation,)g(Cum)n(ulan)n(ts,)h(Consistency)-7 b(,)166 2216 y(Sp)r(ectral)27 b(analysis.)193 2552 y Fm(Estimation)i(sp)s(ectrale)i(p)s(our)h(les)f(pro)s(cessus)g (M-stationnaires)f(par)h(\023)-46 b(ec)m(han)m(tillonnage)31 b(al)n(\023)-46 b(eatoire.)291 2651 y(R)n(\023)g(esum)n(\023)g(e)38 b(:)114 b Fr(Dans)36 b(ce)g(tra)n(v)-5 b(ail,)36 b(l'estimation)g(de)g (la)f(densit)n(\023)-39 b(e)35 b(sp)r(ectrale)g(p)r(our)h(une)g(classe) f(de)g(pro)r(cessus)166 2751 y(non)h(stationnaires)f(est)f(\023)-39 b(etudi)n(\023)g(ee.)61 b(Bien)37 b(que)f(ces)g(pro)r(cessus)f(ne)h (soien)n(t)g(stationnaires)e(relativ)n(emen)n(t)i(\022)-42 b(a)35 b(la)h(loi)166 2851 y(additiv)n(e)29 b(c'est)h(\022)-42 b(a)29 b(dire)g(faiblemen)n(t)h(stationnaire)e(au)h(sens)g(classique,)g (ils)g(son)n(t)g(stationnaires)f(par)h(rapp)r(ort)f(\022)-42 b(a)29 b(la)166 2950 y(loi)e(m)n(ultiplicativ)n(e.)36 b(Ces)27 b(pro)r(cessus)f(existen)n(t)h(naturellemen)n(t)g(comme)g(des) g(enregistremen)n(ts)e(\022)-42 b(a)27 b(temps)g(con)n(tin)n(u.)166 3050 y(P)n(our)e(ten)n(ter)i(de)f(r)n(\023)-39 b(ep)r(ondre)25 b(\022)-42 b(a)26 b(un)h(certain)f(nom)n(bre)g(de)h(questions)f(p)r(os) n(\023)-39 b(ees)25 b(dans)h(des)g(situations)g(pratiques,)h(nous)166 3149 y(adoptons)36 b(l')n(\023)-39 b(ec)n(han)n(tillonnage)35 b(al)n(\023)-39 b(eatoire)35 b(a\014n)i(d'estimer)g(la)g(densit)n(\023) -39 b(e)37 b(sp)r(ectrale)g(en)g(utilisan)n(t)h(un)f(pro)r(cessus)g (\022)-42 b(a)166 3249 y(temps)28 b(discret.)36 b(Des)28 b(sim)n(ulations)f(son)n(t)g(donn)n(\023)-39 b(ees.)166 3448 y Fm(Mots-clefs)44 b(:)60 b Fr(Pro)r(cessus)37 b(non)i (stationnaires,)i(Represen)n(tation)d(sp)r(ectrale,)k(Cum)n(ulan)n(ts,) g(Con)n(v)n(ergence)37 b(en)166 3548 y(mo)n(y)n(enne)27 b(quadratique,)f(Analyse)h(sp)r(ectrale.)166 3823 y Fl(1)135 b(In)l(tro)t(duction)166 4004 y Fr(In)22 b(this)h(pap)r(er,)g(the)g(sp) r(ectral)e(densit)n(y)h(estimation)g(of)h(a)e(class)h(of)g (nonstationary)e(pro)r(cesses)h(is)h(in)n(v)n(estigated.)34 b(The)166 4104 y(considered)g(nonstationary)g(pro)r(cesses)g(are)g (refered)h(to)g(as)f(m)n(ultiplicativ)n(e)i(stationary)d(pro)r(cesses.) 59 b(Although)166 4204 y(these)32 b(pro)r(cesses)e(are)h(not)h (stationary)e(with)i(resp)r(ect)g(to)f(the)h(usual)g(additiv)n(e)f(op)r (eration,)h(they)g(are)f(stationary)166 4303 y(with)e(resp)r(ect)f(to)g (the)g(m)n(ultiplicativ)n(e)g(op)r(eration.)38 b(This)28 b(prop)r(ert)n(y)f(ensures)g(essen)n(tially)g(the)i(same)e(structure)h (as)166 4403 y(is)g(a)n(v)-5 b(ailable)26 b(for)h(stationary)f(pro)r (cesses,)g(esp)r(ecially)h(the)h(sp)r(ectral)f(represen)n(tation.)291 4503 y(The)38 b(concept)g(of)g(stationarit)n(y)e(under)i(a)g(general)f (group)g(comp)r(osition)g(op)r(eration)g(has)h(b)r(een)g(considered)166 4602 y(extensiv)n(ely)31 b(b)n(y)h(Hannan)g(\(1965\).)50 b(It)32 b(ma)n(y)g(ev)n(en)g(b)r(e)g(generalized)f(to)h(a)g(semi-group) f(op)r(eration,)h(see)g(Girardin)166 4702 y(&)i(Senoussi)f(\(2000\).)55 b(The)34 b(M-stationary)e(pro)r(cesses)h(are)g(de\014ned)h(and)g(exist) g(naturally)f(as)g(con)n(tin)n(uous-time)166 4801 y(pro)r(cesses.)h (They)25 b(are)e(studied)i(thoroughly)e(in)i(Gra)n(y)e(and)h(Zhang)g (\(1988\).)34 b(Ho)n(w)n(ev)n(er,)23 b(no)i(particular)e(adv)-5 b(an)n(tage)166 4901 y(seems)33 b(to)f(ha)n(v)n(e)g(b)r(een)i(tak)n(en) e(y)n(et)h(of)f(the)i(generalit)n(y)d(a\013orded)h(b)n(y)h(this)g (generalized)f(concept)h(of)f(stationarit)n(y)-7 b(.)166 5001 y(A)35 b(particular)e(asp)r(ect)i(of)g(their)f(study)h(is)g (considered)f(here:)50 b(for)34 b(practical)g(applications,)i(a)e (sampling)g(of)h(the)166 5100 y(time)27 b(b)r(ecomes)f(suitable)h(for)f (estimating)g(the)h(sp)r(ectral)f(densit)n(y)-7 b(.)37 b(W)-7 b(e)27 b(dev)n(elop)e(a)h(random)g(sampling)g(tec)n(hnique,)166 5200 y(sho)n(wing)f(that)i(the)g(sc)n(heme)f(dev)n(elopp)r(ed)g(b)n(y)g (Masry)g(\(1978\))f(and)h(Messaci)g(\(1986\))f(ma)n(y)h(b)r(e)h(used)f (here.)36 b(Consis-)166 5300 y(tency)26 b(in)f(mean-square)f(of)h(the)h (estimate)g(is)f(established.)36 b(A)26 b(n)n(umeric)f(example)g(is)g (giv)n(en,)h(for)f(an)g(M-stationary)166 5399 y(Gauss-Mark)n(o)n(v)f (pro)r(cess.)p eop %%Page: 2 2 2 1 bop 3778 -149 a Fr(2)291 141 y(This)32 b(pap)r(er)h(is)f(organized) f(as)i(follo)n(ws.)51 b(In)33 b(Section)g(2,)g(w)n(e)g(recall)f(some)g (prop)r(erties)g(and)g(examples)g(of)h(M-)166 241 y(stationary)c(pro)r (cesses.)46 b(The)31 b(sampling)f(tec)n(hnique)h(and)g(the)g(link)n(ed) g(sp)r(ectral)f(densit)n(y)h(estimate)g(construction)166 340 y(are)g(exp)r(osed)h(in)h(Section)f(3.)50 b(W)-7 b(e)33 b(giv)n(e)e(the)i(results)f(of)g(con)n(v)n(ergence)e(of)i(the)h (estimate)f(in)g(Section)h(4.)50 b(A)33 b(short)166 440 y(sim)n(ulation)f(study)h(is)f(giv)n(en)g(in)g(section)h(5.)51 b(The)32 b(pro)r(ofs)g(of)g(the)h(main)g(results)f(are)f(dev)n(elopp)r (ed)h(in)h(Section)f(7.)166 540 y(And)c(a)f(wider)h(concept)f(of)h (stationarit)n(y)e(is)h(in)n(tro)r(duced)h(in)f(Section)h(6,)f (extending)h(the)g(M-stationarit)n(y)-7 b(.)166 814 y Fl(2)135 b(Prop)t(erties)46 b(and)e(examples)i(of)f(M-stationary)h(pro) t(cesses)166 996 y Fm(De\014nition)31 b(1)41 b Fp(L)l(et)28 b Fk(X)i Fr(=)22 b Fj(f)p Fk(X)7 b Fr(\()p Fk(t)p Fr(\))p Fk(;)28 b(t)23 b(>)f Fr(0)p Fj(g)28 b Fp(b)l(e)h(a)g(sto)l(chastic)h (pr)l(o)l(c)l(ess)f(such)g(that,)g(for)h(al)t(l)g(p)l(ositive)g(r)l(e)l (al)f(numb)l(ers)f Fk(t)166 1096 y Fp(and)i Fk(\034)9 b Fp(.)239 1261 y(\(i\))42 b Fk(I)-21 b(E)5 b Fj(f)p Fk(X)i Fr(\()p Fk(t)p Fr(\))p Fj(g)22 b Fr(=)h Fk(\026)29 b Fp(is)h(indep)l(endent)h(of)f Fk(t)p Fp(.)213 1427 y(\(ii\))43 b(var)p Fj(f)p Fk(X)7 b Fr(\()p Fk(t)p Fr(\))p Fj(g)22 b Fk(<)h Fr(+)p Fj(1)p Fp(.)188 1593 y(\(iii\))43 b Fk(I)-21 b(E)5 b Fj(f)p Fr(\()p Fk(X)i Fr(\()p Fk(t)p Fr(\))18 b Fj(\000)g Fk(\026)p Fr(\)\()p Fk(X)7 b Fr(\()p Fk(t:\034)i Fr(\))20 b Fj(\000)e Fk(\026)p Fr(\))p Fj(g)23 b Fr(=)f Fk(R)1552 1605 y Fi(X)1615 1593 y Fr(\()p Fk(\034)9 b Fr(\))31 b Fp(exists)f(and)g(is)g(indep)l(endent)h(of)f Fk(t)p Fp(.)166 1759 y(Then)g Fk(X)36 b Fp(is)31 b(said)g(to)e(b)l(e)h (a)g(we)l(akly)h(multiplic)l(ative-stationary)h(\(or)e(M-stationary\))h (pr)l(o)l(c)l(ess.)166 1941 y Fr(In)k(the)f(classical)f(sense,)j (M-stationary)c(pro)r(cesses)h(are)g(nonstationary)-7 b(,)34 b(but,)j(to)d(eac)n(h)g(M-stationary)e(pro)r(cess)166 2041 y(corresp)r(onds)26 b(a)h(stationary)f(pro)r(cess)g(as)h(stated)h (in)g(the)g(follo)n(wing)e(theorem)h(due)h(to)g(Gra)n(y)e(and)h(Zhang)g (\(1988\).)166 2207 y Fm(Theorem)j(1)42 b Fp(L)l(et)26 b Fk(Y)41 b Fr(=)23 b Fj(f)p Fk(Y)18 b Fr(\()p Fk(u)p Fr(\))p Fk(;)28 b(u)23 b Fj(2)g Fh(R)p Fj(g)32 b Fp(b)l(e)26 b(a)h(sto)l(chastic)g(pr)l(o)l(c)l(ess)g(such)g(that)f Fk(Y)19 b Fr(\()p Fk(u)p Fr(\))k(=)g Fk(X)7 b Fr(\()p Fk(t)p Fr(\))26 b Fp(wher)l(e)h Fk(t)c Fr(=)g(exp)o(\()p Fk(u)p Fr(\))p Fp(.)166 2306 y(Then)30 b Fk(X)36 b Fp(is)31 b(we)l(akly)g(M-stationary)f(if)h(and)f(only)g(if)h Fk(Y)48 b Fp(is)31 b(we)l(akly)f(stationary.)166 2472 y Fk(Y)43 b Fr(is)23 b(referred)g(to)h(as)f(the)h(dual)g(stationary)e(pro)r(cess) g(of)i Fk(X)7 b Fr(.)35 b(Let)24 b(us)g(denote)g(b)n(y)f Fk(C)2739 2484 y Fi(Y)2821 2472 y Fr(the)h(auto)r(co)n(v)-5 b(ariance)22 b(function)166 2572 y(of)28 b Fk(Y)18 b Fr(;)28 b(then)g(for)f(all)g(p)r(ositiv)n(e)h(real)e(n)n(um)n(b)r(er)i Fk(\034)9 b Fr(,)28 b(w)n(e)f(ha)n(v)n(e)1416 2754 y Fk(R)1479 2766 y Fi(X)1542 2754 y Fr(\()p Fk(\034)9 b Fr(\))24 b(=)f Fk(R)1826 2766 y Fg(X)1879 2754 y Fr(\()p Fk(\034)1956 2720 y Ff(\000)p Fq(1)2046 2754 y Fr(\))h(=)e Fk(R)2252 2766 y Fg(Y)2302 2754 y Fr(\(ln)q(\()p Fk(\034)9 b Fr(\)\))p Fk(:)291 2936 y Fr(As)26 b(will)g(b)r(e)h(seen)e(in)i(the)f (follo)n(wing,)g(this)g(dual)g(pro)r(cess)f(will)h(the)h(basis)e(of)h (the)h(dev)n(elop)r(emen)n(t)e(of)h(the)h(theory)166 3036 y(of)h(sp)r(ectral)f(densit)n(y)g(estimation)g(of)h(an)f (M-stationary)f(pro)r(cess.)291 3136 y(Let)h(us)h(giv)n(e)f(some)g (examples)f(of)i(M-stationary)e(pro)r(cesses)g(to)h(set)h(up)g(ideas.) 166 3318 y Fm(Example)i(1)42 b Fr(Consider)26 b(the)i(pro)r(cess)f Fk(X)7 b Fr(,)27 b(de\014ned)h(for)f Fk(t)c(>)f Fr(0)27 b(b)n(y)1518 3500 y Fk(X)7 b Fr(\()p Fk(t)p Fr(\))23 b(=)f Fk(a)28 b Fr(cos)o(\()p Fk(\014)k Fr(ln\()p Fk(t)p Fr(\))19 b(+)f Fk(!)s Fr(\))p Fk(;)166 3683 y Fr(where)37 b Fk(a)g Fr(is)g(a)f(constan)n(t)h(and)g Fk(!)j Fr(is)d(a)f(uniformly)h (distributed)h(on)f([0)p Fk(;)14 b Fr(2)p Fk(\031)s Fr(])36 b(random)g(v)-5 b(ariable;)41 b(then)d Fk(X)43 b Fr(is)37 b(M-)166 3835 y(stationary)-7 b(,)26 b(with)i(zero)f(mean,)g(and)h(co)n (v)-5 b(ariance)25 b(function)j Fk(R)2137 3847 y Fg(X)2191 3835 y Fr(\()p Fk(\034)9 b Fr(\))24 b(=)2422 3779 y Fk(A)2484 3749 y Fq(2)p 2422 3816 100 4 v 2451 3892 a Fr(2)2545 3835 y(cos)o(\()p Fk(\014)33 b Fr(ln\()p Fk(\034)9 b Fr(\)\))29 b(for)e(all)g Fk(\034)33 b(>)23 b Fr(0.)166 4041 y Fm(Example)30 b(2)42 b Fr(Consider)26 b(the)i(pro)r(cess)f Fk(X)7 b Fr(,)27 b(de\014ned)h(for)f Fk(t)c(>)f Fr(0)27 b(b)n(y)1192 4300 y Fk(X)7 b Fr(\()p Fk(t)p Fr(\))23 b(=)1503 4196 y Fi(N)1472 4221 y Fe(X)1479 4398 y Fi(i)p Fq(=1)1606 4300 y Fr([)p Fk(A)1691 4312 y Fi(i)1747 4300 y Fr(cos)o(\()p Fk(a)1934 4312 y Fi(i)1989 4300 y Fr(ln)14 b Fk(t)p Fr(\))19 b(+)f Fk(B)2299 4312 y Fi(i)2355 4300 y Fr(sin)o(\()p Fk(a)2532 4312 y Fi(i)2588 4300 y Fr(ln)c Fk(t)p Fr(\)])g Fk(;)166 4555 y Fr(where,)39 b(for)e(all)g Fk(i)g Fr(and)h Fk(j)5 b Fr(,)39 b Fk(A)1104 4567 y Fi(i)1170 4555 y Fr(and)e Fk(B)1404 4567 y Fi(i)1469 4555 y Fr(are)f(zero-mean)g (uncorrelated)g(random)g(v)-5 b(ariables)36 b(suc)n(h)h(that,)k(for)36 b(all)166 4654 y Fk(i)23 b Fj(2)g(f)p Fr(1)p Fk(;)14 b(:)g(:)g(:)f(;)h(n)p Fj(g)p Fk(;)27 b(I)-21 b(E)5 b Fj(f)p Fk(A)891 4620 y Fq(2)891 4675 y Fi(i)928 4654 y Fj(g)23 b Fr(=)f Fk(I)-21 b(E)6 b Fj(f)p Fk(B)1271 4620 y Fq(2)1267 4675 y Fi(i)1308 4654 y Fj(g)p Fr(.)36 b(Then)28 b Fk(X)34 b Fr(is)27 b(a)h(zero-mean,)d(M-stationary)h(pro)r (cess)h(with)1379 4918 y Fk(R)1442 4930 y Fg(X)1496 4918 y Fr(\()p Fk(\034)9 b Fr(\))24 b(=)1747 4814 y Fi(N)1717 4839 y Fe(X)1723 5016 y Fi(i)p Fq(=1)1851 4918 y Fk(I)-21 b(E)5 b Fj(f)p Fk(A)2036 4884 y Fq(2)2036 4939 y Fi(i)2073 4918 y Fj(g)14 b Fr(cos)n(\()p Fk(a)2315 4930 y Fi(i)2371 4918 y Fr(ln\()p Fk(\034)9 b Fr(\)\))p Fk(:)166 5173 y Fm(Example)30 b(3)42 b Fr(A)21 b(pro)r(cess)f Fk(")g Fr(is)h(said)g(to)g(b)r(e)g(an)g(M-white)g(noise)f(if)i(its)f(dual)g (pro)r(cess)f(is)g(a)h(white)g(noise.)34 b(An)22 b(M-white)166 5273 y(noise)27 b(is)h(th)n(us)f(de\014ned)h(b)n(y)1212 5372 y Fk(I)-21 b(E)6 b Fj(f)p Fk(")p Fr(\()p Fk(t)p Fr(\))p Fj(g)22 b Fr(=)h(0)k(and)g Fk(I)-21 b(E)6 b Fj(f)p Fk(")p Fr(\()p Fk(t)p Fr(\))p Fk(")p Fr(\()p Fk(t\034)j Fr(\))p Fj(g)23 b Fr(=)g Fk(c)14 b(\016)s Fr(\(ln\()p Fk(\034)9 b Fr(\)\))166 5521 y(where)27 b Fk(\016)k Fr(is)c(the)h (Kronec)n(k)n(er)d(sym)n(b)r(ol.)p eop %%Page: 3 3 3 2 bop 3778 -149 a Fr(3)166 186 y Fm(Example)30 b(4)42 b Fr(Consider)26 b(the)i(pro)r(cess)f Fk(X)7 b Fr(,)27 b(de\014ned)h(for)f Fk(t)c(>)f Fr(0)27 b(b)n(y)h Fk(X)7 b Fr(\()p Fk(t)p Fr(\))23 b(=)2573 73 y Fe(Z)2656 94 y Fi(t)2619 262 y Fq(0)2699 186 y Fk(H)7 b Fr(\()2820 130 y(1)p 2817 167 48 4 v 2817 243 a Fk(u)2875 186 y Fr(\))p Fk(")p Fr(\()p Fk(u)p Fr(\))p Fk(d)14 b Fr(ln\()p Fk(u)p Fr(\),)28 b(where)291 461 y Fj(\017)374 348 y Fe(Z)457 369 y Fq(+)p Ff(1)420 537 y Fq(0)592 461 y Fj(j)p Fk(H)7 b Fr(\()p Fk(t)p Fr(\))p Fj(j)808 420 y Fq(2)859 461 y Fk(d)14 b Fr(ln)g Fk(t)23 b(>)g Fj(1)28 b Fr(and)f Fk(H)7 b Fr(\()p Fk(t)p Fr(\))24 b(=)e(0)27 b(for)g(0)c Fk(<)g(t)g(<)f Fr(1)p Fk(:)291 668 y Fj(\017)41 b Fk(")27 b Fr(is)g(an)h(M-white)g(noise.)166 830 y(Then)g Fk(X)34 b Fr(is)27 b(an)h(M-stationary)d(pro)r(cess)i(called)g(M-linear)g(pro)r (cess.)166 1008 y Fm(Example)j(5)42 b Fr(The)23 b(Euler)f(pro)r(cess)g (is)h(a)g(particular)f(case)g(of)h(M-linear)f(pro)r(cesses,)h (sometimes)g(called)g(long)f(mem-)166 1108 y(ory)27 b(pro)r(cess,)f (with)1032 1261 y Fk(H)7 b Fr(\()p Fk(t)p Fr(\))24 b(=)1313 1144 y Fe(\032)1417 1151 y(P)1505 1172 y Fi(M)1505 1238 y(i)p Fq(=1)1630 1151 y Fe(P)1718 1172 y Fi(m)1777 1180 y Fg(i)1718 1238 y Fi(j)s Fq(=1)1851 1213 y Fk(c)1887 1225 y Fi(ij)1945 1213 y Fr(\(ln)14 b Fk(t)p Fr(\))2122 1183 y Fi(j)2158 1213 y Fk(t)2188 1183 y Ff(\000)p Fi(a)2276 1191 y Fg(i)2417 1213 y Fr(for)27 b Fk(t)c Fj(\025)g Fr(1)1841 1319 y(0)534 b(for)27 b(0)22 b Fk(<)h(t)g(<)g Fr(1)p Fk(;)166 1467 y Fr(where)i Fk(c)440 1479 y Fi(ij)526 1467 y Fr(and)j Fk(a)732 1479 y Fi(i)785 1467 y Fr(are)c(complex)i(n)n (um)n(b)r(ers)f(suc)n(h)g(that)h Fj(R)p Fk(e)p Fr(\()p Fk(a)2126 1479 y Fi(i)2154 1467 y Fr(\))d Fj(\025)g Fr(0,)i(where)g Fj(R)p Fk(e)p Fr(\()p Fk(z)t Fr(\))h(denote)f(the)h(real)f(part)g(of)h (the)166 1567 y(the)i(complex)f(n)n(um)n(b)r(er)h Fk(z)t Fr(.)36 b(W)-7 b(e)28 b(ha)n(v)n(e)e(then)1435 1821 y Fk(R)1498 1833 y Fg(X)1552 1821 y Fr(\()p Fk(\034)9 b Fr(\))24 b(=)1798 1717 y Fi(M)1773 1742 y Fe(X)1779 1919 y Fi(i)p Fq(=1)1924 1716 y Fi(m)1983 1724 y Fg(i)1907 1742 y Fe(X)1909 1919 y Fi(j)s Fq(=1)2040 1821 y Fk(b)2076 1833 y Fi(ij)2134 1821 y Fr(\(ln)15 b Fk(\034)9 b Fr(\))2327 1787 y Fi(j)2363 1821 y Fk(\034)2408 1787 y Ff(\000)p Fi(a)2496 1795 y Fg(i)2527 1821 y Fk(;)166 2083 y Fr(where)27 b(the)h(constan)n(ts)f Fk(b)953 2095 y Fi(ij)1038 2083 y Fr(are)g(uniquely)h(determined)g(from)f(the)h(constan)n(ts)e Fk(a)2698 2095 y Fi(i)2754 2083 y Fr(and)h Fk(c)2951 2095 y Fi(ij)3009 2083 y Fk(:)166 2197 y Fr(The)i(in)n(teger)e Fk(k)h Fr(=)773 2134 y Fe(P)861 2155 y Fi(M)861 2221 y(i)p Fq(=1)972 2197 y Fr(\()p Fk(m)1077 2209 y Fi(i)1124 2197 y Fr(+)19 b(1\))28 b(is)h(called)f(the)h(order)e(of)h(the)h(pro)r (cess.)39 b(Note)28 b(that)h(the)g(Euler)f(pro)r(cesses)f(pla)n(y)166 2296 y(the)33 b(same)f(part)g(in)g(the)h(M-stationary)e(pro)r(cesses)g (theory)g(as)h(the)h(auto-regressiv)n(e)c(stationary)i(mo)r(dels)h(in)h (the)166 2396 y(usual)27 b(case.)166 2574 y(Let)22 b Fk(X)29 b Fr(b)r(e)22 b(an)g(M-stationary)f(pro)r(cess.)34 b(The)22 b(M-sp)r(ectral)f(densit)n(y)h Fk(G)2359 2586 y Fg(X)2435 2574 y Fr(of)g Fk(X)29 b Fr(is)22 b(de\014ned)h(as)e(the)i (Mellin)f(transform)166 2673 y(of)28 b(the)g(auto)r(co)n(v)-5 b(ariance)25 b Fk(R)1032 2685 y Fg(X)1113 2673 y Fr(if)j(it)g(do)r(es)f (exists,)h(namely)1351 2905 y Fk(G)1416 2917 y Fg(X)1470 2905 y Fr(\()p Fk(x)p Fr(\))c(=)1693 2792 y Fe(Z)1776 2812 y Fq(+)p Ff(1)1739 2980 y Fq(0)1911 2905 y Fk(\034)1956 2870 y Ff(\000)p Fq(2)p Fi(i\031)r(x)2148 2905 y Fk(R)2211 2917 y Fg(X)2264 2905 y Fr(\()p Fk(\034)9 b Fr(\))p Fk(d)14 b Fr(ln)r(\()p Fk(\034)9 b Fr(\))p Fk(:)1080 b Fr(\(1\))166 3125 y(It)28 b(ma)n(y)f(also)f(b)r(e)i(written)g(as)1409 3278 y Fk(G)1474 3290 y Fg(X)1528 3278 y Fr(\()p Fk(x)p Fr(\))c(=)1750 3165 y Fe(Z)1833 3185 y Fq(+)p Ff(1)1796 3353 y(\0001)1968 3278 y Fk(e)2007 3243 y Ff(\000)p Fq(2)p Fi(i\031)r(xu)2237 3278 y Fk(R)2300 3290 y Fg(Y)2350 3278 y Fr(\()p Fk(u)p Fr(\))p Fk(du;)166 3473 y Fr(where)j Fk(R)469 3485 y Fg(Y)547 3473 y Fr(is)g(the)h(auto)r(co)n(v)-5 b(ariance)25 b(function)k(of)e(the)h(dual)g(pro)r(cess)e Fk(Y)46 b Fr(of)28 b Fk(X)7 b Fr(.)166 3573 y(W)-7 b(e)32 b(ha)n(v)n(e)e(tacitly)h(assumed)g(that)h(the)f(Mellin)h(transform)e (in)i(\(1\))f(exists.)48 b(W)-7 b(e)32 b(mak)n(e)e(this)i(assumption)f (to)r(o)g(for)166 3672 y(the)i(in)n(v)n(erse)e(Mellin)i(transform.)50 b(F)-7 b(or)31 b(details)i(on)f(the)g(Mellin)h(transform,)g(see)f (Morettin)g(\(1974\))f(and)h(\(1981\).)166 3772 y(F)-7 b(rom)27 b(\(1\),)h(it)g(follo)n(ws)f(at)g(once)g(that)h Fk(G)1432 3784 y Fg(X)1486 3772 y Fr(\()p Fk(x)p Fr(\))c(=)f Fk(G)1774 3784 y Fg(Y)1824 3772 y Fr(\()p Fk(x)p Fr(\),)29 b(where)e Fk(G)2292 3784 y Fg(Y)2369 3772 y Fr(is)h(the)g(sp)r(ectral)f (densit)n(y)g(of)h Fk(Y)19 b Fr(.)291 3971 y(In)38 b(the)h(sequel,)h(w) n(e)e(shall)g(use)g(the)h(term)f(sp)r(ectral)g(densit)n(y)g(for)g(M-sp) r(ectral)f(densit)n(y)-7 b(,)41 b(and)d(assume)g(that)166 4071 y(the)d(pro)r(cess)f Fk(X)42 b Fr(is)35 b(zero-mean,)g (M-stationary)e(up)j(to)f(order)e(four)i(i.e.,)i(that,)h(for)c(all)h(p) r(ositiv)n(e)f(real)h(n)n(um)n(b)r(ers)166 4171 y Fk(t)196 4183 y Fq(1)233 4171 y Fk(;)28 b(t)314 4183 y Fq(2)351 4171 y Fk(;)g(t)432 4183 y Fq(3)497 4171 y Fr(and)f Fk(t)e(I)-21 b(E)6 b Fj(f)p Fk(X)h Fr(\()p Fk(t)p Fr(\))p Fk(X)g Fr(\()p Fk(t:t)1198 4183 y Fq(1)1234 4171 y Fr(\))p Fk(X)g Fr(\()p Fk(t:t)1457 4183 y Fq(2)1494 4171 y Fr(\))p Fk(X)g Fr(\()p Fk(t:t)1717 4183 y Fq(3)1754 4171 y Fr(\))p Fj(g)25 b Fr(is)g(indep)r(enden)n(t)h(of)f Fk(t)p Fr(.)36 b(In)26 b(this)f(case)f(w)n(e)h(call)g(M-cum)n(ulan)n(t)f(of)166 4270 y(order)i(four)h(of)h Fk(X)7 b Fr(,)27 b(the)h(function)794 4480 y Fk(Q)p Fr(\()p Fk(t)922 4492 y Fq(1)959 4480 y Fk(;)g(t)1040 4492 y Fq(2)1077 4480 y Fk(;)g(t)1158 4492 y Fq(3)1195 4480 y Fr(\))84 b(=)e Fk(I)-21 b(E)6 b Fj(f)p Fk(X)h Fr(\()p Fk(t)p Fr(\))p Fk(X)g Fr(\()p Fk(t:t)1943 4492 y Fq(1)1979 4480 y Fr(\))p Fk(X)g Fr(\()p Fk(t:t)2202 4492 y Fq(2)2239 4480 y Fr(\))p Fk(X)g Fr(\()p Fk(t:t)2462 4492 y Fq(3)2499 4480 y Fr(\))p Fj(g)18 b(\000)g Fk(R)2737 4492 y Fg(X)2791 4480 y Fr(\()p Fk(t)2853 4492 y Fq(1)2890 4480 y Fr(\))p Fk(R)2985 4492 y Fg(X)3039 4480 y Fr(\()3081 4424 y Fk(t)3111 4436 y Fq(2)p 3081 4461 68 4 v 3081 4537 a Fk(t)3111 4549 y Fq(3)3159 4480 y Fr(\))1458 4690 y Fj(\000)p Fk(R)1586 4702 y Fg(X)1639 4690 y Fr(\()p Fk(t)1701 4702 y Fq(2)1739 4690 y Fr(\))p Fk(R)1834 4702 y Fg(X)1888 4690 y Fr(\()1930 4634 y Fk(t)1960 4646 y Fq(3)p 1930 4671 V 1930 4747 a Fk(t)1960 4759 y Fq(1)2007 4690 y Fr(\))h Fj(\000)f Fk(R)2204 4702 y Fg(X)2257 4690 y Fr(\()p Fk(t)2319 4702 y Fq(3)2357 4690 y Fr(\))p Fk(R)2452 4702 y Fg(X)2506 4690 y Fr(\()2548 4634 y Fk(t)2578 4646 y Fq(2)p 2548 4671 V 2548 4747 a Fk(t)2578 4759 y Fq(1)2625 4690 y Fr(\))p Fk(:)166 4904 y Fm(Remark)31 b(1)41 b Fr(Let)35 b Fk(X)41 b Fr(b)r(e)36 b(an)e(M-stationary)f(pro)r(cess)h (and)h(let)g Fk(Y)54 b Fr(b)r(e)35 b(its)g(dual)g(pro)r(cess.)57 b(By)35 b(the)g(relationship)166 5003 y(b)r(et)n(w)n(een)28 b Fk(X)34 b Fr(and)27 b Fk(Y)19 b Fr(,)27 b(w)n(e)h(get)1212 5181 y Fk(R)1275 5193 y Fg(X)1329 5181 y Fr(\()p Fk(\034)9 b Fr(\))24 b(=)f Fk(R)1613 5193 y Fg(X)1667 5181 y Fr(\()p Fk(\034)1744 5147 y Ff(\000)p Fq(1)1834 5181 y Fr(\))28 b(and)f Fk(R)2118 5193 y Fg(X)2172 5181 y Fr(\()p Fk(\034)9 b Fr(\))24 b(=)f Fk(R)2456 5193 y Fg(Y)2506 5181 y Fr(\(ln\()p Fk(\034)9 b Fr(\)\))p Fk(:)166 5399 y Fr(If)36 b(w)n(e)f(assume)f(that) i Fk(R)932 5411 y Fg(X)1021 5399 y Fr(is)f(con)n(tin)n(uous)f(and)h (that)h Fk(\034)1937 5364 y Ff(\000)p Fi(\013)2037 5399 y Fk(R)2100 5411 y Fg(X)2153 5399 y Fr(\()p Fk(\034)9 b Fr(\))37 b Fj(2)f Fk(L)2447 5364 y Fq(1)p Fi(=\013)2561 5399 y Fr(\()p Fh(R)q Fr(+\))41 b(for)35 b Fk(\013)h Fr(=)3120 5342 y(1)p 3120 5379 42 4 v 3120 5455 a(2)3207 5399 y(or)e(1,)j(then)f Fk(R)3678 5411 y Fg(Y)3763 5399 y Fr(is)166 5521 y(con)n(tin)n(uous,)27 b(absolutely)f(in)n(tegrable)h (and)g(square)g(in)n(tegrable.)p eop %%Page: 4 4 4 3 bop 3778 -149 a Fr(4)166 141 y Fl(3)135 b(Time)45 b(sampling)h(and)e(sp)t(ectral)i(densit)l(y)f(estimation)166 323 y Fr(F)-7 b(or)27 b(sampling)g(the)h(time,)g(w)n(e)f(consider)g (the)h(sc)n(heme)f Fj(f)p Fk(t)1966 335 y Fi(n)2011 323 y Fj(g)2053 335 y Fi(n)p Ff(2)p Fd(N)2212 323 y Fr(prop)r(osed)f(b)n(y) i(Messaci)e(\(1986\),)h(namely)1951 502 y Fk(t)1981 514 y Fq(0)2041 502 y Fr(=)c(1)1624 627 y(and)193 b Fk(t)1981 639 y Fi(n)2049 627 y Fr(=)23 b Fk(t)2167 639 y Fi(n)p Ff(\000)p Fq(1)2297 627 y Fk(\014)2344 639 y Fi(n)3713 627 y Fr(\(2\))166 806 y(where)k Fj(f)p Fk(\014)495 818 y Fi(n)540 806 y Fj(g)582 818 y Fi(n)p Ff(2)p Fd(N)741 806 y Fr(is)g(a)h(sequence)f(of)g(P)n(areto)f(distributed)i(random)e(v) -5 b(ariables,)27 b(with)h(densit)n(y)1351 986 y Fk(f)1392 998 y Fi(\014)1430 1006 y Fg(n)1474 986 y Fr(\()p Fk(x)p Fr(\))c(=)f Fk(\014)t(x)1795 952 y Ff(\000)p Fi(\014)s Ff(\000)p Fq(1)1977 986 y Fr(1)2019 1001 y Fq([1)p Fi(;)p Fq(+)p Ff(1)p Fq([)2231 986 y Fr(\()p Fk(x)p Fr(\))14 b(;)28 b Fk(\014)g(>)22 b Fr(0)p Fk(:)166 1165 y Fr(Consider)36 b(then)i(the)g(sc)n(heme)f Fk(\034)1204 1177 y Fi(n)1289 1165 y Fr(=)i(ln\()p Fk(t)1524 1177 y Fi(n)1569 1165 y Fr(\))f(for)f(all)g Fk(n)i Fj(2)h Fh(N)t Fr(.)72 b(The)38 b(sequence)e Fj(f)p Fk(\034)2846 1177 y Fi(n)2891 1165 y Fj(g)2933 1177 y Fi(n)p Ff(2)p Fd(N)3102 1165 y Fr(v)n(eri\014es)g Fk(\034)3425 1177 y Fq(0)3502 1165 y Fr(=)j(0)e(and)166 1265 y Fk(\034)202 1277 y Fi(n)282 1265 y Fr(=)d Fk(\034)417 1277 y Fi(n)p Ff(\000)p Fq(1)571 1265 y Fr(+)23 b Fk(\013)712 1277 y Fi(n)791 1265 y Fr(for)34 b Fk(n)h Fj(2)g Fh(N)1154 1235 y Ff(\003)1233 1265 y Fr(where)f Fk(\013)1533 1277 y Fi(n)1612 1265 y Fr(=)h(ln\()p Fk(\014)1860 1277 y Fi(n)1905 1265 y Fr(\))g(is)g(a)f(sequence)g(of)g(exp)r(onen)n(tially)g (distributed)h(random)166 1365 y(v)-5 b(ariables)26 b(with)i(parameter) f Fk(\014)g(>)c Fr(0,)k(with)h(densit)n(y)1413 1544 y Fk(f)1454 1556 y Fi(\013)1497 1564 y Fg(n)1541 1544 y Fr(\()p Fk(x)p Fr(\))c(=)f Fk(\014)18 b Fr(exp\()p Fj(\000)p Fk(\014)t(x)q Fr(\)1)2226 1559 y Fq([0)p Fi(;)p Fq(+)p Ff(1)p Fq([)2437 1544 y Fr(\()p Fk(x)p Fr(\))p Fk(:)291 1724 y Fr(Note)27 b(that)h(since)f Fk(t)904 1736 y Fi(n)973 1724 y Fj(\025)22 b Fr(1)28 b(a.s.,)f(w)n(e)g(ha)n(v)n(e)f(also)h Fk(\014)1828 1736 y Fi(n)1896 1724 y Fj(\025)c Fr(1)k(a.s..)291 1823 y(Consider)j(the)h(discrete)g(pro)r(cess)f Fj(f)p Fk(X)7 b Fr(\()p Fk(t)1569 1835 y Fi(k)1609 1823 y Fr(\))p Fj(g)1683 1835 y Fi(k)q Ff(2)p Fd(N)1810 1823 y Fr(,)32 b(whic)n(h)f(is)g(zero-mean)e(to)r(o.)48 b(Since)31 b(the)g(sequence)g Fj(f)p Fk(t)3568 1835 y Fi(n)3613 1823 y Fj(g)f Fr(and)166 1923 y(the)e(con)n(tin)n(uous)f(pro)r(cess)f Fk(X)34 b Fr(are)27 b(indep)r(enden)n(t,)h(the)g(auto)r(co)n(v)-5 b(ariance)25 b(function)k(of)e(the)h(sampled)f(pro)r(cess)g(is)1085 2155 y Fk(\032)1128 2167 y Fg(X)1181 2155 y Fr(\()p Fk(n)p Fr(\))c(=)g Fk(I)-21 b(E)5 b Fj(f)p Fk(X)i Fr(\()p Fk(t)1667 2167 y Fi(k)q Fq(+)p Fi(n)1799 2155 y Fr(\))p Fk(X)g Fr(\()p Fk(t)1969 2167 y Fi(k)2010 2155 y Fr(\))p Fj(g)23 b Fr(=)2195 2042 y Fe(Z)2278 2063 y Fq(+)p Ff(1)2241 2231 y Fq(0)2413 2155 y Fk(R)2476 2167 y Fg(X)2529 2155 y Fr(\()p Fk(t)p Fr(\))p Fk(f)2664 2167 y Fi(n)2710 2155 y Fr(\()p Fk(t)p Fr(\))p Fk(dt:)291 2382 y Fr(Let)h Fk(X)29 b Fr(=)23 b Fj(f)p Fk(X)7 b Fr(\()p Fk(t)p Fr(\))p Fk(;)27 b(t)c Fj(\025)g Fr(0)p Fj(g)g Fr(b)r(e)h(an)g(M-stationary)e(pro)r (cess)h(and)h(let)h Fk(Y)41 b Fr(=)23 b Fj(f)p Fk(Y)18 b Fr(\()p Fk(u)p Fr(\))p Fk(;)28 b(u)22 b Fj(2)i Fh(R)p Fj(g)29 b Fr(b)r(e)c(its)f(dual)g(pro)r(cess,)166 2481 y Fk(Y)52 b Fr(is)33 b(then)h(zero)e(mean,)i(stationary)e(in)h(the)h (classical)e(sens,)i(of)f(order)f(four,)i(and)f(with)h(cum)n(ulan)n(t)f (function)h Fk(C)166 2581 y Fr(giv)n(en)27 b(b)n(y)1180 2680 y Fk(C)6 b Fr(\()p Fk(u)1325 2692 y Fq(1)1363 2680 y Fk(;)14 b(u)1448 2692 y Fq(2)1484 2680 y Fk(;)g(u)1569 2692 y Fq(3)1606 2680 y Fr(\))23 b(=)g Fk(Q)p Fr(\(exp\()p Fk(u)2054 2692 y Fq(1)2091 2680 y Fr(\))p Fk(;)14 b Fr(exp\()p Fk(u)2367 2692 y Fq(2)2404 2680 y Fr(\))p Fk(;)g Fr(exp\()p Fk(u)2680 2692 y Fq(3)2717 2680 y Fr(\)\))p Fk(:)166 2828 y Fr(Since)30 b Fk(X)37 b Fr(and)30 b Fk(Y)49 b Fr(ha)n(v)n(e)29 b(the)h(same)g(sp)r(ectral)f(densit)n(y)-7 b(,)31 b(then)g(the)f(problem)g(of)g(sp)r(ectral)g(densit)n(y)g (estimation)f(can)166 2928 y(b)r(e)f(transfered)f(from)g(an)g (M-stationary)f(pro)r(cess)h(to)g(a)g(stationary)f(one.)166 3168 y Fm(Remark)31 b(2)41 b Fr(F)-7 b(or)27 b(some)g Fk(k)f Fj(\025)d Fr(1,)k(the)h(di\013erence)f Fk(u)1786 3180 y Fi(n)p Fq(+)p Fi(k)1937 3168 y Fj(\000)18 b Fk(u)2068 3180 y Fi(n)2136 3168 y Fr(=)2265 3065 y Fi(k)2224 3089 y Fe(X)2230 3266 y Fi(i)p Fq(=1)2357 3168 y Fk(\013)2410 3180 y Fi(n)p Fq(+)p Fi(i)2530 3168 y Fr(,)28 b(admits)g(as)e (probabilit)n(y)h(densit)n(y)648 3486 y Fk(f)689 3498 y Fi(n)734 3486 y Fr(\()p Fk(t)p Fr(\))d(=)e Fk(\014)1003 3430 y Fr(\()p Fk(\014)t(t)p Fr(\))1148 3400 y Fi(n)p Ff(\000)p Fq(1)p 1001 3467 281 4 v 1001 3543 a Fr(\()p Fk(n)c Fj(\000)g Fr(1\)!)1305 3486 y(exp\()p Fj(\000)p Fk(\014)t(t)p Fr(\)1)1684 3498 y Fd(R)1731 3506 y Fc(+)1777 3486 y Fr(\()p Fk(t)p Fr(\))p Fk(;)42 b Fr(\(Gamma)27 b(of)h(order)e Fk(n)i Fr(and)f(parameter)f Fk(\014)t Fr(\))p Fk(:)291 3715 y Fr(Consider)20 b(the)i(discrete)f(time)h(pro)r (cess)e Fj(f)p Fk(X)7 b Fr(\()p Fk(t)1713 3727 y Fi(n)1757 3715 y Fr(\))p Fj(g)1831 3727 y Fi(n)p Ff(2)p Fd(N)1963 3715 y Fr(.)35 b(This)21 b(pro)r(cess)g(is)g(of)g(zero)g(mean)g(and)g (of)g(auto)r(co)n(v)-5 b(ariance)166 3814 y(function)28 b Fk(\032)534 3826 y Fg(X)615 3814 y Fr(suc)n(h)g(that)840 3994 y Fk(\032)883 4006 y Fg(X)936 3994 y Fr(\()p Fk(n)p Fr(\))84 b(=)e Fk(I)-21 b(E)6 b Fj(f)p Fk(X)h Fr(\()p Fk(t)1543 4006 y Fi(k)q Fq(+)p Fi(n)1675 3994 y Fr(\))p Fk(X)g Fr(\()p Fk(t)1845 4006 y Fi(k)1885 3994 y Fr(\))p Fj(g)23 b Fr(=)g Fk(I)-21 b(E)2151 4009 y Ff(f)p Fi(t)2210 4017 y Fg(n)2251 4009 y Ff(g)2308 3994 y Fj(\016)18 b Fk(I)-21 b(E)2449 4006 y Fi(X)2512 3994 y Fj(f)p Fk(X)7 b Fr(\()p Fk(t)2692 4006 y Fi(k)q Fq(+)p Fi(n)2824 3994 y Fr(\))p Fk(X)g Fr(\()p Fk(t)2994 4006 y Fi(k)3035 3994 y Fr(\))p Fj(g)p Fk(;)1134 4118 y Fr(=)82 b Fk(I)-21 b(E)1363 4133 y Ff(f)p Fi(t)1422 4141 y Fg(n)1462 4133 y Ff(g)1501 4118 y Fj(f)p Fk(R)1606 4130 y Fg(X)1659 4118 y Fr(\()p Fk(t)1721 4130 y Fi(k)q Fq(+)p Fi(n)1854 4118 y Fk(=t)1926 4130 y Fi(k)1966 4118 y Fr(\))p Fj(g)p Fk(;)1134 4309 y Fr(=)1281 4196 y Fe(Z)1364 4217 y Fq(+)p Ff(1)1327 4385 y Fq(0)1499 4309 y Fk(R)1562 4321 y Fg(X)1616 4309 y Fr(\()p Fk(t)p Fr(\))p Fk(f)1751 4321 y Fi(\013)1794 4329 y Fg(n)1839 4309 y Fr(\()p Fk(t)p Fr(\))p Fk(dt:)166 4535 y Fr(since)23 b(the)g(pro)r(cess)f Fk(X)30 b Fr(and)22 b(the)i(sc)n(heme)e Fj(f)p Fk(t)1533 4547 y Fi(n)1578 4535 y Fj(g)h Fr(are)f(indep)r(enden)n(t.)36 b(More,)23 b(the)g(system)g Fj(f)p Fk(f)3004 4547 y Fi(\013)3047 4555 y Fg(n)3091 4535 y Fr(\()p Fk(t)p Fr(\))p Fj(g)3227 4547 y Fi(n)p Ff(2)p Fd(N)3356 4531 y Fb(\003)3417 4535 y Fr(is)g(complete)166 4635 y(in)28 b Fk(L)320 4601 y Fq(2)357 4635 y Fr(\(0)p Fk(;)14 b Fr(+)p Fj(1)p Fr(\),)27 b(and,)g(if)i(orthonormalized,)c(it)j(b)r(ecomes)1271 4831 y Fk(g)1311 4843 y Fi(n)1356 4831 y Fr(\()p Fk(t)p Fr(\))23 b(=)1561 4756 y Fe(p)p 1644 4756 93 4 v 75 x Fr(2)p Fk(\014)t(L)1794 4787 y Fq(\(0\))1794 4853 y Fi(n)p Ff(\000)p Fq(1)1924 4831 y Fr(\(2)p Fk(\014)t(t)p Fr(\))14 b(exp\()p Fj(\000)p Fk(\014)t(t)p Fr(\)1)2504 4843 y Fd(R)2551 4851 y Fc(+)2597 4831 y Fr(\()p Fk(t)p Fr(\))p Fk(;)166 5026 y Fr(where)26 b Fk(L)462 4991 y Fq(\(0\))462 5046 y Fi(n)577 5026 y Fr(is)g(the)g Fk(n)p Fr(-th)g(Laguerre)e(p)r (olynomial.)36 b(F)-7 b(or)26 b(details)f(on)h(Laguerre)e(p)r (olynomials,)i(see)g([26)o(])g(for)g(notions)166 5125 y(on)h(these)h(p)r(olynomials.)291 5225 y(Con)n(v)n(ersely)-7 b(,)25 b(the)j(expansion)f(of)g Fk(g)1390 5237 y Fi(n)1435 5225 y Fr(\()p Fk(t)p Fr(\))h(in)g(the)g(basis)f Fj(f)p Fk(f)2084 5237 y Fi(k)2124 5225 y Fr(\()p Fk(t)p Fr(\))p Fj(g)2260 5191 y Fi(n)2260 5245 y(k)q Fq(=1)2385 5225 y Fr(,)h(is)1016 5471 y Fk(g)1056 5483 y Fi(n)1101 5471 y Fr(\()p Fk(t)p Fr(\))c(=)1346 5367 y Fi(n)1307 5392 y Fe(X)1306 5571 y Fi(k)q Fq(=1)1441 5471 y Fk(\022)1480 5483 y Fg(n;k)1572 5471 y Fk(f)1613 5483 y Fi(k)1653 5471 y Fr(\()p Fk(t)p Fr(\))29 b(where)e Fk(\022)2055 5483 y Fg(n;k)2170 5471 y Fr(=)2257 5396 y Fe(p)p 2340 5396 135 4 v 75 x Fr(2)p Fk(=\014)t Fr(\()p Fj(\000)p Fr(2\))2646 5437 y Fi(k)q Ff(\000)p Fq(1)2771 5471 y Fj(C)2820 5435 y Fi(k)q Ff(\000)p Fq(1)2815 5493 y Fi(n)p Ff(\000)p Fq(1)2946 5471 y Fk(:)p eop %%Page: 5 5 5 4 bop 3778 -149 a Fr(5)166 206 y(Hence)28 b(w)n(e)f(obtain)g Fk(R)856 218 y Fg(X)910 206 y Fr(\()p Fk(t)p Fr(\))d(=)1119 102 y Fq(+)p Ff(1)1118 127 y Fe(X)1115 303 y Fi(n)p Fq(=1)1254 206 y Fk(a)1298 218 y Fi(n)1343 206 y Fk(g)1383 218 y Fi(n)1428 206 y Fr(\()p Fj(j)p Fk(t)p Fj(j)p Fr(\))k(in)g Fk(L)1750 172 y Fq(2)1787 206 y Fr(\()p Fh(R)p Fr(\))34 b(,)28 b(with)780 540 y Fk(a)824 552 y Fi(n)892 540 y Fr(=)979 427 y Fe(Z)1026 615 y Fd(R)1073 623 y Fc(+)1132 540 y Fk(R)1195 552 y Fg(X)1249 540 y Fr(\()p Fk(t)p Fr(\))p Fk(g)1383 552 y Fi(n)1428 540 y Fr(\()p Fk(t)p Fr(\))p Fk(dt)c Fr(=)1747 436 y Fi(n)1707 461 y Fe(X)1707 640 y Fi(k)q Fq(=1)1841 540 y Fk(\022)1880 552 y Fg(n;k)1986 427 y Fe(Z)2032 615 y Fd(R)2079 623 y Fc(+)2139 540 y Fk(R)2202 552 y Fg(X)2255 540 y Fr(\()p Fk(t)p Fr(\))p Fk(f)2390 552 y Fi(k)2432 540 y Fr(\()p Fk(t)p Fr(\))p Fk(dt)f Fr(=)2750 436 y Fi(n)2710 461 y Fe(X)2710 640 y Fi(k)q Fq(=1)2845 540 y Fk(\022)2884 552 y Fg(n;k)2975 540 y Fk(\032)3018 552 y Fg(X)3072 540 y Fr(\()p Fk(k)s Fr(\))p Fk(:)166 797 y Fr(And)28 b(the)g(sp)r(ectral)f(densit)n(y)h (can)f(b)r(e)h(written)g(as)1314 1017 y Fk(G)1379 1029 y Fg(X)1433 1017 y Fr(\()p Fk(\025)p Fr(\))84 b(=)1777 904 y Fe(Z)1823 1093 y Fd(R)1883 1017 y Fr(exp\()p Fj(\000)p Fr(2)p Fk(i\031)s(\025u)p Fr(\))p Fk(R)2419 1029 y Fg(Y)2468 1017 y Fr(\()p Fk(u)p Fr(\))p Fk(du)1629 1274 y Fr(=)1781 1171 y Fq(+)p Ff(1)1779 1195 y Fe(X)1777 1371 y Fi(n)p Fq(=1)1916 1274 y Fk(a)1960 1286 y Fi(n)2005 1274 y Fk(G)2070 1286 y Fi(n)2115 1274 y Fr(\()p Fk(\025)p Fr(\))p Fk(;)166 1608 y Fr(with)28 b Fk(G)420 1620 y Fi(n)466 1608 y Fr(\()p Fk(\025)p Fr(\))c(=)690 1495 y Fe(Z)736 1684 y Fd(R)783 1692 y Fc(+)843 1608 y Fr(exp)o(\()p Fj(\000)p Fr(2)p Fk(i\031)s(\025u)p Fr(\))p Fk(g)1355 1620 y Fi(n)1400 1608 y Fr(\()p Fj(j)p Fk(u)p Fj(j)p Fr(\))p Fk(du)p Fr(.)37 b(But)28 b Fk(t)1904 1620 y Fi(k)q Fq(+1)2047 1608 y Fj(\000)18 b Fk(t)2160 1620 y Fi(k)2225 1608 y Fj(\030)k Fk(f)2353 1620 y Fi(\013)2396 1628 y Fg(n)2469 1608 y Fr(and)27 b Fk(t)2660 1620 y Fi(k)q Fq(+)p Fi(n)2812 1608 y Fj(\000)18 b Fk(t)2925 1620 y Fi(k)2989 1608 y Fr(=)3116 1504 y Fi(n)3077 1529 y Fe(X)3083 1706 y Fi(i)p Fq(=1)3197 1608 y Fr(\()p Fk(t)3259 1620 y Fi(k)q Fq(+)p Fi(i)3393 1608 y Fj(\000)g Fk(t)3506 1620 y Fi(k)3547 1608 y Fr(\))28 b(where)166 1785 y(for)36 b(eac)n(h)h Fk(i)h Fj(2)h(f)p Fr(1)p Fk(;)14 b(:)g(:)g(:)f(;)h(n)p Fj(g)p Fr(,)38 b(the)g(random)e(v)-5 b(ariables)35 b(\()p Fk(t)1963 1797 y Fi(k)q Fq(+)p Fi(i)2103 1785 y Fj(\000)25 b Fk(t)2223 1797 y Fi(k)2264 1785 y Fr(\))37 b(are)f(i.i.d)h(with)h (distribution)f Fk(E)5 b(xp)p Fr(\()p Fk(\014)t Fr(\).)66 b(The)166 1885 y(c)n(haracteristic)27 b(function)j(of)e(an)h(exp)r (onen)n(tial)g(distribution)g(of)f(parameter)g Fk(\014)t Fr(,)i(is)f Fk(\036)p Fr(\()p Fk(\025)p Fr(\))d(=)f Fk(\014)t(=)p Fr(\()p Fk(\014)f Fj(\000)19 b Fk(i\025)p Fr(\),)30 b(th)n(us)f(the)166 1984 y(c)n(haracteristic)d(function)i(of)f Fk(t)1126 1996 y Fi(k)q Fq(+)p Fi(n)1278 1984 y Fj(\000)18 b Fk(t)1391 1996 y Fi(k)1459 1984 y Fr(is)1181 2232 y Fk(\036)1230 2244 y Fi(t)1255 2253 y Fg(k)q Fc(+)p Fg(n)1371 2244 y Ff(\000)p Fi(t)1448 2253 y Fg(k)1488 2232 y Fr(\()p Fk(\025)p Fr(\))24 b(=)1745 2128 y Fi(n)1713 2153 y Fe(Y)1712 2330 y Fi(i)p Fq(=1)1833 2232 y Fk(\036)1882 2244 y Fi(t)1907 2253 y Fg(k)q Fc(+)p Fg(i)2009 2244 y Ff(\000)p Fi(t)2086 2253 y Fg(k)2126 2232 y Fr(\()p Fk(\025)p Fr(\))h(=)2350 2115 y Fe(\022)2510 2176 y Fk(\014)p 2421 2213 230 4 v 2421 2289 a(\014)e Fj(\000)18 b Fk(i\025)2661 2115 y Fe(\023)2722 2132 y Fi(n)2781 2232 y Fk(:)166 2486 y Fr(Hence)1171 2715 y Fk(G)1236 2727 y Fi(n)1282 2715 y Fr(\()p Fk(\025)p Fr(\))84 b(=)1625 2602 y Fe(Z)1672 2790 y Fd(R)1732 2715 y Fr(exp)o(\()p Fj(\000)p Fr(2)p Fk(i\031)s(\025u)p Fr(\))2258 2611 y Fi(n)2218 2636 y Fe(X)2218 2815 y Fi(k)q Fq(=1)2352 2715 y Fk(\022)2391 2727 y Fg(n;k)2483 2715 y Fk(f)2524 2727 y Fi(k)2565 2715 y Fr(\()p Fj(j)p Fk(u)p Fj(j)p Fr(\))p Fk(du)1478 3001 y Fr(=)e(2)p Fj(R)p Fk(e)1790 2859 y Fe( )1895 2898 y Fi(n)1856 2922 y Fe(X)1856 3101 y Fi(k)q Fq(=1)1990 3001 y Fk(\022)2029 3013 y Fg(n;k)2121 3001 y Fk(\036)2170 3013 y Fi(t)2195 3022 y Fg(k)q Fc(+)p Fg(n)2311 3013 y Ff(\000)p Fi(t)2388 3022 y Fg(k)2429 3001 y Fr(\(2)p Fk(\031)s(\025)p Fr(\))2633 2859 y Fe(!)1478 3288 y Fr(=)g(2)p Fj(R)p Fk(e)1790 3146 y Fe( )1895 3184 y Fi(n)1856 3209 y Fe(X)1856 3388 y Fi(k)q Fq(=1)1990 3288 y Fk(\022)2029 3300 y Fg(n;k)2135 3171 y Fe(\022)2341 3232 y Fk(\014)p 2206 3269 322 4 v 2206 3345 a(\014)23 b Fj(\000)18 b Fr(2)p Fk(i\031)s(\025)2538 3171 y Fe(\023)2599 3188 y Fi(n)2644 3146 y Fe(!)166 3523 y Fr(or)860 3688 y Fk(G)925 3700 y Fi(n)970 3688 y Fr(\()p Fk(\025)p Fr(\))24 b(=)f(2\()p Fj(\000)p Fr(1\))1407 3653 y Fi(n)p Ff(\000)p Fq(1)1536 3538 y Fe(s)p 1619 3538 425 4 v 1788 3631 a Fk(b\014)p 1629 3668 405 4 v 1629 3744 a(\014)1680 3720 y Fq(2)1736 3744 y Fr(+)18 b(4)p Fk(\031)1911 3720 y Fq(2)1948 3744 y Fk(\025)1996 3720 y Fq(2)2058 3688 y Fr(cos)13 b(\(\(2)p Fk(n)18 b Fj(\000)g Fr(1\))c(arctan)o(\(2)p Fk(\031)s(\025=\014)t Fr(\)\))g Fk(:)166 3972 y Fm(Remark)31 b(3)41 b Fr(Some)30 b(simple)h(algebra)d(sho)n(ws)i(that)1821 3859 y Fe(Z)1867 4047 y Fd(R)1927 3972 y Fk(G)1992 3984 y Fi(n)2038 3972 y Fr(\()p Fk(\025)p Fr(\))p Fk(G)2215 3984 y Fi(p)2254 3972 y Fr(\()p Fk(\025)p Fr(\))p Fk(d\025)g Fr(=)d(2)p Fk(\016)2658 3984 y Fi(n;p)2756 3972 y Fr(,)32 b(where)d Fk(\016)3090 3984 y Fi(n;p)3220 3972 y Fr(is)h(the)h(Kronec)n(k)n(er) 166 4113 y(sym)n(b)r(ol.)166 4279 y(The)d(expansion)e(of)i(the)g(sp)r (ectral)f(densit)n(y)g(in)h(this)g(basis)f(b)r(ecomes)1413 4535 y Fk(G)1478 4547 y Fg(X)1531 4535 y Fr(\()p Fk(\025)p Fr(\))d(=)1759 4431 y Fq(+)p Ff(1)1758 4456 y Fe(X)1755 4632 y Fi(n)p Fq(=1)1894 4535 y Fk(a)1938 4547 y Fi(n)1983 4535 y Fk(G)2048 4547 y Fi(n)2094 4535 y Fr(\()p Fk(\025)p Fr(\))k(in)g Fk(L)2388 4501 y Fq(2)2425 4535 y Fr(\()p Fh(R)p Fr(\))p Fk(;)166 4868 y Fr(with)g Fk(a)399 4880 y Fi(n)467 4868 y Fr(=)595 4765 y Fi(n)555 4790 y Fe(X)555 4968 y Fi(k)q Fq(=1)690 4868 y Fk(\022)729 4880 y Fg(n;k)820 4868 y Fk(\032)863 4880 y Fg(X)917 4868 y Fr(\()p Fk(k)s Fr(\))g(and)f Fk(\022)1255 4880 y Fg(n;k)1370 4868 y Fr(=)1458 4793 y Fe(p)p 1541 4793 135 4 v 75 x Fr(2)p Fk(=\014)s Fr(\()p Fj(\000)p Fr(2\))1846 4834 y Fi(k)q Ff(\000)p Fq(1)1972 4868 y Fj(C)2021 4833 y Fi(k)q Ff(\000)p Fq(1)2016 4891 y Fi(n)p Ff(\000)p Fq(1)2146 4868 y Fr(.)291 5085 y(F)-7 b(or)28 b(estimating)h Fk(G)914 5097 y Fg(X)968 5085 y Fr(,)g(let)g(us)g(b)r(e)h(giv)n(en)e(the)h(observ)-5 b(ations)28 b Fk(X)7 b Fr(\()p Fk(t)2340 5097 y Fi(k)2380 5085 y Fr(\))p Fk(;)28 b(k)g Fr(=)e(1)p Fk(;)14 b(:)g(:)g(:)f(;)h(N)37 b Fr(for)29 b Fk(N)34 b(>)25 b Fr(0.)41 b(First,)29 b Fk(\032)3651 5097 y Fg(X)3705 5085 y Fr(\()p Fk(n)p Fr(\))166 5184 y(ma)n(y)e(b)r(e)h(estimated)g(b)n(y)874 5418 y Fe(b)-51 b Fk(\032)912 5430 y Fg(X;n)1018 5418 y Fr(\()p Fk(N)9 b Fr(\))23 b(=)1269 5301 y Fe(\032)1396 5340 y Fq(1)p 1383 5354 59 4 v 1383 5402 a Fi(N)1465 5311 y Fe(P)1553 5331 y Fi(N)6 b Ff(\000)p Fi(n)1553 5398 y(k)q Fq(=1)1723 5373 y Fk(X)h Fr(\()p Fk(t)1861 5385 y Fi(k)q Fq(+)p Fi(n)1993 5373 y Fr(\))p Fk(X)g Fr(\()p Fk(t)2163 5385 y Fi(k)2204 5373 y Fr(\))111 b(if)g(1)23 b Fj(\024)f Fk(n)h(<)g(N)1373 5473 y Fr(0)1119 b(otherwise)o Fk(:)p eop %%Page: 6 6 6 5 bop 3778 -149 a Fr(6)166 141 y(Then,)28 b(w)n(e)f(estimate)h Fk(a)905 153 y Fi(n)977 141 y Fr(b)n(y)1452 386 y Fe(b)-45 b Fk(a)1497 398 y Fg(n)1537 386 y Fr(\()p Fk(N)9 b Fr(\))24 b(=)1828 282 y Fi(n)1789 307 y Fe(X)1788 486 y Fi(k)q Fq(=1)1923 386 y Fk(\022)1962 398 y Fg(n;k)2059 386 y Fe(b)-51 b Fk(\032)2097 398 y Fg(X;n)2203 386 y Fr(\()p Fk(N)9 b Fr(\))p Fk(:)166 643 y Fr(Th)n(us,)27 b(the)h(sp)r(ectral)f (densit)n(y)h(estimate)f(is)h(\014nally)f(giv)n(en)g(b)n(y)1310 882 y Fe(b)1294 903 y Fk(G)1359 915 y Fg(X;n)1465 903 y Fr(\()p Fk(\025)p Fr(\))d(=)1693 798 y Fi(M)1756 806 y Fg(N)1691 824 y Fe(X)1689 1000 y Fi(n)p Fq(=1)1828 903 y Fk(y)1869 915 y Fi(n)1914 903 y Fr(\()p Fk(N)9 b Fr(\))o Fe(b)-45 b Fk(a)2098 915 y Fg(n)2139 903 y Fr(\()p Fk(N)9 b Fr(\))p Fk(G)2344 915 y Fi(n)2389 903 y Fr(\()p Fk(\025)p Fr(\))p Fk(;)1189 b Fr(\(3\))166 1174 y(with)32 b(the)g(bandwidth)g Fk(M)1006 1186 y Fg(N)1088 1174 y Fr(=)d([)p Fk(b)14 b Fr(ln\()p Fk(N)9 b Fr(\))p Fk(=)p Fr(\(2)p Fk(a)p Fr(\)])21 b(+)f(1)31 b(and)g(with)h Fk(y)2257 1186 y Fi(n)2302 1174 y Fr(\()p Fk(N)9 b Fr(\))30 b(=)f Fk(h)2628 1107 y Fe(\000)2666 1174 y Fr(exp\()p Fk(\013n)p Fr(\))p Fk(=)-5 b(N)3073 1140 y Fi(b)3106 1107 y Fe(\001)3144 1174 y Fr(,)33 b(where)e Fk(\013)f(>)f Fr(ln\(3\),)166 1274 y(0)23 b Fk(<)f(b\013=)p Fr(\(2)14 b(ln\(3\)\),)27 b(and)e(the)i(real)e(function)i Fk(h)f Fr(is)g(a)f(Lipsc)n(hitz)h(function)h(of)f(order)e(1)i(suc)n(h)g(that)g Fk(h)p Fr(\()p Fk(u)p Fr(\))d Fj(\024)g Fk(h)p Fr(\(0\))g(=)f(1,)166 1374 y Fj(8)p Fk(u)g Fj(2)h Fh(R)p Fr(.)166 1648 y Fl(4)135 b(Mean)45 b(square)g(consistency)166 1830 y Fr(F)-7 b(or)29 b(studying)f(the)i(asymptotic)f(b)r(eha)n(viour)e(of)i(the)h(sp)r (ectral)e(densit)n(y)h(estimate)g(\(3\),)h(w)n(e)f(will)g(assume)f (that)i(the)166 1930 y(follo)n(wing)d(conditions)g(are)f(ful\014lled.) 374 2096 y Fa(Condition)31 b(1.)36 b Fr(Let)28 b Fk(AC)1214 2061 y Fi(r)r Ff(\000)p Fq(1)1336 2096 y Fr([1)p Fk(;)14 b Fr(+)p Fj(1)p Fr([)27 b(b)r(e)h(the)h(set)e(of)h Fk(r)r 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Fk(a)586 5182 y Fg(n)627 5170 y Fr(\()p Fk(N)9 b Fr(\))p Fj(g)22 b(\024)h Fk(A)981 5182 y Fq(3)1029 5113 y Fr(3)1071 5083 y Fq(2)p Fi(n)p 1029 5151 120 4 v 1051 5227 a Fk(N)1188 5170 y Fp(with)30 b Fk(A)1430 5182 y Fq(3)1491 5170 y Fr(=)1589 5113 y(2)p Fk(A)1693 5125 y Fq(2)p 1589 5151 142 4 v 1613 5227 a Fr(9)p Fk(\014)1740 5170 y Fp(.)166 5398 y Fr(The)e(mean-square)d(error)h(of)i(the)g (estimate)1610 5377 y Fe(b)1593 5398 y Fk(G)1658 5410 y Fg(X;n)1792 5398 y Fr(is)g(giv)n(en)f(in)g(the)h(follo)n(wing)f (theorem.)p eop %%Page: 7 7 7 6 bop 3778 -149 a Fr(7)166 141 y Fm(Theorem)30 b(4)42 b Fp(If)30 b(Condition)h(1)f(and)h(Condition)g(2)f(ar)l(e)g(satis\014e) l(d,)h(we)f(have)1233 386 y Fk(I)-21 b(E)5 b Fj(f)1372 365 y Fe(b)1356 386 y Fk(G)1421 398 y Fg(X;n)1527 386 y Fr(\()p Fk(\025)p Fr(\))19 b Fj(\000)f Fk(G)1806 398 y Fg(X)1860 386 y Fr(\()p Fk(\025)p Fr(\))p Fj(g)2014 352 y Fq(2)2075 386 y Fr(=)23 b Fk(O)2242 269 y Fe(\022)2397 330 y Fr(1)p 2313 367 210 4 v 2313 443 a(ln\()p Fk(N)9 b Fr(\))2533 269 y Fe(\023)2594 286 y Fi(r)r Ff(\000)p Fq(2)2729 386 y Fk(:)166 614 y Fr(And)28 b(\014nally)g(the)g(mean)f (square)f(error)g(is)h(giv)n(en)g(b)r(elo)n(w.)166 780 y Fm(Theorem)j(5)42 b Fp(Under)31 b(the)h(same)g(c)l(onditions,)i(the)d (inte)l(gr)l(ate)l(d)h(me)l(an)g(squar)l(e)f(err)l(or)h(of)h(the)e (estimate)3499 759 y Fe(b)3482 780 y Fk(G)3547 792 y Fg(X;n)3685 780 y Fp(sat-)166 879 y(is\014es)1089 1041 y Fk(I)-21 b(E)1171 974 y Fe(\002)1219 928 y(Z)1302 949 y Fq(+)p Ff(1)1265 1117 y(\0001)1437 1041 y Fj(j)1477 1020 y Fe(b)1460 1041 y Fk(G)1525 1053 y Fg(X;n)1632 1041 y Fr(\()p Fk(\025)p Fr(\))19 b Fj(\000)f Fk(\036)p Fr(\()p Fk(\025)p Fr(\))p Fj(j)2030 1007 y Fq(2)2069 1041 y Fk(d\025)2160 974 y Fe(\003)2218 1041 y Fr(=)23 b Fk(O)2385 924 y Fe(\022)2541 985 y Fr(1)p 2457 1022 V 2457 1098 a(ln\()p Fk(N)9 b Fr(\))2676 924 y Fe(\023)2737 941 y Fi(r)r Ff(\000)p Fq(1)2873 1041 y Fk(:)166 1340 y Fl(5)135 b(Short)45 b(sim)l(ulation)h(study)166 1522 y Fr(The)27 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Fk(\025)2726 2076 y Fq(2)2764 2100 y Fr(\))2806 2043 y Fk(;)884 b Fr(\(4\))166 2274 y(Eq.)55 b(\(4\))33 b(is)h(the)g(lo)n(wpass)e(sp)r(ectral)h(densit)n(y)h(of)f(the)h (Gauss-Mark)n(o)n(v)d(pro)r(cess.)54 b(W)-7 b(e)34 b(will)g(use)f(t)n (w)n(o)g(n)n(um)n(b)r(ers)g(of)166 2374 y(samples)27 b Fk(N)32 b Fr(=)22 b(500)27 b(and)g Fk(N)32 b Fr(=)23 b(1000,)i(and)j(consider)e(P)n(arzen's)g(sp)r(ectral)h(windo)n(w)g(for) g(whic)n(h)1092 2652 y Fk(h)p Fr(\()p Fk(t)p Fr(\))c(=)1345 2481 y Fe(8)1345 2556 y(<)1345 2706 y(:)1460 2551 y Fr(1)18 b Fj(\000)g Fr(6)p Fk(t)1675 2521 y Fq(2)1730 2551 y Fr(+)g(6)p Fj(j)p Fk(t)p Fj(j)1931 2521 y Fq(3)1968 2551 y Fk(;)111 b Fr(if)g Fj(j)p Fk(t)p Fj(j)23 b(\024)2458 2518 y Fq(1)p 2458 2532 34 4 v 2458 2580 a(2)2501 2551 y Fk(;)1460 2651 y Fr(2\(1)18 b Fj(\000)g(j)p Fk(t)p Fj(j)p Fr(\))1785 2621 y Fq(3)1822 2651 y Fk(;)257 b Fr(if)2271 2619 y Fq(1)p 2271 2633 V 2271 2680 a(2)2337 2651 y Fk(<)23 b Fj(j)p Fk(t)p Fj(j)g(\024)f Fr(1)p Fk(;)1460 2751 y Fr(0)p 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3739 y Fi(k)3543 3727 y Fj(g)3585 3739 y Fi(k)3655 3727 y Fr(w)n(ere)166 3826 y(obtained)d(b)n(y)1445 3960 y Fk(\013)1498 3972 y Fi(k)1562 3960 y Fr(=)22 b Fj(\000)1729 3904 y Fr(1)p 1724 3941 52 4 v 1724 4017 a Fk(\014)1799 3960 y Fr(ln\()p Fk(\022)1939 3972 y Fi(k)1980 3960 y Fr(\))p Fk(;)28 b(k)e Fr(=)d(1)p Fk(;)14 b(:)g(:)g(:)f(;)h(N)t(;)166 4166 y Fr(where)32 b(the)g Fj(f)p Fk(\022)639 4178 y Fi(k)680 4166 y Fj(g)722 4178 y Fi(k)794 4166 y Fr(are)f(uniform)i(on)e ([0)p Fk(;)14 b Fr(1])32 b(random)f(n)n(um)n(b)r(ers.)50 b(The)32 b(observ)-5 b(ations)31 b Fj(f)p Fk(X)7 b Fr(\()p Fk(t)3096 4178 y Fi(k)3136 4166 y Fr(\))p Fj(g)3210 4131 y Fi(N)3210 4186 y(k)q Fq(=1)3367 4166 y Fr(are)31 b(then)i(ob-)166 4265 y(tained)28 b(via)f(the)h(dynamical)f(represen)n(tation)f(for)h (pro)r(cess)f Fk(X)34 b Fr(\(see,)28 b([)p Fm(?)p Fr(]\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e)760 4506 y Fk(X)7 b Fr(\()p Fk(t)p Fr(\))23 b(=)g(exp)o(\()p Fj(\000)p Fr(\()p Fk(t)c Fj(\000)f Fk(\034)9 b Fr(\)\))p Fk(X)e Fr(\()p Fk(\034)i Fr(\))20 b(+)1825 4433 y Fj(p)p 1894 4433 42 4 v 73 x Fr(2)1950 4393 y Fe(Z)2033 4413 y Fi(t)1996 4581 y(\034)2076 4506 y Fr(exp\()p Fj(\000)p Fr(\()p Fk(t)e Fj(\000)g Fk(s)p Fr(\)\))p Fk(dW)12 b Fr(\()p Fk(s)p Fr(\))p Fk(;)38 b(t)23 b(>)g(\034)33 b Fj(\025)22 b Fr(0)p Fk(;)166 4735 y Fr(where)32 b Fj(f)p Fk(W)12 b Fr(\()p Fk(t)p Fr(\))p Fk(;)46 b(t)31 b Fj(\025)h Fr(0)p Fj(g)f Fr(is)i(the)g(real)f(v)-5 b(alued)33 b(Wiener)f(pro)r(cess)g(with)h(zero)f(mean,)i(and)e(co)n(v) -5 b(ariance)31 b Fk(R)3499 4747 y Fg(X)3552 4735 y Fr(\()p Fk(t;)14 b(s)p Fr(\))32 b(=)166 4834 y(min\()p Fk(t;)14 b(s)p Fr(\).)p eop %%Page: 8 8 8 7 bop 3778 -149 a Fr(8)103 1475 y @beginspecial 17 @llx 180 @lly 593 @urx 611 @ury 2267 @rwi 1700 @rhi @setspecial %%BeginDocument: cn500.ps % MathWorks dictionary /MathWorks 120 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc /setrgbcolor ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % bounding box in default coordinates %initialize variables used in binded macros /llx 0 def /lly 0 def /urx 0 def /ury 0 def /bbox {/ury xdef /urx xdef /lly xdef /llx xdef} bdef % orientation switch /portraitMode (op) def /landscapeMode (ol) def /Orientation portraitMode def /portrait {/Orientation portraitMode def} bdef /landscape {/Orientation landscapeMode def} bdef % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize dpi2point mul 0 0 FontSize dpi2point mul neg 0 0] makefont setfont }bdef /setPortrait { 1 dpi2point div -1 dpi2point div scale llx ury neg translate } bdef /setLandscape { 1 dpi2point div -1 dpi2point div scale urx ury neg translate 90 rotate } bdef /csm {Orientation portraitMode eq {setPortrait} {setLandscape} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { % LineTo lineto stroke } bdef /MP { % MakePoly 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { % AddPoly {rlineto} repeat } bdef /PP { % PaintPoly closepath fill } bdef /DP { % DrawPoly closepath stroke } bdef /MR { % MakeRect 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { % FrameRect MR stroke } bdef /PR { % PaintRect MR fill } bdef /L1i { % Level 1 Image { currentfile picstr readhexstring pop } image } bdef /half_width 0 def /half_height 0 def /MakeOval { newpath /ury xstore /urx xstore /lly xstore /llx xstore /half_width urx llx sub 2 div store /half_height ury lly sub 2 div store llx half_width add lly half_height add translate half_width half_height scale .5 half_width div setlinewidth 0 0 1 0 360 arc } bdef /FO { gsave MakeOval stroke grestore } bdef /PO { gsave MakeOval fill grestore } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def %dictionary MathWorks begin % fonts for text, standard numbers and exponents %line width, line cap, and joint spec 0 setlinecap 1 setlinejoin end MathWorks begin bpage bplot /dpi2point 12 def 0204 2160 7116 7332 bbox portrait csm 0 0 6912 5183 MR c np % Colortable Dictionary /colortable 76 dict begin /c0 { 0 0 0 sc} bdef /c1 { 1 1 1 sc} bdef /c2 { 1 0 0 sc} bdef /c3 { 0 1 0 sc} bdef /c4 { 0 0 1 sc} bdef /c5 { 1 1 0 sc} bdef /c6 { 1 0 1 sc} bdef /c7 { 0 1 1 sc} bdef currentdict end def % Colortable colortable begin /Helvetica 12 FMS c1 0 0 6912 5183 PR 6 setlinewidth DO SO c0 898 4612 mt 6220 4612 L 898 414 mt 6220 414 L 898 4612 mt 898 414 L 6220 4612 mt 6220 414 L 898 414 mt 898 414 L 6220 4612 mt 6220 4612 L 898 4612 mt 6220 4612 L 898 4612 mt 898 414 L 898 4612 mt 898 4612 L 898 4612 mt 898 4559 L 898 414 mt 898 467 L 858 4773 mt (0) s 1658 4612 mt 1658 4559 L 1658 414 mt 1658 467 L 1618 4773 mt (1) s 2419 4612 mt 2419 4559 L 2419 414 mt 2419 467 L 2379 4773 mt (2) s 3179 4612 mt 3179 4559 L 3179 414 mt 3179 467 L 3139 4773 mt (3) s 3939 4612 mt 3939 4559 L 3939 414 mt 3939 467 L 3899 4773 mt (4) s 4699 4612 mt 4699 4559 L 4699 414 mt 4699 467 L 4659 4773 mt (5) s 5460 4612 mt 5460 4559 L 5460 414 mt 5460 467 L 5420 4773 mt (6) s 6220 4612 mt 6220 4559 L 6220 414 mt 6220 467 L 6180 4773 mt (7) s 898 4612 mt 951 4612 L 6220 4612 mt 6167 4612 L 791 4665 mt (0) s 898 3912 mt 951 3912 L 6220 3912 mt 6167 3912 L 671 3965 mt (0.1) s 898 3213 mt 951 3213 L 6220 3213 mt 6167 3213 L 671 3266 mt (0.2) s 898 2513 mt 951 2513 L 6220 2513 mt 6167 2513 L 671 2566 mt (0.3) s 898 1813 mt 951 1813 L 6220 1813 mt 6167 1813 L 671 1866 mt (0.4) s 898 1114 mt 951 1114 L 6220 1114 mt 6167 1114 L 671 1167 mt (0.5) s 898 414 mt 951 414 L 6220 414 mt 6167 414 L 671 467 mt (0.6) s 898 4612 mt 6220 4612 L 898 414 mt 6220 414 L 898 4612 mt 898 414 L 6220 4612 mt 6220 414 L 898 4612 mt 898 4612 L 6220 414 mt 6220 414 L gs 898 414 5323 4199 MR c np DO 10 2 9 1 10 2 9 1 10 2 9 1 10 2 9 1 10 1 10 2 9 1 10 1 9 1 10 1 9 1 10 1 9 1 10 1 10 0 9 1 10 0 9 1 10 0 9 0 10 0 9 0 10 0 9 -1 10 0 10 0 9 -1 10 -1 9 0 10 -1 9 -1 10 -1 9 -1 10 -1 10 -2 9 -1 10 -1 9 -1 10 -2 9 -1 10 -1 9 -2 10 -1 10 -1 9 -1 10 -1 9 -2 10 -1 9 0 10 -1 9 -1 10 -1 10 0 9 0 10 -1 9 0 10 0 9 0 10 1 9 0 10 1 10 0 9 1 10 1 9 1 10 1 9 1 10 2 9 1 10 1 10 2 9 1 10 2 9 1 10 2 9 1 10 2 9 1 10 2 10 1 9 1 10 1 9 1 10 1 9 1 10 0 9 1 10 0 10 0 9 0 10 -1 9 0 10 -1 9 -1 10 -1 4729 4283 100 MP stroke 9 -2 10 -2 9 -2 10 -2 10 -2 9 -3 10 -3 9 -3 10 -4 9 -3 10 -4 9 -4 10 -4 10 -5 9 -4 10 -5 9 -5 10 -5 9 -6 10 -5 9 -6 10 -6 10 -6 9 -6 10 -6 9 -6 10 -6 9 -7 10 -6 9 -6 10 -7 10 -7 9 -6 10 -7 9 -7 10 -6 9 -7 10 -7 9 -6 10 -7 10 -6 9 -7 10 -6 9 -7 10 -6 9 -6 10 -6 9 -6 10 -5 10 -6 9 -5 10 -6 9 -5 10 -4 9 -5 10 -4 9 -4 10 -4 10 -3 9 -4 10 -3 9 -2 10 -2 9 -2 10 -2 9 -1 10 -1 10 0 9 0 10 0 9 0 10 1 9 1 10 2 9 2 10 2 9 3 10 2 10 3 9 4 10 3 9 4 10 4 9 4 10 4 9 4 10 4 10 5 9 4 10 4 9 5 10 4 9 4 10 4 9 4 10 3 10 4 9 3 10 3 3783 4508 100 MP stroke 9 2 10 3 9 2 10 1 9 2 10 1 10 0 9 1 10 0 9 -1 10 0 9 -1 10 -2 9 -2 10 -2 10 -2 9 -2 10 -3 9 -3 10 -3 9 -3 10 -3 9 -4 10 -3 10 -4 9 -3 10 -3 9 -4 10 -3 9 -3 10 -2 9 -3 10 -2 10 -2 9 -2 10 -1 9 -1 10 -1 9 0 10 0 9 0 10 1 10 1 9 2 10 2 9 3 10 3 9 3 10 4 9 4 10 4 9 5 10 5 10 6 9 6 10 6 9 6 10 6 9 7 10 7 9 7 10 7 10 7 9 8 10 7 9 8 10 7 9 8 10 7 9 7 10 8 10 7 9 8 10 7 9 7 10 7 9 7 10 7 9 7 10 7 10 7 9 7 10 6 9 7 10 7 9 6 10 7 9 6 10 7 10 6 9 7 10 6 9 7 10 7 9 6 10 7 9 7 10 6 10 7 2837 4212 100 MP stroke 9 7 10 7 9 7 10 7 9 7 10 7 9 7 10 7 10 7 9 7 10 6 9 7 10 7 9 7 10 6 9 7 10 6 10 6 9 7 10 6 9 5 10 6 9 6 10 5 9 5 10 5 9 5 10 4 10 5 9 4 10 4 9 4 10 4 9 4 10 3 9 4 10 3 10 4 9 3 10 4 9 3 10 3 9 4 10 3 9 4 10 4 10 4 9 4 10 5 9 4 10 5 9 5 10 6 9 6 10 6 10 6 9 7 10 7 9 7 10 8 9 8 10 8 9 9 10 9 10 10 9 10 10 10 9 10 10 11 9 11 10 11 9 12 10 12 10 12 9 12 10 13 9 12 10 13 9 13 10 13 9 13 10 13 10 14 9 13 10 14 9 14 10 13 9 14 10 14 9 14 10 14 10 14 9 14 10 14 9 14 10 15 9 14 10 15 9 15 1892 3405 100 MP stroke 10 14 9 15 10 16 10 15 9 16 10 15 9 16 10 17 9 16 10 17 9 17 10 17 10 18 9 18 10 18 9 19 10 18 9 20 10 19 9 20 10 20 10 21 9 21 10 21 9 21 10 22 9 22 10 23 9 23 10 23 10 23 9 24 10 23 9 25 10 24 9 25 10 25 9 25 10 26 10 26 9 26 10 26 9 27 10 27 9 27 10 27 9 28 10 28 10 28 9 29 10 29 9 29 10 30 9 30 10 30 9 31 10 31 10 31 9 32 10 32 9 32 10 32 9 33 10 34 9 33 10 34 10 34 9 34 10 34 9 35 10 34 9 35 10 34 9 35 10 34 9 34 10 34 10 33 9 34 10 32 9 32 10 32 9 31 10 30 9 29 10 29 10 27 9 26 10 25 9 24 10 23 9 21 10 20 9 18 10 17 10 15 9 14 10 12 9 10 946 914 100 MP stroke 10 8 9 6 10 5 9 3 908 892 5 MP stroke SO 10 0 9 0 10 1 9 0 10 0 9 0 10 1 9 0 10 0 10 0 9 0 10 1 9 0 10 0 9 0 10 1 9 0 10 0 10 0 9 1 10 0 9 0 10 0 9 1 10 0 9 0 10 0 9 1 10 0 10 0 9 0 10 1 9 0 10 0 9 0 10 1 9 0 10 0 10 0 9 1 10 0 9 0 10 1 9 0 10 0 9 0 10 1 10 0 9 0 10 1 9 0 10 0 9 0 10 1 9 0 10 0 10 1 9 0 10 0 9 1 10 0 9 0 10 0 9 1 10 0 10 0 9 1 10 0 9 0 10 1 9 0 10 0 9 1 10 0 10 0 9 1 10 0 9 0 10 1 9 0 10 1 9 0 10 0 10 1 9 0 10 0 9 1 10 0 9 0 10 1 9 0 10 1 10 0 9 0 10 1 9 0 10 1 9 0 10 0 4729 4528 100 MP stroke 9 1 10 0 9 1 10 0 10 0 9 1 10 0 9 1 10 0 9 1 10 0 9 0 10 1 10 0 9 1 10 0 9 1 10 0 9 1 10 0 9 1 10 0 10 1 9 0 10 0 9 1 10 0 9 1 10 0 9 1 10 0 10 1 9 0 10 1 9 1 10 0 9 1 10 0 9 1 10 0 10 1 9 0 10 1 9 0 10 1 9 1 10 0 9 1 10 0 10 1 9 0 10 1 9 1 10 0 9 1 10 1 9 0 10 1 10 0 9 1 10 1 9 0 10 1 9 1 10 0 9 1 10 1 10 0 9 1 10 1 9 0 10 1 9 1 10 1 9 0 10 1 9 1 10 0 10 1 9 1 10 1 9 0 10 1 9 1 10 1 9 1 10 0 10 1 9 1 10 1 9 1 10 1 9 0 10 1 9 1 10 1 10 1 9 1 10 1 3783 4467 100 MP stroke 9 0 10 1 9 1 10 1 9 1 10 1 10 1 9 1 10 1 9 1 10 1 9 1 10 1 9 1 10 1 10 1 9 1 10 1 9 1 10 1 9 1 10 1 9 2 10 1 10 1 9 1 10 1 9 1 10 1 9 2 10 1 9 1 10 1 10 1 9 2 10 1 9 1 10 2 9 1 10 1 9 1 10 2 10 1 9 1 10 2 9 1 10 2 9 1 10 1 9 2 10 1 9 2 10 1 10 2 9 1 10 2 9 2 10 1 9 2 10 1 9 2 10 2 10 1 9 2 10 2 9 1 10 2 9 2 10 2 9 1 10 2 10 2 9 2 10 2 9 2 10 2 9 2 10 2 9 2 10 2 10 2 9 2 10 2 9 2 10 2 9 2 10 2 9 3 10 2 10 2 9 2 10 3 9 2 10 2 9 3 10 2 9 3 10 2 10 3 2837 4315 100 MP stroke 9 2 10 3 9 2 10 3 9 3 10 2 9 3 10 3 10 3 9 3 10 2 9 3 10 3 9 3 10 3 9 3 10 4 10 3 9 3 10 3 9 3 10 4 9 3 10 4 9 3 10 3 9 4 10 4 10 3 9 4 10 4 9 3 10 4 9 4 10 4 9 4 10 4 10 4 9 4 10 5 9 4 10 4 9 5 10 4 9 5 10 4 10 5 9 5 10 4 9 5 10 5 9 5 10 5 9 5 10 5 10 6 9 5 10 6 9 5 10 6 9 5 10 6 9 6 10 6 10 6 9 6 10 6 9 6 10 7 9 6 10 7 9 6 10 7 10 7 9 7 10 7 9 7 10 7 9 8 10 7 9 8 10 7 10 8 9 8 10 8 9 8 10 9 9 8 10 9 9 8 10 9 10 9 9 9 10 9 9 10 10 9 9 10 10 10 9 9 1892 3790 100 MP stroke 10 10 9 11 10 10 10 11 9 10 10 11 9 11 10 11 9 11 10 12 9 11 10 12 10 12 9 12 10 12 9 13 10 13 9 12 10 13 9 13 10 14 10 13 9 14 10 14 9 14 10 14 9 14 10 15 9 14 10 15 10 15 9 16 10 15 9 16 10 15 9 16 10 16 9 16 10 17 10 16 9 17 10 17 9 17 10 17 9 17 10 17 9 18 10 17 10 18 9 18 10 18 9 18 10 18 9 18 10 18 9 18 10 18 10 18 9 18 10 19 9 18 10 18 9 18 10 18 9 18 10 18 10 17 9 18 10 17 9 17 10 17 9 17 10 17 9 16 10 16 9 16 10 15 10 15 9 15 10 14 9 14 10 14 9 13 10 13 9 12 10 12 10 11 9 11 10 10 9 9 10 9 9 9 10 7 9 7 10 7 10 6 9 5 10 5 9 3 946 2394 100 MP stroke 10 3 9 3 10 2 9 1 908 2385 5 MP stroke gr 1736 299 mt end % pop colortable dictionary eplot epage end showpage %%EndDocument @endspecial 1890 w @beginspecial 17 @llx 180 @lly 593 @urx 611 @ury 2267 @rwi 1700 @rhi @setspecial %%BeginDocument: cn1000.ps % MathWorks dictionary /MathWorks 120 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc /setrgbcolor ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % bounding box in default coordinates %initialize variables used in binded macros /llx 0 def /lly 0 def /urx 0 def /ury 0 def /bbox {/ury xdef /urx xdef /lly xdef /llx xdef} bdef % orientation switch /portraitMode (op) def /landscapeMode (ol) def /Orientation portraitMode def /portrait {/Orientation portraitMode def} bdef /landscape {/Orientation landscapeMode def} bdef % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize dpi2point mul 0 0 FontSize dpi2point mul neg 0 0] makefont setfont }bdef /setPortrait { 1 dpi2point div -1 dpi2point div scale llx ury neg translate } bdef /setLandscape { 1 dpi2point div -1 dpi2point div scale urx ury neg translate 90 rotate } bdef /csm {Orientation portraitMode eq {setPortrait} {setLandscape} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { % LineTo lineto stroke } bdef /MP { % MakePoly 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { % AddPoly {rlineto} repeat } bdef /PP { % PaintPoly closepath fill } bdef /DP { % DrawPoly closepath stroke } bdef /MR { % MakeRect 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { % FrameRect MR stroke } bdef /PR { % PaintRect MR fill } bdef /L1i { % Level 1 Image { currentfile picstr readhexstring pop } image } bdef /half_width 0 def /half_height 0 def /MakeOval { newpath /ury xstore /urx xstore /lly xstore /llx xstore /half_width urx llx sub 2 div store /half_height ury lly sub 2 div store llx half_width add lly half_height add translate half_width half_height scale .5 half_width div setlinewidth 0 0 1 0 360 arc } bdef /FO { gsave MakeOval stroke grestore } bdef /PO { gsave MakeOval fill grestore } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def %dictionary MathWorks begin % fonts for text, standard numbers and exponents %line width, line cap, and joint spec 0 setlinecap 1 setlinejoin end MathWorks begin bpage bplot /dpi2point 12 def 0204 2160 7116 7332 bbox portrait csm 0 0 6912 5183 MR c np % Colortable Dictionary /colortable 76 dict begin /c0 { 0 0 0 sc} bdef /c1 { 1 1 1 sc} bdef /c2 { 1 0 0 sc} bdef /c3 { 0 1 0 sc} bdef /c4 { 0 0 1 sc} bdef /c5 { 1 1 0 sc} bdef /c6 { 1 0 1 sc} bdef /c7 { 0 1 1 sc} bdef currentdict end def % Colortable colortable begin /Helvetica 12 FMS c1 0 0 6912 5183 PR 6 setlinewidth DO SO c0 898 4612 mt 6220 4612 L 898 414 mt 6220 414 L 898 4612 mt 898 414 L 6220 4612 mt 6220 414 L 898 414 mt 898 414 L 6220 4612 mt 6220 4612 L 898 4612 mt 6220 4612 L 898 4612 mt 898 414 L 898 4612 mt 898 4612 L 898 4612 mt 898 4559 L 898 414 mt 898 467 L 858 4773 mt (0) s 1658 4612 mt 1658 4559 L 1658 414 mt 1658 467 L 1618 4773 mt (1) s 2419 4612 mt 2419 4559 L 2419 414 mt 2419 467 L 2379 4773 mt (2) s 3179 4612 mt 3179 4559 L 3179 414 mt 3179 467 L 3139 4773 mt (3) s 3939 4612 mt 3939 4559 L 3939 414 mt 3939 467 L 3899 4773 mt (4) s 4699 4612 mt 4699 4559 L 4699 414 mt 4699 467 L 4659 4773 mt (5) s 5460 4612 mt 5460 4559 L 5460 414 mt 5460 467 L 5420 4773 mt (6) s 6220 4612 mt 6220 4559 L 6220 414 mt 6220 467 L 6180 4773 mt (7) s 898 4612 mt 951 4612 L 6220 4612 mt 6167 4612 L 791 4665 mt (0) s 898 3912 mt 951 3912 L 6220 3912 mt 6167 3912 L 671 3965 mt (0.1) s 898 3213 mt 951 3213 L 6220 3213 mt 6167 3213 L 671 3266 mt (0.2) s 898 2513 mt 951 2513 L 6220 2513 mt 6167 2513 L 671 2566 mt (0.3) s 898 1813 mt 951 1813 L 6220 1813 mt 6167 1813 L 671 1866 mt (0.4) s 898 1114 mt 951 1114 L 6220 1114 mt 6167 1114 L 671 1167 mt (0.5) s 898 414 mt 951 414 L 6220 414 mt 6167 414 L 671 467 mt (0.6) s 898 4612 mt 6220 4612 L 898 414 mt 6220 414 L 898 4612 mt 898 414 L 6220 4612 mt 6220 414 L 898 4612 mt 898 4612 L 6220 414 mt 6220 414 L gs 898 414 5323 4199 MR c np DO 5 0 5 1 4 1 5 0 5 1 5 1 4 0 5 1 5 1 5 1 5 0 4 1 5 1 5 0 5 1 4 1 5 1 5 0 5 1 5 1 4 0 5 1 5 0 5 1 4 1 5 0 5 1 5 0 5 1 4 0 5 0 5 1 5 0 4 0 5 1 5 0 5 0 5 0 4 0 5 1 5 0 5 0 4 0 5 0 5 0 5 0 5 0 4 -1 5 0 5 0 5 0 4 0 5 -1 5 0 5 0 4 -1 5 0 5 -1 5 0 5 0 4 -1 5 0 5 -1 5 0 4 -1 5 -1 5 0 5 -1 5 0 4 -1 5 0 5 -1 5 -1 4 0 5 -1 5 -1 5 0 5 -1 4 0 5 -1 5 -1 5 0 4 -1 5 0 5 -1 5 0 5 -1 4 0 5 -1 5 0 5 -1 4 0 5 -1 5 0 5 -1 5 0 4 0 5 -1 5 0 5202 4442 100 MP stroke 5 0 4 -1 5 0 5 0 5 0 5 -1 4 0 5 0 5 0 5 0 4 0 5 0 5 -1 5 0 5 0 4 0 5 0 5 0 5 0 4 0 5 0 5 0 5 0 5 0 4 1 5 0 5 0 5 0 4 0 5 0 5 0 5 0 5 0 4 0 5 0 5 0 5 1 4 0 5 0 5 0 5 0 5 0 4 0 5 0 5 0 5 0 4 0 5 0 5 0 5 0 5 0 4 0 5 0 5 0 5 0 4 0 5 0 5 0 5 0 5 0 4 0 5 0 5 0 5 -1 4 0 5 0 5 0 5 0 5 0 4 0 5 -1 5 0 5 0 4 0 5 0 5 0 5 -1 5 0 4 0 5 0 5 0 5 -1 4 0 5 0 5 0 5 0 5 -1 4 0 5 0 5 0 5 0 4 -1 5 0 5 0 5 0 5 0 4 -1 5 0 5 0 4729 4450 100 MP stroke 5 0 4 0 5 -1 5 0 5 0 4 0 5 0 5 0 5 -1 5 0 4 0 5 0 5 0 5 0 4 -1 5 0 5 0 5 0 5 0 4 0 5 -1 5 0 5 0 4 0 5 0 5 -1 5 0 5 0 4 0 5 -1 5 0 5 0 4 -1 5 0 5 0 5 -1 5 0 4 -1 5 0 5 -1 5 0 4 -1 5 0 5 -1 5 0 5 -1 4 0 5 -1 5 -1 5 -1 4 0 5 -1 5 -1 5 -1 5 -1 4 0 5 -1 5 -1 5 -1 4 -1 5 -1 5 -1 5 -2 5 -1 4 -1 5 -1 5 -1 5 -1 4 -2 5 -1 5 -1 5 -2 5 -1 4 -1 5 -2 5 -1 5 -2 4 -1 5 -2 5 -1 5 -2 5 -1 4 -2 5 -1 5 -2 5 -2 4 -1 5 -2 5 -2 5 -1 5 -2 4 -2 5 -1 5 -2 5 -2 4 -1 5 -2 5 -2 5 -1 4256 4531 100 MP stroke 5 -2 4 -2 5 -1 5 -2 5 -1 4 -2 5 -2 5 -1 5 -2 5 -1 4 -2 5 -2 5 -1 5 -2 4 -1 5 -2 5 -1 5 -1 5 -2 4 -1 5 -2 5 -1 5 -1 4 -2 5 -1 5 -1 5 -1 5 -1 4 -2 5 -1 5 -1 5 -1 4 -1 5 -1 5 -1 5 -1 5 -1 4 0 5 -1 5 -1 5 -1 4 0 5 -1 5 -1 5 0 5 -1 4 -1 5 0 5 -1 5 0 4 0 5 -1 5 0 5 0 4 -1 5 0 5 0 5 0 5 0 4 -1 5 0 5 0 5 0 4 0 5 0 5 0 5 1 5 0 4 0 5 0 5 0 5 1 4 0 5 0 5 1 5 0 5 0 4 1 5 0 5 1 5 0 4 1 5 0 5 1 5 0 5 1 4 0 5 1 5 1 5 0 4 1 5 1 5 0 5 1 5 1 4 0 5 1 5 1 5 1 3783 4576 100 MP stroke 4 0 5 1 5 1 5 1 5 0 4 1 5 1 5 1 5 1 4 0 5 1 5 1 5 1 5 0 4 1 5 1 5 1 5 0 4 1 5 1 5 0 5 1 5 1 4 1 5 0 5 1 5 1 4 0 5 1 5 0 5 1 5 1 4 0 5 1 5 0 5 1 4 0 5 1 5 0 5 1 5 0 4 1 5 0 5 1 5 0 4 1 5 0 5 0 5 1 5 0 4 1 5 0 5 1 5 0 4 0 5 1 5 0 5 1 5 0 4 1 5 0 5 1 5 0 4 1 5 0 5 1 5 0 5 1 4 0 5 1 5 1 5 0 4 1 5 1 5 1 5 1 5 0 4 1 5 1 5 1 5 1 4 1 5 1 5 2 5 1 5 1 4 1 5 2 5 1 5 2 4 1 5 2 5 2 5 1 4 2 5 2 5 2 5 2 5 2 3310 4498 100 MP stroke 4 2 5 2 5 2 5 3 4 2 5 2 5 3 5 2 5 3 4 2 5 3 5 3 5 3 4 3 5 3 5 3 5 3 5 3 4 3 5 3 5 3 5 4 4 3 5 3 5 4 5 3 5 4 4 3 5 4 5 3 5 4 4 4 5 3 5 4 5 4 5 4 4 3 5 4 5 4 5 4 4 4 5 3 5 4 5 4 5 4 4 4 5 4 5 3 5 4 4 4 5 4 5 4 5 3 5 4 4 4 5 4 5 3 5 4 4 3 5 4 5 4 5 3 5 4 4 3 5 4 5 3 5 4 4 3 5 3 5 3 5 4 5 3 4 3 5 3 5 3 5 4 4 3 5 3 5 3 5 2 5 3 4 3 5 3 5 3 5 3 4 2 5 3 5 3 5 3 5 2 4 3 5 2 5 3 5 2 4 3 5 2 5 3 5 2 5 3 2837 4182 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b Ff(\000)p Fi(n)1573 5326 y Fe(X)1552 5504 y Fi(k)q(;l)p Fq(=1)1728 5405 y Fk(cum)p Fj(f)p Fk(X)h Fr(\()p Fk(t)2065 5417 y Fi(k)q Fq(+)p Fi(n)2196 5405 y Fr(\))p Fk(X)g Fr(\()p Fk(t)2366 5417 y Fi(k)2407 5405 y Fr(\))p Fk(;)14 b(X)7 b Fr(\()p Fk(t)2614 5417 y Fi(l)p Fq(+)p Fi(n)2731 5405 y Fr(\))p Fk(X)g Fr(\()p Fk(t)2901 5417 y Fi(l)2927 5405 y Fr(\))p Fj(g)p Fk(;)p eop %%Page: 9 9 9 8 bop 3778 -149 a Fr(9)166 141 y(where)30 b Fk(cum)h Fr(means)f(the)i(cum)n(ulan)n(t)e(function.)48 b(By)31 b(some)f(simple)h(manipulation)g(of)g(the)g(cum)n(ulan)n(t)g (functions,)166 241 y(the)d(quan)n(tit)n(y)f(\(3\))h(b)r(ecomes)765 434 y(1)p 730 471 114 4 v 730 547 a Fk(N)806 523 y Fq(2)872 387 y Fi(N)6 b Ff(\000)p Fi(n)887 412 y Fe(X)867 590 y Fi(k)q(;l)p Fq(=1)1042 490 y Fk(I)-21 b(E)1124 505 y Ff(f)p Fi(t)1183 513 y Fg(n)1223 505 y Ff(g)1275 490 y Fj(f)p Fk(R)1380 502 y Fg(Y)1430 490 y Fr(\()p Fk(t)1492 502 y Fi(k)q Fq(+)p Fi(n)1643 490 y Fj(\000)18 b Fk(t)1756 502 y Fi(k)1797 490 y Fr(\))p 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y Fr(denotes)27 b(the)i(mean)e(conditonal)h(to)f (the)i(distribution)f(of)g(the)g(sc)n(heme)f Fj(f)p Fk(t)3018 886 y Fi(n)3063 874 y Fj(g)p Fr(.)37 b(T)-7 b(ogether)27 b(with)i(the)166 974 y(stationarit)n(y)d(up)i(to)g(order)e(four)h(of)h (the)f(pro)r(cess)427 1142 y Fk(Q)p Fr(\()p Fk(t)555 1154 y Fi(k)q Fq(+)p Fi(n)688 1142 y Fk(;)14 b(t)755 1154 y Fi(k)796 1142 y Fk(;)g(t)863 1154 y Fi(l)p Fq(+)p Fi(n)980 1142 y Fk(;)g(t)1047 1154 y Fi(l)1072 1142 y Fr(\))p Fj(g)23 b Fr(=)g Fk(Q)p Fr(\()p Fk(t)1385 1154 y Fi(k)q Fq(+)p Fi(n)1518 1142 y Fk(=t)1590 1154 y Fi(k)1630 1142 y Fk(;)14 b(t)1697 1154 y Fi(l)p Fq(+)p Fi(n)1814 1142 y Fk(=t)1886 1154 y Fi(k)1927 1142 y Fk(;)g(t)1994 1154 y Fi(l)2019 1142 y Fk(=t)2091 1154 y Fi(k)2131 1142 y Fr(\))p Fj(g)23 b Fr(=)g Fk(cum)p Fr(\()p Fk(t)2535 1154 y Fi(k)q Fq(+)p Fi(n)2686 1142 y Fj(\000)18 b Fk(t)2799 1154 y Fi(k)2840 1142 y Fk(;)c(t)2907 1154 y Fi(l)p Fq(+)p Fi(n)3042 1142 y Fj(\000)19 b Fk(t)3156 1154 y Fi(k)3196 1142 y Fk(;)14 b(t)3263 1154 y Fi(l)3307 1142 y Fj(\000)k Fk(t)3420 1154 y Fi(k)3461 1142 y Fr(\))p Fj(g)p Fk(;)166 1360 y Fr(the)28 b(tec)n(hnique)g(used)f(in)h(Messaci)f(\(1986\),)f (pages)h(18-25,)e(ensures)i(that)h Fk(v)s(ar)r Fj(f)5 b Fe(b)-51 b Fk(\032)2712 1372 y Fg(X;n)2818 1360 y Fr(\()p Fk(N)9 b Fr(\))p Fj(g)23 b Fr(=)g Fk(O)3190 1243 y Fe(\022)3278 1304 y Fr(1)p 3261 1341 76 4 v 3261 1417 a Fk(N)3347 1243 y Fe(\023)3422 1360 y Fk(:)194 1560 y Fp(Pr)l(o)l(of)31 b(of)f(The)l(or)l(em)h(3.)166 1736 y Fr(Recall)c(that)g Fe(b)-45 b Fk(a)640 1748 y Fg(n)704 1736 y Fr(=)822 1632 y Fi(N)792 1657 y Fe(X)791 1836 y Fi(k)q Fq(=1)926 1736 y Fk(\022)965 1748 y Fg(n;k)1062 1736 y Fe(b)-51 b Fk(\032)1100 1748 y Fg(X;k)1202 1736 y Fr(\()p Fk(N)9 b Fr(\).)37 b(F)-7 b(rom)27 b(Theorem)g(2,)g(w)n(e)g(get)h(b)r(oth)g(follo)n(wing)e (p)r(oin)n(ts.)166 1915 y(\(1\))1312 2164 y Fk(I)-21 b(E)6 b Fj(f)o Fe(b)-45 b Fk(a)1480 2176 y Fg(n)1520 2164 y Fr(\()p Fk(N)9 b Fr(\))p Fj(g)83 b Fr(=)1963 2061 y Fi(N)1933 2085 y Fe(X)1932 2264 y Fi(k)q Fq(=1)2067 2164 y Fk(\022)2106 2176 y Fg(n;k)2198 2164 y Fk(I)-21 b(E)5 b Fj(f)g Fe(b)-51 b Fk(\032)2364 2176 y Fg(X;k)2465 2164 y Fr(\()p Fk(N)9 b Fr(\))p Fj(g)1785 2457 y Fr(=)81 b Fe(b)-45 b Fk(a)1976 2469 y Fg(n)2035 2457 y Fj(\000)2145 2401 y Fr(1)p 2128 2438 V 2128 2514 a Fk(N)2259 2354 y Fi(N)2228 2379 y Fe(X)2228 2557 y Fi(k)q Fq(=1)2362 2457 y Fk(\022)2401 2469 y Fg(n;k)2493 2457 y Fk(\032)2536 2469 y Fi(k)2602 2401 y Fk(k)p 2587 2438 V 2587 2514 a(N)166 2705 y Fr(Denote)28 b(b)n(y)f Fk(B)t(ias)p Fr(\()o Fe(b)-45 b Fk(a)821 2717 y Fg(n)862 2705 y Fr(\()p Fk(N)9 b Fr(\)\))28 b(the)g(bias)f(of)f Fe(b)-44 b Fk(a)1515 2717 y Fg(n)1555 2705 y Fr(\()p Fk(N)9 b Fr(\).)37 b(W)-7 b(e)28 b(get)g(then)g(from)f(\(5\))413 2955 y Fk(B)t(ias)p Fj(f)o Fe(b)-45 b Fk(a)678 2967 y Fg(n)718 2955 y Fr(\()p Fk(N)9 b Fr(\))p Fj(g)82 b Fr(=)h Fj(\000)1222 2899 y Fr(1)p 1205 2936 V 1205 3012 a Fk(N)1335 2851 y Fi(N)1305 2876 y Fe(X)1304 3055 y Fi(k)q Fq(=1)1439 2955 y Fk(\022)1478 2967 y Fg(n;k)1570 2955 y Fk(k)1629 2842 y Fe(Z)1712 2862 y Fq(+)p Ff(1)1675 3030 y Fq(0)1847 2955 y Fk(R)1910 2967 y Fg(Y)1960 2955 y Fr(\()p Fk(t)p Fr(\))p Fk(f)2095 2967 y Fi(k)2136 2955 y Fr(\()p Fk(t)p Fr(\))p Fk(dt)982 3248 y Fr(=)g Fj(\000)1222 3192 y Fr(1)p 1205 3229 V 1205 3305 a Fk(N)1304 3135 y Fe(Z)1387 3155 y Fq(+)p Ff(1)1350 3324 y Fq(0)1522 3106 y Fe( )1619 3144 y Fi(N)1588 3169 y Fe(X)1588 3348 y Fi(k)q Fq(=1)1723 3248 y Fk(\022)1762 3260 y Fg(n;k)1853 3248 y Fk(k)s(f)1940 3260 y Fi(k)1981 3248 y Fr(\()p Fk(t)p Fr(\))2075 3106 y Fe(!)2155 3248 y Fk(R)2218 3260 y Fg(Y)2268 3248 y Fr(\()p Fk(t)p Fr(\))p Fk(f)2403 3260 y Fi(k)2444 3248 y Fr(\()p Fk(t)p Fr(\))p Fk(dt)24 b Fr(=)e Fj(\000)2814 3192 y Fr(1)p 2797 3229 V 2797 3305 a Fk(N)2896 3135 y Fe(Z)2979 3155 y Fq(+)p Ff(1)2943 3324 y Fq(0)3115 3248 y Fk(R)3178 3260 y Fg(Y)3228 3248 y Fr(\()p Fk(t)p Fr(\))p Fk(q)3359 3260 y Fi(n)3404 3248 y Fr(\()p Fk(t)p Fr(\))p Fk(dt)166 3570 y Fr(where)446 3467 y Fi(n)407 3492 y Fe(X)406 3670 y Fi(k)q Fq(=1)541 3570 y Fk(k)s(\022)626 3582 y Fg(n;k)717 3570 y Fk(f)758 3582 y Fi(k)799 3570 y Fr(\()p Fk(t)p Fr(\))i(=)e Fk(nq)1091 3582 y Fi(n)1136 3570 y Fr(\()p Fk(t)p Fr(\))d Fj(\000)f Fr(\()p Fk(n)h Fj(\000)f Fr(1\))p Fk(q)1627 3582 y Fi(n)p Ff(\000)p Fq(1)1757 3570 y Fr(\()p Fk(t)p Fr(\),)28 b(with)g Fk(q)2128 3582 y Fi(n)2174 3570 y Fr(\()p Fk(t)p Fr(\))23 b(=)2419 3467 y Fi(n)2379 3492 y Fe(X)2379 3670 y Fi(k)q Fq(=1)2513 3570 y Fk(\022)2552 3582 y Fg(n;k)2644 3570 y Fk(f)2685 3582 y Fi(k)2726 3570 y Fr(\()p Fk(t)p Fr(\).)37 b(Th)n(us)699 3884 y Fk(B)t(ias)p Fj(f)o Fe(b)-45 b Fk(a)964 3896 y Fg(n)1003 3884 y Fr(\()p Fk(N)9 b Fr(\))p Fj(g)83 b Fr(=)g Fj(\000)1503 3828 y Fk(n)p 1491 3865 V 1491 3941 a(N)1590 3771 y Fe(Z)1673 3792 y Fq(+)p Ff(1)1636 3960 y Fq(0)1808 3884 y Fk(R)1871 3896 y Fg(Y)1921 3884 y Fr(\()p Fk(t)p Fr(\))p Fk(g)2055 3896 y Fi(n)2100 3884 y Fr(\()p Fk(t)p Fr(\))19 b(+)2306 3828 y Fk(n)g Fj(\000)f Fr(1)p 2306 3865 193 4 v 2365 3941 a Fk(N)2523 3771 y Fe(Z)2606 3792 y Fq(+)p Ff(1)2569 3960 y Fq(0)2741 3884 y Fk(R)2804 3896 y Fg(Y)2854 3884 y Fr(\()p Fk(t)p Fr(\))p Fk(g)2988 3896 y Fi(n)p Ff(\000)p Fq(1)3118 3884 y Fr(\()p Fk(t)p Fr(\))p Fk(dt)1268 4103 y Fr(=)83 b Fj(\000)1503 4047 y Fk(n)p 1491 4084 76 4 v 1491 4160 a(N)1575 4103 y Fe(b)-45 b Fk(a)1620 4115 y Fg(n)1679 4103 y Fr(+)1772 4047 y Fk(n)18 b Fj(\000)h Fr(1)p 1772 4084 193 4 v 1831 4160 a Fk(N)1975 4103 y(a)2019 4115 y Fi(n)p Ff(\000)p Fq(1)1268 4303 y Fr(=)82 b Fe(b)-45 b Fk(a)1460 4315 y Fg(n)1519 4303 y Fj(\000)1629 4246 y Fr(1)p 1612 4283 76 4 v 1612 4359 a Fk(N)1712 4303 y Fr(\()p Fk(n)o Fe(b)g Fk(a)1838 4315 y Fg(n)1897 4303 y Fj(\000)18 b Fr(\()p Fk(n)g Fj(\000)g Fr(1\))p Fk(a)2281 4315 y Fi(n)p Ff(\000)p Fq(1)2411 4303 y Fr(\))c Fk(:)166 4499 y Fr(\(2\))28 b(The)f(auto)r(co)n(v)-5 b(ariance)26 b(function)i(of)f(the)h(estimate) f Fe(b)-45 b Fk(a)1975 4511 y Fg(n)2016 4499 y Fr(\()p Fk(N)9 b Fr(\))28 b(b)r(eha)n(v)n(es)e(as)h(follo)n(ws)1170 4759 y Fk(v)s(ar)r Fj(f)o Fe(b)-45 b Fk(a)1382 4771 y Fg(n)1423 4759 y Fr(\()p Fk(N)9 b Fr(\))p Fj(g)23 b(\024)1715 4617 y Fe( )1821 4655 y Fi(n)1782 4680 y Fe(X)1781 4859 y Fi(k)q Fq(=1)1916 4759 y Fj(j)p Fk(\022)1978 4771 y Fg(n;k)2070 4759 y Fj(j)2093 4684 y Fe(p)p 2176 4684 500 4 v 75 x Fk(v)s(ar)r Fj(f)5 b Fe(b)-51 b Fk(\032)2387 4771 y Fg(X;n)2493 4759 y Fr(\()p Fk(N)9 b Fr(\))p Fj(g)2675 4617 y Fe(!)2741 4634 y Fq(2)2792 4759 y Fk(;)166 5002 y Fr(and)27 b(from)h(theorem)f(2,)g(this)h(b)r(ecomes,)f(b)n(y)h(some)f (simple)g(calculation)1208 5268 y Fk(v)s(ar)r Fj(f)o Fe(b)-45 b Fk(a)1420 5280 y Fg(n)1461 5268 y Fr(\()p Fk(N)9 b Fr(\))p Fj(g)83 b(\024)1874 5126 y Fe( )1979 5164 y Fi(n)1940 5189 y Fe(X)1940 5368 y Fi(k)q Fq(=1)2074 5119 y Fe(s)p 2157 5119 161 4 v 2167 5212 a Fr(2)p Fk(A)2271 5224 y Fq(2)p 2167 5249 142 4 v 2174 5325 a Fk(\014)t(N)2318 5268 y Fr(2)2360 5234 y Fi(k)q Ff(\000)p Fq(1)2485 5268 y Fj(C)2534 5232 y Fi(k)q Ff(\000)p Fq(1)2529 5290 y Fi(n)p Ff(\000)p Fq(1)2660 5126 y Fe(!)2726 5133 y Fq(2)1726 5522 y Fj(\024)g Fr(3)1916 5488 y Fq(2)p Fi(n)2004 5466 y Fk(A)2066 5478 y Fq(3)p 2004 5503 100 4 v 2015 5579 a Fk(N)2141 5522 y Fr(with)28 b Fk(A)2392 5534 y Fq(3)2452 5522 y Fr(=)2550 5466 y(2)p Fk(A)2654 5478 y Fq(2)p 2550 5503 142 4 v 2600 5579 a Fr(9)2701 5522 y Fk(;)p eop %%Page: 10 10 10 9 bop 3736 -149 a Fr(10)166 141 y(whic)n(h)28 b(completes)f(the)h (pro)r(of)f(of)g(the)h(theorem.)194 340 y Fp(Pr)l(o)l(of)j(of)f(The)l (or)l(em)h(4.)166 440 y Fr(Recall)c(that)h(the)g(mean)g(square)e(error) f(of)1536 419 y Fe(b)1520 440 y Fk(G)1585 452 y Fg(X;n)1719 440 y Fr(is)j(expanded)f(as)936 611 y Fk(I)-21 b(E)5 b Fj(f)1075 590 y Fe(b)1059 611 y Fk(G)1124 623 y Fg(X;n)1230 611 y Fr(\()p Fk(\025)p Fr(\))19 b Fj(\000)f Fk(G)1509 623 y Fg(X)1563 611 y Fr(\()p Fk(\025)p Fr(\))p Fj(g)1717 577 y Fq(2)1778 611 y Fr(=)23 b Fk(v)s(ar)r Fj(f)2051 590 y Fe(b)2034 611 y Fk(G)2099 623 y Fg(X;n)2206 611 y Fr(\()p Fk(\025)p Fr(\))p Fj(g)c Fr(+)f Fk(B)t(ias)2641 577 y Fq(2)2677 611 y Fr(\()2726 590 y Fe(b)2709 611 y Fk(G)2774 623 y Fg(X;n)2881 611 y Fr(\()p Fk(\025)p Fr(\)\))p Fk(:)166 772 y Fr(The)28 b(pro)r(of)f(of)g(the)h(theorem)f (will)h(tak)n(e)f(t)n(w)n(o)g(main)g(steps.)166 972 y Fp(First)j(step:)37 b Fr(Ma)5 b(joration)26 b(of)i(the)g(terms)f Fk(v)s(ar)r Fj(f)1664 951 y Fe(b)1647 972 y Fk(G)1712 984 y Fg(X;n)1819 972 y Fr(\()p Fk(\025)p Fr(\))p Fj(g)p Fr(.)291 1071 y(F)-7 b(rom)27 b(the)h(expression)e(of)h(the)h(estimate) g(and)f(theorem)g(3,)h(w)n(e)f(get)844 1218 y Fe(\020)894 1310 y Fk(v)s(ar)r Fj(f)1079 1289 y Fe(b)1062 1310 y Fk(G)1127 1322 y Fg(X;n)1234 1310 y Fr(\()p Fk(\025)p Fr(\))p Fj(g)1388 1218 y Fe(\021)1438 1235 y Fq(1)p Fi(=)p Fq(2)1625 1310 y Fj(\024)1777 1205 y Fi(M)1840 1213 y Fg(N)1775 1231 y Fe(X)1773 1407 y Fi(n)p Fq(=1)1912 1310 y Fj(j)p Fk(y)1976 1322 y Fi(n)2021 1310 y Fr(\()p Fk(N)9 b Fr(\))p Fj(jj)p Fk(G)2272 1322 y Fi(n)2318 1310 y Fr(\()p Fk(\025)p Fr(\))p Fj(j)2467 1193 y Fe(\022)2538 1254 y Fk(A)2600 1266 y Fq(3)2638 1254 y Fr(3)2680 1223 y Fq(2)p Fi(n)p 2538 1291 220 4 v 2610 1367 a Fk(N)2768 1193 y Fe(\023)2829 1207 y Fq(1)p Fi(=)p Fq(2)1625 1604 y Fj(\024)83 b Fr(2)1832 1499 y Fi(M)1895 1507 y Fg(N)1831 1525 y Fe(X)1829 1701 y Fi(n)p Fq(=1)1967 1604 y Fk(h)2029 1487 y Fe(\022)2100 1548 y Fr(exp\()p Fk(na)p Fr(\))p 2100 1585 286 4 v 2188 1661 a Fk(N)2264 1637 y Fi(b)2395 1487 y Fe(\023)2470 1455 y(s)p 2553 1455 566 4 v 2679 1548 a Fr(2)p Fk(A)2783 1560 y Fq(3)2820 1548 y Fk(\014)t Fr(3)2913 1524 y Fq(2)p Fi(n)p 2563 1585 546 4 v 2563 1661 a Fk(N)9 b Fr(\()p Fk(\014)2722 1637 y Fq(2)2778 1661 y Fr(+)18 b(4)p Fk(\031)2953 1637 y Fq(2)2990 1661 y Fk(\025)3038 1637 y Fq(2)3076 1661 y Fr(\))3118 1604 y Fk(:)572 b Fr(\(5\))166 1934 y(F)-7 b(rom)35 b(the)g(expression)f(of) h(the)h(bandwidth)f Fk(M)1706 1946 y Fg(N)1759 1934 y Fr(,)i(the)f(quan)n(tit)n(y)e(\(5\))i(is)f(less)f(than)i(9)2946 1785 y Fe(s)p 3028 1785 161 4 v 3038 1878 a Fr(2)p Fk(A)3142 1890 y Fq(3)p 3038 1915 142 4 v 3083 1991 a Fk(\014)3189 1934 y(N)3265 1900 y Ff(\000)p Fq(1)p Fi(=)p Fq(2\(1)p Ff(\000)3558 1878 y Fg(b)p 3538 1887 67 3 v 3538 1920 a Fc(2)p Fg(\013)3625 1900 y Fq(ln\(3\)\))3796 1934 y Fr(,)166 2153 y(th)n(us)28 b Fk(v)s(ar)r Fj(f)534 2132 y Fe(b)517 2153 y Fk(G)582 2165 y Fg(X;n)689 2153 y Fr(\()p Fk(\025)p Fr(\))p Fj(g)23 b Fr(=)g Fk(O)r Fr(\()p Fk(N)1127 2119 y Ff(\000)p Fi(p)1218 2153 y Fr(\))28 b(with)g Fk(p)23 b Fr(=)f(1)c Fj(\000)1802 2097 y Fk(b)p 1772 2134 95 4 v 1772 2210 a Fr(2)p Fk(\013)1891 2153 y Fr(ln\(3\).)166 2276 y Fp(Se)l(c)l(ond)30 b(step:)37 b Fr(Bound)28 b(for)f(the)h(bias.) 291 2376 y(W)-7 b(e)28 b(ha)n(v)n(e)642 2598 y Fk(B)t(ias)p Fj(f)879 2577 y Fe(b)863 2598 y Fk(G)928 2610 y Fg(X;n)1034 2598 y Fr(\()p Fk(\025)p Fr(\))p Fj(g)83 b Fr(=)1423 2494 y Fi(M)1486 2502 y Fg(N)1422 2519 y Fe(X)1419 2695 y Fi(n)p Fq(=1)1558 2598 y Fk(y)1599 2610 y Fi(n)1644 2598 y Fr(\()p Fk(\025)p Fr(\))p Fk(G)1821 2610 y Fi(n)1867 2598 y Fr(\()p Fk(\025)p Fr(\))p Fk(a)2023 2610 y Fi(n)2088 2598 y Fj(\000)2175 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Fk(r)p 1614 5127 183 4 v 1614 5203 a(r)d Fj(\000)d Fr(2)1834 5146 y(and)28 b Fk(M)2077 5158 y Fg(N)2152 5146 y Fj(\024)2280 5090 y Fk(b)p 2250 5127 95 4 v 2250 5203 a Fr(2)p Fk(\013)2369 5146 y Fr(ln\()p Fk(N)9 b Fr(\))19 b(+)f(1,)27 b(w)n(e)g(deduce)h(that)629 5369 y Fi(M)692 5377 y Fg(N)627 5395 y Fe(X)625 5571 y Fi(n)p Fq(=1)764 5474 y Fj(j)p Fk(G)852 5486 y Fi(n)897 5474 y Fr(\()p Fk(\025)p Fr(\))14 b Fk(a)1067 5486 y Fi(n)1127 5474 y Fr(\(1)k Fj(\000)g Fk(y)1343 5486 y Fi(n)1388 5474 y Fr(\()p Fk(N)9 b Fr(\)\))p Fj(j)84 b(\024)1815 5351 y Fe(r)p 1898 5351 159 4 v 1956 5418 a Fr(2)p 1908 5455 139 4 v 1908 5531 a Fk(\031)1958 5507 y Fq(2)1995 5531 y Fk(\014)2056 5474 y(k)2099 5486 y Fq(2)2137 5474 y Fk(A)2199 5486 y Fq(1)2236 5474 y Fr(\()p Fk(r)r Fr(\))2422 5418 y Fk(r)p 2349 5455 183 4 v 2349 5531 a(r)23 b Fj(\000)18 b Fr(2)2557 5474 y Fk(e)c(N)2686 5440 y Ff(\000)p Fi(b=)p Fq(2)2861 5474 y Fr(=)23 b Fk(O)3028 5382 y Fe(\020)p 3088 5455 177 4 v 3088 5532 a Fk(N)3164 5508 y Fi(b=)p Fq(2)3274 5382 y Fe(\021)3337 5474 y Fk(;)p eop %%Page: 11 11 11 10 bop 3736 -149 a Fr(11)166 202 y(whic)n(h)28 b(is)f(of)h(order)e (of)h(magnitude)h Fk(O)1383 84 y Fe(\022)1538 145 y Fr(1)p 1454 182 210 4 v 1454 259 a(ln)q(\()p Fk(N)9 b Fr(\))1674 84 y Fe(\023)1735 102 y Fq(\()p Fi(r)r Ff(\000)p Fq(2\))p Fi(=)p Fq(2)1976 202 y Fr(.)194 351 y(\(ii\))28 b(By)f(simple)h (calculations,)f(w)n(e)g(ha)n(v)n(e)f(that)217 560 y(1)p 200 597 76 4 v 200 673 a Fk(N)304 511 y Fi(M)367 519 y Fg(N)302 537 y Fe(X)300 713 y Fi(n)p Fq(=1)439 616 y Fj(j)p Fk(y)503 628 y Fi(n)548 616 y Fr(\()p Fk(N)9 b Fr(\))p Fk(G)753 628 y Fi(n)799 616 y Fr(\()p Fk(\025)p Fr(\))p Fj(j)14 b(j)p Fk(na)1065 628 y Fi(n)1129 616 y Fr(+)k(\()p Fk(n)g Fj(\000)g Fr(1\))p Fk(a)1513 628 y Fi(n)p Ff(\000)p Fq(1)1643 616 y Fj(j)83 b(\024)1924 560 y Fr(2)p 1907 597 V 1907 673 a Fk(N)1993 493 y Fe(r)p 2076 493 159 4 v 2134 560 a Fr(2)p 2086 597 139 4 v 2086 673 a Fk(\031)2136 649 y Fq(2)2173 673 y Fk(\014)2252 512 y Fq(+)p Ff(1)2251 537 y Fe(X)2248 713 y Fi(n)p Fq(=1)2387 616 y Fk(M)2468 628 y Fg(N)2521 616 y Fk(A)2583 628 y Fq(1)2621 616 y Fr(\()p Fk(r)r Fr(\))p Fk(n)2774 582 y Ff(\000)p Fi(r)r(=)p Fq(2)1749 902 y Fj(\024)g Fr(2)p Fk(A)2001 914 y Fq(1)2038 902 y Fr(\()p Fk(r)r Fr(\))2141 780 y Fe(r)p 2225 780 159 4 v 2284 846 a Fr(2)p 2235 883 139 4 v 2235 959 a Fk(\031)2285 935 y Fq(2)2323 959 y Fk(\014)2402 799 y Fq(+)p Ff(1)2401 823 y Fe(X)2398 999 y Fi(n)p Fq(=1)2537 902 y Fk(n)2587 868 y Ff(\000)p Fi(r)r(=)p Fq(2)2756 785 y Fe(\022)2857 846 y Fk(b)p 2827 883 95 4 v 2827 959 a Fr(2)p Fk(\013)2946 902 y Fr(ln\()p Fk(N)9 b Fr(\))19 b(+)f(1)3299 785 y Fe(\023)3552 846 y Fr(1)p 3384 883 379 4 v 3384 959 a Fk(N)3460 935 y Ff(\000)p Fi(p)p Fq(+1)3648 959 y Fk(N)3724 935 y Fi(p)3772 902 y Fk(;)166 1213 y Fr(where)25 b Fk(p)e Fr(=)f(1)14 b Fj(\000)709 1157 y Fk(b)p 700 1194 54 4 v 700 1270 a(\013)777 1213 y Fr(ln)q(\(3\).)36 b(The)25 b(imp)r(osed)h(h)n(yp)r (othesis)e(on)i Fk(b)f Fr(and)g Fk(\013)h 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Fk(:)193 1967 y Fr(\(iii\))28 b(F)-7 b(rom)27 b(lemma)g(2.1)f(of)i(Masry)d(\(1980\),)i(under)g(Condition)g (1)f(and)h(since)g Fj(j)p Fk(a)2723 1979 y Fi(n)2768 1967 y Fj(j)d(\024)e Fk(A)2964 1979 y Fq(1)3002 1967 y Fr(\()p Fk(r)r Fr(\))p Fk(n)3155 1933 y Ff(\000)p Fi(r)r(=)p Fq(2)3339 1967 y Fr(with)28 b Fk(A)3590 1979 y Fq(1)3627 1967 y Fr(\()p Fk(r)r Fr(\))c(=)166 2003 y Fe(r)p 249 2003 113 4 v 285 2069 a Fr(1)p 259 2106 93 4 v 259 2182 a(2)p Fk(\014)362 2125 y Fj(jj)p Fk(t)438 2091 y Fi(r)r(=)p Fq(2)574 2069 y Fk(d)617 2039 y Fi(r)p 552 2106 124 4 v 552 2182 a Fk(d)14 b(t)639 2158 y Fi(r)699 2125 y Fr(\()q Fk(R)795 2137 y Fg(Y)845 2125 y Fr(\()p Fk(t=)p Fr(2)p Fk(\014)t Fr(\))g(exp)o(\()p Fj(\000)p Fk(t=)p Fr(2\)\))f Fj(jj)1548 2140 y Fi(L)1594 2124 y Fc(2)1627 2140 y Fq(\()p Fd(R)1700 2148 y Fc(+)1741 2140 y Fq(\))1771 2125 y Fr(,)28 b(the)g(term)g(\(6\))f(is)h(less)f(than)898 2316 y Fe(r)p 981 2316 159 4 v 1039 2382 a Fr(2)p 991 2419 139 4 v 991 2495 a Fk(\031)1041 2471 y Fq(2)1078 2495 y Fk(\014)1140 2438 y(A)1202 2450 y Fq(1)1239 2438 y Fr(\()p Fk(r)r Fr(\))1445 2335 y Fq(+)p Ff(1)1444 2359 y Fe(X)1356 2538 y Fi(n)p Fq(=)p Fi(M)1511 2546 y Fg(N)1566 2538 y Fq(+1)1664 2438 y Fk(n)1714 2404 y Ff(\000)p Fi(r)r(=)p Fq(2)1952 2438 y Fj(\024)2100 2316 y Fe(r)p 2183 2316 159 4 v 2242 2382 a Fr(2)p 2193 2419 139 4 v 2193 2495 a Fk(\031)2243 2471 y Fq(2)2281 2495 y Fk(\014)2342 2438 y(A)2404 2450 y Fq(1)2441 2438 y Fr(\()p Fk(r)r Fr(\))2558 2325 y Fe(Z)2643 2346 y Fq(+)p Ff(1)2606 2514 y Fi(M)2669 2522 y Fg(N)2778 2438 y Fk(x)2825 2404 y Ff(\000)p Fi(r)r(=)p Fq(2)2981 2438 y Fk(dx)1952 2716 y Fj(\024)2100 2594 y Fe(r)p 2183 2594 159 4 v 2242 2660 a Fr(2)p 2193 2697 139 4 v 2193 2773 a Fk(\031)2243 2749 y Fq(2)2281 2773 y Fk(\014)2342 2716 y(A)2404 2728 y Fq(1)2441 2716 y Fr(\()p Fk(r)r Fr(\))2626 2660 y(2)p 2554 2697 183 4 v 2554 2773 a Fk(r)c Fj(\000)18 b Fr(2)2748 2716 y Fk(M)2829 2728 y Fg(N)2892 2655 y Fb(\000)p Fg(r)q Fc(+2)p 2892 2668 146 3 v 2951 2700 a(2)1952 2976 y Fj(\024)2100 2854 y Fe(r)p 2183 2854 159 4 v 2242 2920 a Fr(2)p 2193 2957 139 4 v 2193 3033 a Fk(\031)2243 3009 y Fq(2)2281 3033 y Fk(\014)2342 2976 y(A)2404 2988 y Fq(1)2441 2976 y Fr(\()p Fk(r)r Fr(\))2558 2859 y Fe(\022)2713 2920 y Fr(2)p Fk(\013)p 2631 2957 259 4 v 2631 3033 a(b)c Fr(ln)o(\()p Fk(N)9 b Fr(\))2900 2859 y Fe(\023)2971 2851 y Fg(r)q Fb(\000)p Fc(2)p 2971 2864 103 3 v 3008 2897 a(2)166 3300 y Fr(th)n(us)436 3196 y Fq(+)p Ff(1)435 3221 y Fe(X)349 3399 y Fi(n)p Fq(=)p Fi(M)504 3407 y Fg(N)557 3399 y Fq(+1)655 3300 y Fj(j)p Fk(a)722 3312 y Fi(n)767 3300 y Fk(G)832 3312 y Fi(n)878 3300 y Fr(\()p Fk(\025)p Fr(\))p Fj(j)24 b Fr(=)f Fk(O)1204 3183 y Fe(\022)1359 3244 y Fr(1)p 1275 3281 210 4 v 1275 3357 a(ln\()p Fk(N)9 b Fr(\))1495 3183 y Fe(\023)1566 3174 y Fg(r)q Fb(\000)p Fc(2)p 1566 3187 103 3 v 1603 3220 a(2)1682 3300 y Fr(.)37 b(F)-7 b(rom)27 b(the)h(tree)g(substeps,)f(w)n(e)g(deduce)h(directly)g(the)g(result.) 194 3524 y 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