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(1),. A Criterion for the Equivalence of two Stationary Gaussian.Processes with, Not Necessariiy ･ Rational, Spectral Densities t. By. .. Ken-ichi YOSHIHARA. ,. 1. Summary. The question concerning the equivale'nce of stationary Gaussian processes defined on the finite interval EO, T] was fully solved in the case where their spectral densities are rational in 2, (cf. [1], [5] and [7]). The object of this. paper is to consider a criterion for the equivalence of two one-dimensionql stationary Gaussian processes on [O, T] with,not necessarily rational, spectral densities (Theorem 2): The main result is that P6,g and P,-,,T are equivalent. iff,(2)andh(R)arepositiveforalmostall2andifi , , i. /. ' o'f'S-OO..{-,kE:',-:i-iog-,h(,:l-}d2<cx). S-oo..{-2[:l--i-iog-2g,hZl-}d2<oo. 2. Terminologies and nbtationS. , We shall use the terminologies and notations in [5]. Let g,={8,(t)} and ' g2 ={e2(t)} be one-dimensional real-valued continuous-parameter stationary Gaussian processes which are continuousin quadratic mean. We assume that '. Ec',(t) = Eg,(t) == o. ,. Let F,(2) and F,(1) be spectral functions of the stationary processes 6,. and 6,, respectively. We put -. (i)･ H(e,g;8,g)'=Hk,g(6,g)-sup;.:pe,g(Ez)iog-i]i.li.[iitlS)-.. ' ' supremum is taken over all partitions where the {E.i}, of the measurable space (9, S) on which probability measures Pe,g and P,-,g are defined, and '. ' -2- -((:l-}d2 ' <2) s(g,, 6,5 -= s6f,･- i S-OO.{!2il3i--iTiog. '. where f,(R) and f>(2) are spectral densities of F,(R) and F,(k), respectively..

(2) K. YosHIHARA. 20. 3. Lammas.. We put. ･:･. (3) g(x)=:x-1;logx (x>O), g(O)=oo. LEMMA 1. if the series S) a.(a.>=O,n=O, ±L±2,･･･) and S b. (b.>O.. n=-oo n=-oo. n == O, ±1, ±2, ･･･) are convergent, then. .. (4) g://li.l.i--o.ok)-<-max{q(.=s-u,p,,k)･q(.=i-n,f,,b"."ff)} v. -` 9(.=.,S-U,P,,,,,.. ba." )+g(.=,..,i.nS,,,,... b".n').. PRooF. The lemma is obvious from the inequalities'. :;ii an ..,...,i-"ifo,i,,,.-ba:T ;$ "=2o-o epb. ;ll ...,,.9",?,,,,.,. ba.". '. ' n==-co. and the property of the function g(O.. 'COROLLARY. (s) gEbill:i:i=`==max{q(.s-.-3tL,ilil;[:'))･g(.!.eE,.'iis((:)'ny)}. a LEMMA 2. IllfA(R)>O for almost all 2 in a closed interval [a, b] 'and iT O<c$.il,(R) ;E; d for all R in [a, b] where c and d ore constants, then. (6) (b-a)gp i%b"f'ii((li:i :s ,l,S:{p(--:A(li[i/ll-)dR l. a PRooF. ,Using the inequality'. .. S.bgi(oiog gg,iEoX) dxiS.bgi(odxiog sSaiggli'S:l. a for nonnegative integrab ,le. functions g,(O and g,(x) on [a, b], we have. ' -li-S:{A(z)-h(2)m(z) iog k[:i }dR Sk(-h-((ig/-)d2 )'. ''. '. ' ' illl,Af>((ii,d,R i-ii-l2- S:fi(2)dR-S:h(R)dR+S:n(R)dRiog '.

(3) A Criterion for the Equivalence of two Statioriary Gaussian Processes. 21. =-ld-=s.bh(R)dR･g''I"b,2:i:l >= c(bia) g'Iab,2((liddi. "a. .tta. So,wehavethelemma. - '. Next, we :EZi put <7) ･2iz:E)-,I5. r. '. '. and for any positive integer Tt. ' (s) g(1,Ti)=sx(z,Is,,)g g"L)fi(R)dz. 'i. r :- oo. ' J"t)h(R)d2. ' where Isb denotes the semiclosed interval (2IZ-, , 2(rT+,1)g ]. Then, ,,. ' (9) .' -21i'i-.=1200])-.9 lii',lli:iiSi = 4i.S-O.O..g(Z:i)dZ'. '. a t. '. .L r. '. ' Now, we shall prove the fgllowing' LEMMA 3. Let {Ti} be any sequence ofPositiveintegers. Let f>(2) be 2ositive for almost all 2. if 8(6,, 8,) is finite, then for arbitrary numbers K and. Ki (Ki<K) ' (10) ,t'. ' d2. .. 41. S.K,g(2:Tt)dz$ 41. S.K,q(iri-((-22-l-). ,m. PRooF. We shall divide the pr,oof into three steps. Step. I. At first, we shall prove (10) in the case where O<c;S]1,(R)$.d for all 2 in [K', K]. Since 8(g,, 8,) is finite, so for any E>O there exists a sum of semi-closed intgrval E, in [K', K] such that j. ;. ' <ii.) ,,..'-4'ii-S.,g(2[:)))d2<3... and-,''''''"'-･,., t. <i2)''''' ''6::;lt.sD(-Lf'lt[:]);s.M''":'''. "" L. for all 2 in [K', K]--E,. Furthermore, we choose a subset E, of the interval. [K', K] such that EocEi bnd ' '' a3),,, ....''. ,'. ',.'. ' , 4iitS.,go<-ffl>-((:i/-)dR<e.'.. ' ForanYTt'siufficientlylarge,let ' Ez･{i) .= {I;i)II￡t) C E,}. and gSi) = {ISi)lJ;e) c [K/, K]-E,, I8t) e g54)}. ,. ,.

(4) 22 ･. .K. YosHIHARA If we choose E, suitably, we can divide the class of all semi-closed interva!s I;b contained in [K',K] into Sf`' and gS`' for'all Tt sufliciently large. We. remark that , . (i4) ･ '''. ;}-m.g(z:T,)=:g(-2-g(il-). ' for almost all Z in [K/, K]-E, and from Corollary to Lemma 1 and (12). (is) o;gziei-g SsJ i`,l'.iilliiliddl .sM fori;bEssi). ,;. ,/. v. and consequently. (16) '' '. O;:;l--41gifg(R:Ti);sM. '. for all 2 in [K', K]-E,. Accordingly, from Lemma 2 and (16) we have }il}[.l il N4ilnS.,g(R: Ti)dR K. '. '. =Tlii-M･oo 41T {,￡t)]l.ll.-..Et)Si5`'g(i:Ti)dZ+,s.t)?,,.fe)Si￡`'g(R:T`)dR}. '. (17). iSl ;,i.m,.. zl;li-S,.,,.,-.,g,(2: Te)dR+-4--I-llJET, S.,go( ;l;[:i-)d2. $ -4L,, S,.,,.,H.,gt)(-Zff;-((f-l-)dR+-4d,,E,. ' si .i,,-S.K,,o(-2- g,:i )dR+ ,d,,e, .. . ･, ･. As E>O. is arbitrary, we have ''EMmo 4in S.",g(R:Ti)dR :S 4i. S.",9D( 2Xi )dZ' L. . Step. '. II. Next, we shla11 p' rove (8) in the case where O<k(2) ,<= d for almost. all Z in [K', K]. Let 4.= {4.(t)} be real-valued continuous-parameter station-. ary Gaussian process with zero mean which is independent of gi={gi(t)} and whose spectral density is given by (18). ' ' ' '(tOm'>O)･ fElzl)(R)llP+MRI,. Similarly, let v.={v.(t)} be a real-va'lued continuous-pararpeter stationary. Gaussian process with zero mean which is independent of g2=={6,(t)} and whose spectraldensity is given by (19). Define a. == {ev.(t)} a,nd. fSY)(2) = IP+M7-ii-,.. P.= {P.(t)} as. `.

(5) ' ACriterionfortheEquivalenceoftwoStationaryGaussianProcesses 23 '. (20) . ,' a.(t)=6,(t)+C.(t) '. '. '. (21) P.(t) =6,(t)+v.(t)･ /(h,h g,e?5i:i,1,m.(g). '. =il)P,.mf(ti)===Oflra-.".d.(ih)e=irfSi(P,e)iltX,,le,":l')i'2;)11Ml(i./.,:nd fSM'(2) satisfy. and' ''.'. ' '. (23) fE7n)(2) == fi,.p.(R) = '.11,(2)+fS?Z?)(R) =fl,(2)+ lllllt'n,ili2- .. '. Since, .. ' ･,･i$ff.i,:l(,B,<=.A,i,X,) if:AzlS(f-l.).i and ' ･ '1).,hffl,-M.;IE/ll:))->..ftX)) if!hil-((-B-<1, '. so ･ (24) ' '. ' go(i[22]-)i-igp(-ilili,-M.lg(:-]-) foraii2. ' and so( ifi,'."',[:]-) increases monotonically to sD( i2X))-) for aii R when p.-->-Fo. By the same reason as above, we see that. f (2s) . ･･ig ii;i,ill,:',[li:l gg lfl,i',2[il,di. xgdthTet afin. !aeEk-hr9nShSeinde,phnlSlrftaoS.eS Monotonicaily to the right-hand side for any. 2. '. We put ''. '. ' ' (26) ' h(m)(z:T',)=Sit(2,T,)qS4`)･flt(2)dR.. . . r-'oo SIy,fl,(R)dR . L. ,. Then,fromt.heresuliofStepl ' ' . '. O:;l,tl,i:mee,4i.S.K,hCm)(R:Ti)d2 .. ' '. ' (27) - s･.,Ei.tn.,-.i,i.m..4i.S.",h(M'(R:Tt)dR$,Ii.m,,4i.S.5g(ll(,M.',X]-)dR =- t,, S.K,sp(-￡/t[(illYZi ) dR :g s･(g,, 6,) < oo.. Thus, there exist subsequences {Tt･} and {p.,} such that. p. tJt. 1.

(6) 24 ･･････････-･K.YosHIHARA .1,III}.,esM.., 41z S.",h(M''(2iTt')dR=,.li.M.,,.ll'.M.. 41. S.K,h(M"(2:Tt')d2. and so we have. ' 'ii .1-j.m.., 4i. S.K,g(2:Ti)aR:s ,i3m-,..,scm.., '4i. Sf,h(m')(z:Tt･)d2. tt. (28) N'. ==,l!Ii,I!}.,}l',rp.zl'..S'ilih,(M''(R:Tt･)d2. 'i. s ,Ii,rp., ,'. S.K,g( fl,:llX]-)dz== ,i. S.K,g(-A,･ fB:)dz ". ' tt. which implies (8).. '. Step III. Finally, we shall consider the case where h(Z)>O for almost all 2 in [K', K]. For any positive integer N, let 6iN ={6iN(t)} and g2N=={g2N(t)}. be real-valued continuous-parameter stationary Gaussian processes with zero mean whose spectral densities are defined, respectively, by. '. tt flN(2)==kiNe,.(2),.IA(R) (f>(R);sN). ,N (.11,(R)>N) ' .1`liN(R)=k,.e,.(2).,lfli(R) (fl,(Z);$N). ･･･ {N･. (A(Z)> N).. Weput , , ,,.･,.,･ ...'. ' gN(R:T,)=. :l3 x(2,Ist)){,,･ Si5`)flN(2)dR . (29)'. ,r=-oo ,...S",).fl,.(2)dZ Then, O<kN(2) E,N for almost all 2 in [K',,K] and. '. (3o) , .g.(z:T,)=Ig(o2:Tt) ((ffii:ER2))'//NN)).' v. ' Since, from the assumption that 8(8･,, g,) < oo, fl,(R)>O for almost all R, so tt. ' S,y)A(R)dR>O and S,st).(l.N(2)dR>O .,. ,,. for any fixed Tt and for all r. Similarly. t-. `. J. ' ' tt S"t)hN(R)d2>O'' S,;L)h(R)dZ>O and for and fixed Tt an' d for all r: Thus, for any fixed Ti, gN(R:Ti) and g(R:Tt). are bounded on [K'i K] and. limgN(R:Tt)'=g(2:Ti) ,. N"oo holdsuniformlyinZE[K7,K].Accordingly ' ･. : 1.

(7) "-. ACriterionfortheEquivalenceoftwoStationaryGaussianProcesses 25. (3i) .Si.m..4i.S.{fgN(R:Tt)dR=4i.SI,g(Z:Ti)d2･ From the resuit of step II and the above properties of g(-fflhX[ZRI-), we. have. Og.-im,.m..4i.S.K,gN(z:T,)dR/1.'' ,. ". '. N-co. (32) ' :;iftl-Im...l"j.m. 4i.S.K,gN(z:Tt)d2:iiY.m. 4i..S'I5,sp(-hXXI-)dR '. ･ =- ,i.S.K,g(1[{l')dzss(g,,6,)<oo･'''' ''. e. Therefore, we can choose subsequences {Ti"} and {N.} such that Tl:'-M.. iJ.ill}.. 41T S.",g"n(R:Tt")d2 =.1.i"M..' -.1,l!I}.. 41. S.K,gNn(2:Tt")dR. ' andso'`ro ilSl,...3.i).Wse.ia(V,e,,15,,,,.y,il}.s￡g}ts,i..s'2'i,gyn(ii)dZ'. (33). ..'. '. ==,il.ilP..FiM...-4tLii-SE,gNn(R:T`")dZ.,-' . s.ILrp.. ,'. S.K,g(t=X#Xl=)d4-s ,i. S.K,g( ftX), )d2. which implies (8). Thus, we have the lemma, LEMMA 4. Let ]1,(2) be Positive for almost all 2. Ilf 8(8,, g,) is .fZnite, then. foranyseqeqence{Tt}ofintegers.., ,, , ... (34) ;1-m.-41ii-S-co.g.(Z:Ti)d2s-S(g,,6,)･-'･ ' ''' ' i. PRooF.:...LetKbeanyinteger. SincefromLemma3 ' '･''''･･"'O;S},i-mds'41.S-K.g(2:T,)d2 ･ .,,･. ;. K-+oo '. ･ i:illP-lil.{ig.itt.m. 4i.S-K.g(2':T,)dR '- ･. . ?%, frg-.,"i',z.it}6n6' t/1//tV9'82iZsllK;' [ll`lidld;"'2hi,f,(/1 i" 2't) l./t'gg,.,h,,g {,,,,,} '.ld. t ttt tt. T,･l')M-.. kLM--.. 4iz SEZ.g(2:T`'")d2=.I'-M.. .i,l.Il.i. 4ih ,SEZ.g(Z:Ti"')dZ '. and so, we obtain , , '. '.

(8) 26 K, YosHIHARA TI,i--M.. 41T S-oo.g(2:T`dRiiSl.,J,tM-.Ii.M-.. 41T S5Z'.g(2:Ti''')d2. =KL`-MooT2'IM･-.m4in-S5Z.g(R:Tt'")dZ. '' 'f. :l2hlll.. 4iT SE;.g(-'hf' i" <(A' l!)d2 ==8(gi･ s2)<oo･. ' Thus,wehavethelemma. ' -"' '. -. tt. '. '. 1 4. The relation between, llee. r i7H(6i8;g28) and 8(ei, 62,)･･. .. THEoREM 1. Let 6,={6,(t)} and e,=={g,(t)} be one-dimensional continuous-. ParameterstationaryGaussianProcessesfor.which ' Ee,(t) == E8,(t) = O. SzaPPose that the sPectral functions F,(2) and F,(2) are absolutelg continuous and have sPectral densities A(2) and .IZ,(2), resPectively. Lf fl,(2)>O for almost. all 2･, then ' '. ' (3s) ILgg-l;-H(8,5;6,s)S-8(&･62)･ ' '. '. '. '. '. We shall Prove this theorem usingthe comPl2tety anagogous method used in. theProofofTheorem1inPartlof[10]. . ' ' PRooF. At first, we assume that '. ' ' ,liLm..-lH(e,s;g,6)<oo･. ' .t In this case, it is enough to prove (35) when 8(6.g,)<oo. Let {Ti} be 'a sequence of real numbers such that Tt->oo (lto6) and. (36) ''. l-ine+H(6,5;g,5)S;,H(g,ift;8,gt)+e (l==1,2,･･･) s. '. where E>O is given arbitrarily. Let T be any member of the sequence {Tt}. In general,. `. H(eioT; 6,g) = sup H((gi(ti), ･･･ , 6,(t.)) ; (6,(t,),-;･･ , 82(t.))). where the supremum is taken over all integers n and finite sets. of points tj(O <tj.S T, 1'---1, -･･,n) (cf. [3]). Since g,(tj) and 4,(tj) are Gaussian random. variables and thus H((6i(ti),･･･,6i(t.))'; (e2(ti),･･･,6,(t.))) is written by the determin,ants whose entries are correla･t･ion functiens p,A,e,(･) and pe,e2(･), SO,. using the continuity Of pe,e,(･) and p.-,6,(･), for any e,>O we.can choose rational numbers t9<O <tg･ $ T, 1' ---'- 1, ･･･, n) such that. '. (37) H(8,,T;6,ir):IillH((6,(t?),･･t,6,(t2));(6,'(t?),･･･,6,(tg))).

(9) '. '. ACriterionfortheEquivalenceoftwoSt.ationaryGaussianProcesses 27 '. (38) ' ' t2･=sjh=-3Z-･ (1'=1,･･･,n) '. Ilrlh12ri9 .e.?n.d) ,S.g'li,'k.,,i,,".) ai,e i"tegers Thus･ 6i,(tg) (ii-i, ,n) and g,(tg). 'i'S39). '. ,6,･ t.iP;lv9),ll,8¥.'ISg'IIi,S,f.n,elS'ld,Z.eih'.(Rli',.]("., =.=.".,2 '5].,[l i' Tl'n).; ,,.. .. (4o) Re- 2le0[illili, (o-o,±i･±2･･･')･ , '''. We put. ,. .. k[Z. '. (41) gShk'(Sj')=,=-.]li,lih....,e`ZeS"('zgsh)(2e)'zesh)(2e-i)) (P=1,2;]'=1,･'･,n) tt. Since for each P(P = 1, 2) and each 7'(1' :1, ･･･,n), 8Shic'(s,) converges toX6S?i(sj). in quadratic mean, as k-oo, so - H((e,(t2),-･･,g,(ta));(6,(t9),-･･,Ng,(t,?))). (42) ==H(("h'(s,),･･･,g'fh'(s.));(8Sh'(s,),･･･,gSh)(s.))) '. '. '. iS IYt H((gR'(si), ･･･,8R'(s.)); (g52'(s,), ･･･,8S2'(s.)))･. (cf. [2] and [5]).. For any positive integer k, let ASh)(z:le)==z6s,)(2)-26s,)(R--fe-2[hTn]--) (p..1,2) IIIeden' afh'(Re le) and aShb/2Aes,,(leR)s.rlee)c=ogl,ple>:pvi:1:gd2)random varlables for whlch. EaSh)(2,:k)･aSh)(2e:fe)==O . '. (p =- 1, 2;l# o; l, o= - le[2T.hnv]..-+1, ..., fe[2Ti]i-l). -.. EgmO:i$loLts 'ZfOPgnl X,/i-. I.hv,l/lrial,g,Sile,i:.!,i -tX' &1/iPlf-'iSll. ,Yl,i.-,Ti]L': iS'Sh'g,gi"far. Fo f. ]. each p(p=="2.',e.n=diase ,'(.1:. 1':i/'rri]i', .'.91 -[ew.Li}. ･. [;gehn.svfeOr eaCh P(P =1,2) Upe(i==1, ,n) are mutually subordinate and so.

(10) 28. -,･ ., K.YosHIHARA ,,i (43) lkh) == H<(eSte)(s,), ･･･,gve(s.));(gSit'(s,), ･･･,gSte'(sn))). :ll H((Uii, "', Uin); (U2i, '", U2n)) = H(Uii; U2i)･. AS(Uiir,U2i.)(r=----[gh]-,･･･,-[S-h]---1) are independent complex-valued Gaussian random variables for which. EUplr = O,. ic. '. r.. ts. El Upiri2 = ￡ ElASh'(2kr+e: le)l2. e==1 ./,. -.. -,S.l,{Ffah'(2(le,r[+.e3hT)-Ffah)(2(lerl;[0i]i)hz)} . . ..,,,. ･illl'i -Fsh)(-2i(zztTIi-l-ii!kz'g-)-,Ffoh)(-2[r.h]7F-)' '. .･' ..,,.:ooE)-.{F,(-2L<r[-+."]i)-. +a,,LZL)"LL F,(-[2.Iz]-+ 2i,z )} (p .. i, 2). so, we have (44). '. [T] --2h-1 H(Uii;U2i)== E) H(Uiir;U2ir). m r=-L,tl '' ' ' ' '=- -l}- -[ttt ,i'g(-SlUu;i; Ii2,-) == J4T-.]-S Tf'.- g*(2:[T])d2. 'where 2h h. r= --- -tt. (4s) g"(2:[T])=.;oi..z(2:ir)g(fii:`{E.''i`(i-'Yl'.Z.l)//'lli･lfli-m;ll'IETi･illi･liiil)Iill)'.'. ,u. E. Thus, from Lemmal .. .. H(U,,; U,,). jli,lllii'ilii]"i(i).i,,li111iil.Ii',,,l/lli-$'iiillllill/ll･ll,,it,illllli',i'i'ii). ･･i･r- $ ',(46)･. J' '1, ''i.)i'.i' srzoo-e(;;1:'IIimSiiZ,T.:E{,illiiii).,'. As from Lemma4 '' ''1.,.. ,.

(11) ACriterionfortheEquivalenceoftwoStationaryGaussianProcesses 29 ',,irie-.. t, ,.2.oo...g 2(( i((l,[i`li.T ii;,iE[32/;l'l )) '. i. '. '. ' i S-oo.-.g(Z:[Ti])d2 ;:$ 28(&, g,) < .., == .lp,--M.. so for all Ti of a subsequence {Ti} of {Ti}, l. t, . . ..rr. ' Tl:']-]H(Uii;U2i)=41itSh.g*(2:[Tit])dR$L .. -. , -Lh". ''. + where L is a constant independent of h and Ti. But ･'. . ･ limg*(R:[Ti･])=g(2:[Tit])'. '. . h-+O. on any finite interval, and therefore. (47) ,ttlp,4i.S':i,g*(R:[T,,])d2-t.s-oo.,(R,[T,,])b2･. t.'. '. d-'hJ ･ '. Combining (36), (37), (44) and (47), we have. Iitillm. -i.-H(ei6; 626) ;-i{ ..im,-moo -li! I-- H(g,,Ti; s,,Tt) +e. '. (4g) S.l;/le'-M.. TSiJ,1 (:-!ll}, 2411.dsJkh)) +e ;:l.l']--M-.. [T,,]1 (,1-i{l}, l'-'-El H(Uii; U2i))+e. ==.1-i-m.. 4i. S.co.,.g(2:[Tt･])dR+E;s8(s,, g,)+e. since e>O is arbitrary, we have (35) when Iitlli'm.. -1; H(6i8;828) is finite･ t. It remains to prove that 8(6,, 8,) < oo implies LiLm. -lll=H(gi6;g25) < oo. On ". the contrary, we assume that 8(6i, g,)<oo and tLtifim. -IT H(gi5;628)== oo. Then. the possible cases would be the following two: .. (a) H<6,5;6,6)<oo forallT>O, '' (b) H(6i5;g25)='oo forallT>TowhereT,isapositivenumber. In the case (a), for an arbitrarily large number K there exists a sequence of positive numbers such that Ti->oo (l->oo) a･nd. K5--;H,(g,gt;g,gt)<cx) (l=1,2,･･･). So, proceed as before and we have that for any Tt(l == 1, 2, ･･･).

(12) '. 30' ･ K.'YosH{HARA. '. ' .n. K ;:$ -6! i-H(g,gi ; g,,Ti) !$ -4ttSmhl.g*(R : [T,])dR + L,. h'. = I where Lo isaconstant. Consequently, ''-. Kg }l-m. -I; H(6,,Ti; 6,gt) S 8(6,, g,)+ L,,< oo ,. whichcontradictsthearbitrarinessofK. '. `. In the case (b), for an arbitrarily large number Kand for any T>To there exist rational numbers t,, ･･･,t. in (O, T] such that .. KS"b-H((6i(ti),''',8i(tn));(g2(ti),'''-`,62(tn))). (As ei and 62 are stationary Gaussian processes, we can always choose･such rational numbers (cf. [3]).) Accordingly, we can use-theabove method, which. leads the contradiction that Kis finite. .. Thus, we can cdfi'clud6 that if 8(g,, 62) < oo, then '1.i-m. 1. H<gi8, 826) < oo･. Hence,wehavethetheorem. . . ･'･･. tt. ' '. 5. A criterion for the equivalence of tWo stationary Gaussian processes.. THEoREM 2. Let g,=={g,(t)}andg,={6,(l)}beone-dimensionalreal-valued continuous-Parameter stationary GausSian Proce$ses with 2ero mean whose sPectral functions F,(2) and F2(2) are absolutely continuous. 11f A(2) and ]1,(R) are positive for almost all 2 an`d if 8(g,, 8,) < oo or 8(6,, 6,) < oo, then the Probability measures P6,T and P,-,T are equivalent for all T> O.. PRooF. From Theorem 1, if the conditions of Theorem 2 are satisfied, then. (4g) ''. IMg--"'m..-i.H(6,sllehs)<oo, 6r'i'. r,. ltn-l;-H(g,6;6,65<'oo,. '. '. '. which implies that .. s,,. t'/ ' "i. H(6,,T;6,g)<oo,or H(6,g;eo<oo ' ,. .. for all T>O. So, from the result in [3], P,･,,T and P6,f are equivalent for all. T>O. Thus, we have the theorem. CoRoLLARy 1. (cf. Section 2 in [5]>. Let .11,(1) be the spectral density fbr. which ,.,. ･2-oo ,･ , (50) lim2crh(2)>O '. -. '. t.. '. for som2 cr > O. 111e the sPectr,al density f,(2) is such that for some P> a+12. (51) lim2P(.1`l,(R)-.fl,(Z))=O R.oo. '. then the probability measures P,-,s and P,-,T are equivalent.. J,. L.

(13) ACriterionfortheEquivalenceoftwoStationaryGaus$ianProcesses 31 '. ' PRooF. Sincechangingthespectrumonafiniteintervaldoesnotchange Zh(e,)e.q,Ug",a,ie,n,,C,e.,re,i,a,tiO.i,ihggt f,r8u 85.0a and (5i) we can assume that A(2) and. (52) S,Lq(:f71iE-((:-l-)d2<oo foranyfinitepositivenumberL. If the conditions are satisfied, then for all 2 suMcient･ly large A,(,lg,--i== 2cr(A,(.R2i.?(i)' -o(-J[,tL-iF). '. ,. tt t. (s3) zA [:i-i-iog-,"`,i,[{]--:'7llXl--i-iog(iL(i-:IIIXi))==O(z2(}-ar)),. t/ Since P-a>-}, from (53), we have ' 't 'J. tt. (s4) - S.OO. ,g( 2[:]-)dR<oo.'. '. for an L, sufficiently large. Thus, from (52) and (54) Sgeg( fAh((RR)) )dx<cx) and. it follows from Theorem 2 that Pe,T and P.-,T are equivalent for all T>O.. Thus, the proof isSuppose completed. CoRoLLARy 2. (cf. Theorem 2'in [8]) thdt there exists an ･ entire analytic function g(R) of exPonential tyPe which-is real for real Z, and for. whichthe･following･conditionsaresatisy7ed: ' - . '. (i) .S'ffl..ga)2dR'< oo, ' '. '. '. (ii) .,li,-m..22(:(iiiliis,fli(2)),,.o, ' s . '. (iii) ･ .S-OO..R`(fi(i3)(is)-,fli(R))2'd2<oo. '. . '' le. (iv) forsomd>OandC>O, . . A(R)sh(2) ,.(s,(2. 2'3,) lo ny. tforall2suchthatIRIIIilC･. ' '. , (v) forsomem>O. , ' ','x.m.. 'kk'2(} >o (]'-i,2) '. Then,forallT>O,Pe,TandP,-,Tareequtvalent.･･ / PRooF. From (ii) and (iv), there exists a positive constant A such that. g' (2)2. .li,(2) >-- A -2-2'for all Z sufficiently large for which IRIl.l) C. So. 1.

(14) 32 ,' K.YosHIHARA R2(A(R)-f,(Z)). ･. Ah((:,i-i== Sli,l2, ..o(22(A(,Rlf,,h(i))). ' g(b2'. '. ' arid consequently,' '. '. '. -h"'`is[/B--i-iog -'hi;2{3- -o( R`(A(2)(ii,h, (2))2 ). .. ' Thus,from(iii),wehave. ' ' ' ･ Si,[..9(ifii,tf3-)dR.oo .. for a K sufficiently large. On the other hand, from Theorem 1 in Section 4 of [8], we can deduce the fact that a change in spectral densities which satisfy Condition (v) on any finite interval does not disturb the equivalence of the measures. Hence, from Theorem 2, we have the' corollary. '. ' References 1) FELDMAN, J. Equivalence and perpendicularity of Gaussian processes, Pacific J. Math., 8, 699-708 (1958). 2) DoBRusHiN, R.L., Passage to the limit under theinformation and entropy signs, Th. Prob. and Applic. 5, 25-32 (1960). 3) HAJEK, J., On a property of normal distributions of any stochastic process, Selected translations in Mathematical statistics and probability, 1, 245-252, Providence, R.I., 1961. 4) PiNsKER, M.S., The entropy, the rate of establishment of entropy, and entropic stability of Gaussian random variables and processes. Dokl. Akad. Nauk SSSR 133, No. 3, 531-533 (1960).. '. 5) ,Informationandinformationstabilityofrandomvariablesandprocesses, Holden Day Inc., (1964).. 6) RozANov, Yu. A., On the problem of the equivalence of probability measures corresponding to stationary Gaussian processes, Th. Prob. and Applic. 8, 223-231 (1963).. h. 7) ,Onprobabilitymeasuresinfunctionalspacescorrespondingtostationary Gaussian process, Th. Prob. and Applic. 9, 404-420 (1964). 8) SKoRoHoD, A.V., On the densities of probability measures in functional spaces, Proc. Fifth Berk. Symp. Math. Statisti. Prob. 2, part I, 163-182 (1967).. 9) YAGLoM, A.M., On the equivalence and perpendicularity of two Gaussian probability measures in function space, Proceedings of the Symposium on Time Series Analysis, edited by M. Rosenblatt, New York, Wiley (1963). 10) YosHiHARA, K., The information rate for continuous-parameter stationary Gaussian processes I, Science Reports of the Yokohama National Univ. Section I, No.. 15,1-8(1969). . ,. , The information,rate for continuous-parameter Stationary Gaussian processes II, Science Reports of the Yokohama National Univ. Section I, No. 16, 1-8 (1970).. `.

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