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(1)A Criterion for the Equivalence of two Multi-Dimensional Stationary Gaussian Processes By. Ken-ichi YOSHIHARA .. 1. Summary. In [5], the author considered a criterion for the equivalence of two onedimensional stationary Gaussian processes with, not necessarily rational, densities. In this paper, we shall generalize the result to a criterion for the equivalence of two n-dimensional Gaussian processes (Theorem 2).. 2. NOtations and definitions.. We shall use the same terminologies and notations in [3]. Let &={&(t)} = {(6ii(t),'", &n(t))} (l = 1, 2) be n-dimensional real-valued continuous-parameter. stationary Gaussian processes with spectral and joint spectral functions J7ktsijt(2) (l･=1,2;7',7'i=1,･･･,n). We assume that gij(t) (l==1,2;]'=1,･･i,n). are continuous in quadratic mean and. Ee,j(t)==O (l==1,2;1'=:1,･-.,n).. We put (i) H(gioT;6,gy =='Hkrs (s,g)=sup >i.) pers (Ei) iog pP. i. ,6. 1i' [EEi.i. where the supremum is taken over all Partitions of the measurable space (9, S.) on which probability measure P6,T and Pe,T are defined. Let. --. a･. (2) 8(gi,s,)-S,,,S,OO{,,;#=;fo6,,a)fsi>),(R)-n-iogg2t,l21;X]}d2. '. where A6e(2)=Uftisele(2) IIj,k=i,...,n (l == 1, 2) are the matrices of derivatives of the. spectral and joint spectral functions of the processes 6i =(gti, ･･･,6in)(l =1, 2) and fS;.},,(2) is a typical element of the inverse Ae-,i(2) of Ae,(2),. 3･ Relations between ttt 1H(6,g;6,O and 8(g,, g,)･ ... T In this section, we shall prove the following theorem which is the generalization of Theorem 1 in [5]. The method' of the proof depends heavily on the one used in [4]..

(2) 34 KYosHIHARA. THEOREM 1. Let gt=={6i(t)}=={(&,(t),･･･,g,.(t))} (l =1, 2) be n-dimensional. continuous-Parameter station,ary Gazassian Processes with sPectral and 1'oint. sPectral fLtnciions 4,,.,-,,(R) and ' Eet,(t)l=O (l=1,2;7'--1,･･･,n). SuPPose that Fti,.,-i,(2) (l =1,2;7', fe= 1,･･･,n) are absotutely continuous. if 62 is regntar of full ranfe, then. '. ,. (3) , }i-m..-I-H(g,g;g,g)$8(e.e,). , PRooF. It is enough to prove (3) only when 8(gi,e,)<oo. At first, we assume that tt 1 Iim H(6,g; g,,T) < oo . T-oo T. .. For an arbitrary positive number e, let {Ti} be a sequence of positive num-. bers such that Te-oo as l-oo and. <4) l!n+H(6,5;g,5)S. ;, H(g,3i;e,gi)+s (l==1,2,･･･). Let Tt be fixed and write T in stead of Tt, for brevity. In general, '. '. ttH(g,g; / 8,g) == sup H((g,(t,), ･･･, 6,(t.)); (g,(t,), ･･･, g,(t.))). where the supremum is taken over all t,, ･･･,t. in (O, T] (m=1, 2, ･･･).. Since H((g,(t,),･･･,6,(t.)); (e,(ti),･･･,8,(t.))) is a continuous function of p6,s,,(t.-t.) (l == 1, 2; ]', le = 1, ･･･,L'n; pt, v= 1, ･･･, m) and each p6isik(t) is a continuous func-. tion of t, so, there exists a family of positive numbers {t,, ･･･,t.} (tj E (O, T],. ]'=1, ･･･,m) such that for any positive number ei (5) H(8i8;62if) ;!I H((6i(ti), ''',6i(t7n)); (6'2(ti), ''',62(tm)))+Ei'. Without loss of generality, we may take t,, ･･･,t. as. i.. ti=1'ih (i=1,･･･,m) where h== (Ll}-)g and g and ]'i (i=1, ･･･,m) are positive inie' gers. Let. `. '. (6) 6i.(t)=S-oo..e"2dxg,.(2) (l=1,2;v==1,･･･,n) and'. '. <7) et.(t.)==gSB'(1' .)==SHn..eZi'ptdzg,(tc)(R) (l=1,2;v==1,･･･,n;ict==1,･･･,m).. tt Let [T] be the integer such .that TIS [T]<T+1, and put ' (s)' '. tt t. . zo==20[ge],,(o-=o,±i,±2,･･･)., .. Define. y.

(3) e. ACriterionfortheEquivalenceoftwoMulti-Dimensional .35 '. "(9)'. '. -,"ki [T] -1 ･. '6Shvi(7'pe)=. '. ,..-ill]h]-[.]42qd'"(2g",(l}'('lq+i)-2g'iB'(llg)) ". (v == 1, ･･･,n; pt = 1, ･･･,m; le = 1, 2, ･･･).. Then, for each v (v =1, ･･･,n) and l (l =1, 2), gEk.k converges in the mean to ･6Ete', and 6Sh.h (1' .)(pt L. == 1, ･･･, m) is subordinate to the family of random variables. c<10) VVi.(fe)={Wi.,(k):v=-I2T-h]-,･･･,I2Th] -1} 'where. .. vvi..(k) =,.$,exp (-iliE-ge-]Z: ) ×. "(11) . ×{2e-ih.)(L2[VThL]Z-+mfe2-[rcThZ-)Hxe,(z)(tu2[VThi+ 2(rci[TTI)]hT )}. ' ,(l==1,2;v==1,･･･,n;pt==1,･･･,m;v==--[2T-hl-,･･･,-[2T-h]--1)･' We rernark that VVt..(k) are complex-valued Gaussian random variables such that for each le (le =1, 2, ･･･). {(Wiiv(le),''',1)Vinv(fe))} (l==1,2:v=-m[2T'h]m,''',n[2T-h]--1). -are mutually independent and. EPVt..(k)=O .. t'. <12)' EWtj.(k)IVtj･,(k)FF?E?.).-,(e.)(2("[+Tl])h")-Fts?).-ie.?(2[VTh]n) ' L,.l..{jF?i,ev('2(Z'Tl)T+2ZLZi;-)-i?,,g',,t(3.Vrc]+2￡LZ!-)} tt. tt .. (l=1,2;1',7'i=1,･･･,n;v==--[2Th]-,･･･,-[2Th]-um1).. ' .Accordingly, we have. t. +'. L ,. '. t<13) ' J(ic'==H(((6{?k(7'i)･'''･g{nhh(1'i)),''',(4￡?k(2'm)･'",6fnh}(im)));. -((6E･?k(i',),･･･,gSigk(i',)),･･･,(gM]'.),･･･,6Sk%(i'.)))). ,. : H((Wii(le), ''', Win(le)); (W2i(k), '" , ur2n(le))) "[Ti]fip1. = 2h= H((J7V,,,(le), ''', Winv(le)); (W2iv(fe), '''',1 17V2nv(le))). [T]. v=--2h- '. ;:s./[tpt-'lilii[.l{ i ,,i;.l:},=,rsx? (v)ri"(v)--z---5-iog gg:l:;,ilf;,it(z,)l,i;l;.1:-iii:.# ,}'. 'where 2h rtJ･j･i(v)==EJ71Zt,･.(k)17V,,･,.(le) (l..i,2) 'i :and rS];･,) (v) are the elements of the inverse of the matrix, ll r23,,(v)Uj,,J=r,..,,n･. '.

(4) 36 K. YosHmARA. ' As`is known in [2] ' and [3], for each v(v =- [2Th] ,･･･,-[2't-h] -1), there. exist non-singular linear transformations . (14) Utj.(le)=tr.,cj,([2TVT]-)Wi,.(k) (l=1,2;]'=1,･･･,n). t/. such that. ' .. (15) cj,(iTY-T]---)$1 (]',q.=1,･･･,n;v=--[21Th],･･･,[2Th]-1). and the random variables (U,..(k), U,..(le))(v==--[2T-h]-,･･･,-[tTh]--1) are. .. independent.Sousing(12),wehave ' ''. '. ,,#, ..,rsti? (?)ri"(v)-n-iog g,el iA Zf.f.:[:l:i;.;:'i:iiii".. =i t#,{ Sl 21I;f.:[ii?li-F22---i-'og Zl 2-l;;'.:((￡2)l-II-} -= tY.,g(-S-1Uu;;.g[lek,l'l 22-). ' a6) -]:ll]'gq,;･.,iCjg(-[2'4ger)Cjq'(?g-Z]-)EWiq.(k)vv,,,.(k). d=1. == nZ9 .i=1. q,il,ll..icjq( [2TVT]. )c,･,,(-[2-ij-ni)EW2,.(le)VV2q･v(le). e.Z.OO. )-.. q,tr, ..ICI'q( -3-liiiZ,L)cj,･(2-l7iZ:･-). ,.]E,OO. ]-.,,li,ill=,Cjq(-[2miiLiZ]iiq)cjq･(-[2-lisl'"Z]-). ' 2(V[-+TTIL' + 20hZ )-ftiaeiq'(2-VT-4r+ 20hZ {Ftiq6iq'( {Ft2,62q'( 2(V[+T-1:)T + 2ZT )-jPle2,e2,r( 2[VTZ] 'i-. )}. 2ZT )}. '. s,. where g(x) == x-1-log x(x> O) and g(O)=:c>o. Thus from Lemma 1 in [5] and.. (13)and(16) ' ,. (17) limJ(ic'. ic-oo. '. n ''1 s ,,il,ll=,cjq([2iZ])cjq'(gliLZ!]-){Fei,6i,t(2(V[+Tl])rr)-Ef,,e,,･(. =< Z. ' j'--1. a,! )} [T]. 2 v=fioo,9 ,,;.,..,cjg([2' gZ])cjq･(r2'--ipa']-){I7?2,s2q'(2(V[+{7;,Tl?/Z!Z)-E?iq6iq'([2TVZ])}, ". hand, from (15), if we Suitably choose cj,(･) such that for On the other almost (18). all. ' ' t tt (]',q=1,･･･,n) limcj,(R+A2)=c,"･,(2). dZ-o. where. ' ' ,q ,･･･,n) are measurable functions, then we have.. c,*･,(Z)(ic* (R)i;Sl 1 ll', q == 1. ,.

(5) A Criterion for the Equivalence of two Multi-Dimensional. 37. 2'III}, iz ,,;,.,,Cjq(2+a2)Cjq'(1+aZ){Ftiqg"iq'(2+a2)-4i.,ei,･(Z)} f(19). '. '. '. '. n ==,,ill,Il..,Cj"q'(2)Cj"'Q'('il)ktqg"iq'('il) (l=1,2;.1'=1,''',n) for almost all 2. .Since 6, is regular of full rank, so from (19) (20) .. lfor `. !(17). n ,,:l,1=,Cj*q(2)Cj*q'(Z)k2ae2q'(Z)>O (1'--1,'･･,n) almost all R.. ' Thus, the conditions' of Theorem 1 in '[5] are satisfied, and from (4),. and (20) we have '. (5),. li-m,.. I H(6i5;g28)E,l,i-m..-7ii,E-H(g,ifi;6,gi)+E. !.,,-. iimTi-{H((gi(ti), -1. ･･･, 6i(tm)); 6,(t,), ･･･, g,(t.)))+e,}+e. i;il ;' tlEoto -;' ;' i- { ]II'ts J(k'}+S. ,(21) g Tlii-M'oo j]￡i -2Tl'7..￡.Oi... i g,;,..,cj,(-[2-8Z]-)cj,t(L[2TVZ]){.IF?,,e,,･(2!/l9V[m+Tlll')"Z)--Egi,6i,t(ig"t])} 9 ,,lil,il..,cjq(38Z-] ) cjqt(t2Lg/'+) {1 l#2,g'2,'(2(V[+TTI])Lt) m'F.e2qg-2q' (. 2vT )}. [T]. '. '. `te., iS,oo9 :ilil,q:i,Cl,qilifl[l.l[ili)d2+e. +e ,. By construction, it is obvious that r. n. '. Sl)log q,iil'Il-iC'*'q(2)Cj"'q'(2)1iqg"-i,･(2) .. == log det A6i(2). "=' ,,i;,il=,Ci"'q(2)Cj"'q'(R)1?,,,-,,,(R) detA62(Z). ' "･. ,#, iog :l/ill::l".lizZi,`1.*,q izJi))klig.iig ((,Jzi)) =,ll.},,fgnt}2,(R)fei,6i,(jR). ･Consequently, from (21), we have. ' li. EL Hg6,s;8,s) S 8(e,･ e2)+e' ･･ T-oo T. ' As e>' O is arbitrary, we have (3), if both 8(ei, e2) and'.lilx I H(g. '. ,5･ ; 8,5). are.

(6) 38 ' K. YosHIHAR'A. finite. ･- .. It remains to prove that 8(6i, 62) < oo implies ILg] -li H(8i5;g25) < oo･ On/. the contrary, we assume that 8(6,, 6,) < oo and lim !H<g,g;g,6) == oo.･ Then,. r-oo T. the following alternatives would be the possible cases:. ' forallT>Q; (a) H(g,g;6,g)<oo .. (b)/ H(6ig;82g)=-oo forallT>To,,whereT,isanon-negativenumber. In the case (a), for an arbitrarily large number K there exists a sequence of positive numbers {Ti} such that Ti-oo (l--> oo) and. '. ,. Kg-I,-H(g,ge;8,gi) (l==1,2,･･･). By assumption, H(g,8t;g,ffZ)<oo (l =:1,2,･･･). So, proceed as before and we. ' , Ks-{ .1,i-m.. -71s7 H(g,gi;g,3i) :;l 8(g,, 6,) < ･c>o, which contradicts the arbitrariness of K. In the case (b), for an arbitrarily large number K and for any T>T,, there exist positive numbers ti,･･･,tve in (O, T] of the form tj=-tti (d and. sj(7'=1,･･･,m)beingintegers)suchthat ･. '. ' ' 1 K;;lTH((g,(t,),･･･,g,(t.));(g,(t,),･･･,e,(t.)))<oo･ ,. Accordingly, we can use the abQve method, which leads the contradiction that. KS8(8,, g,)<oo, too. Thus, we can conclude that 8(6,, g,)<oo, implies. lim H(g,5;6,5)･ ' 1. T-+oo T. Thus, we have the theorem. ,. CoRoLLARy. Under the assumPtions of Theorem 1, Si[m.-g-H(6,&6,g,>. exists and ' ,. -. ' e,,T) == 8(g,, g,)･ (22) ;Lm.. -}-H(8,Z. PRooF. Since for two n-dimensional stationary Gaussian processes. 1 , }L/Elil T H(8,g;8,3) >=8(g,, g,), ' tt. '. (cf. Remark to Theorem 10. 5.2 in [3]), so, from the theorem, we have (22)... f of multi-dimensio'nal stationary 4. A criterion for the equivalence. Gaussian processes. ' Using Theorem 1, above, we can generalize Theorem 2 in [5] as follows: THEoREM 2, Let gt={&(t)}=={8t,(t),･･･,et.(t))} (l==1,2) be n-dim`ensional. ,.

(7) A Criterion for the Equivalence of two Multi-Dimensional 39 real-valued continuous-Parameter stationary Gaussian Processes with zero mean. Lf g, and g, are regular of .futl ranle and if 8(e,, 6,) < oo or 8(8,, 8i) < c)o, then. the Probability measures Pe,T and Pg,T are eq"ivalent for all T> O.. PRooF. From Theorem 1, if the conditions of Theorem 2 are satisfied. then ･･ ''. ,. }i-m.. iH(6,3;6,g)<oo or }l,!m.. Tl H<6,g;6,g)<oo,. which 'implies. H(s,g;6,g)<oo or ll<6,g;g,g)<oo ' for all T>O. So, from the result in [2], P.-,T and P6,{ are equivalent for all. ,. T>O. Thus, we have the theorem. '. References 1) DoBRusHiN, R.L., Passage to the limit under the information and entropy signs. Theory of Probability and its applications 5, 25-32 (1960). 2) HAJEK, J., On a property of normal distributions of any stochastic process. Selected transla`tions in Mathematical statistics and probability, 1, 245-252, Providence, R.I., 1961.. 3) PiNsKER, M.S., Information and information stability of random variables and processes, translated and edited by. A. Feinsteln, Holden-day Inc. (1964).. 4) YosHiHARA, K., The information rate for continuous-parameter stationary Gaussian random processes III, Science Reports of the Yokohama National University,. Section l, No. 16. 9-18 (1970). .･. 5) ,A'criterionfortheequivalenbeoftwostationaryGaussianprocesses with, not necessarily rational, spectral densities, Science reports of the i Yokohama National University, Section I, No. 16. 19-32 (1970).. p t. '. .. ..

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