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(1)The Information Rate for Continuous-Parameter Stationary Gaussian Processes III By r. Ken-ichi YOSHIHARA '. 1. Summary. In this part, we shall prove the same formulas as the one in [4] for the information rate of multi-dimensional continueus-parameter stationary Gaussian processes under less restrictive conditions, using the fesults 'in:[7] and [8]:. 2. The main results. , Let 6== {6(t)}=={(8,(t), t･･, 6.(t))} and ny =={rp(t)}= {(rpi(t), ,'", rpm(t))} = {(6n+i(t), ･･･. ,en+m(t))} be real-valued stationary and stationarily correlated random. processes, which form an (n+m)-dimensional stationary Gaussian process (e, rp). Let Egj(t) == O(1' = 1, ･･･, n+m). We assume that each gj(t) is a stationafy. process continuous in quadratic mean. Let A6(Z)==`11.1?,.-d(2)11, i,j'--1,'",n,. An(Z)=(l.1?,g,(2)11', i,1'--n+1,･･･,m+n, Ag"rp(Z)==IIk,6,(2)II, i,1'==1,･･･,m+n ir. be the matrices of spectral and joiint spectral densities of the random pro-. cesses 6, rp and,(6, rp). Eurthermore, let , (i) ･'. 46,-= ･i,, S,coi-og detA,gERIIIt.d,sl?,fi!E'(2) .d2... v. whtere de･t Ae'v(2)(det A,--(R)) is a non-vanishing principal'mimor of highest order. P(P') Q･f the determinant detAg(2)(detArp(R)) and de'tAg'v"(R) i's the princiPal. minor of order P+P' of the determinant detAetp(R) which contains detAs'v(R). and det A,'-(2). ., Weshallprovethefollowingtheorems. '' x. THEoREM 1. SuPPose that the continuous-Parameter Processes 8=={6(t)}== {(6i(t), ･･･, g.(t))}'and rp == {rp(t)} == {<rp･,(t), ･･･, rpnt(t))} == {(g.l',(t), ･･`, g.÷'-.(t))} for･m. an (n+m)-dimensional stationary Gaussian Process (g, rp). SuPPose that I7t&.(2) are absolutely continuozas and have densities .1?,,-,.(2). if6 (or rp) is regular, then. I(6,rp)isdofned,and ' ' -. ..

(2) 10 K. YosHIHARA (2) , . I(6, rp) == Lsv. where. ''. ' '. ''''/'lii''',','i(g･'77)=9',}itt,.''{}lii(ij(i,,j)'.･''1t.,. 'J. ''. THEoREM 2. Und2r the same conditions, 7(g, rp) is dofned and. (3) 7(8, rp) =Lg'v where. v. tt. -t; i""(e, ty)== y-m. Ti i(gg,trp). kl. '. 3.Proofsofthemainresults. . . ploo7hoefOrTehMeoireCmtnl.be PrOVed on iines simiiar to those used in [7] for the. PRoOF OF THEoREM 1. From Theorem 10. 4.1 and Remark to Theorem 7. 4.1 in [4] the inequality 'I.VMoo wu1,. I(6oT, o7E') >= L,-v. '. holds for any 6 and rp. So, if Lerp == cx), then (3) holds obviously. Therefore,. it is enough to prove that . , (4) li-m,..+I(g5,rp5)-<-L6v ' ', ' when L.-rp is finite. '. '. (I) At first, we assume that g is regular of full rank, i.e.,P==n. Let. l.i.m.. {i-I(88, rpg) be finite･ Let {Ti} be a sequence of positive numbers such. that Tt-oo as l-->c>o and for an arbitrary positive number E. (5) '' fi/illm.I･i(ss,rpg)-<.-TiTi(gsi,rp,TL)+e (i=i,2,･t･) SL. For a while, we fix Tt and, for brevity, write T instead of Tt. In general. , I(6s,rpg)=supl((g(t,),･･･,6(t.)),(rp(s,),･･･,rp(s,))) ,. where the supremum is taken over all t. ･･･, t., s. ･･･, spt in (O, T] (pt, v =1, 2, ･･･).. Since p.-iej.(t) are continuous, so there exists a family of rational numbers,. {t?,t･･,tP,} (t9E(O,T],v==1,･･･,v,). such that ･. ･ J(gg',)7g)S.I((8(tg),･･･,e(t2,)),(,7(t?),･･･,)7(t9,)))+E･'. ' ' without 'loss of generality, We assume that for some h= (-21Dg ' ' t,O･=1'ih (i=1,･･･,vo) '. " wheregand]' i(i=1,･･･,vo)arepositiveintegers. Let ･ 'tt '. '. x.

(3) ' ' ' TheInformationRateofStationaryGaussianRandomProcessesIII 11. tt. ' (b - ,8,(t)=S-OOooei`Zd2.-.(2) (pt==1,･･･,n+m).;. Put - .･'.･ .. ,. (s)- ･, - '''6.(tg) == 62h)(2'.) ==･.S' I･he`j"2dzese)(R) .' '' ('pe -- i, ''` ,"rl,,, iv = i, '",., ilo).': '.. and (9) '7pt'(t2)=='72'(7'v)=4?se'+n(1'v)==S-X.e`'V2d2ege..(R)' li '. '. '. ' (pti==1,･･･,m;v=1,･･･,vo)･. Let[T]betheintegersuchtliatTS.[T]<T+1,andput ., ･.. f. . Ro=-le2-[0-4Z]---.(o=o,±i,±2,･･･).' ' '. Define -2ich-[T]-i. (10) 6Zig'ic(1'p)=,.L-iiihl.,,,,eiAq""(26zh)(']lqg"i)-Zezh?(Rq)) '. ' .. ' (pt =1, ･･･,m+n;v== 1, ･･･,vo;le=1, 2, 3, ･･?. Then, for each pt (pt==1,･･･,m+n), 8$'k(]'.) converges in the mean to 6hh'(]'.) and 6$'k(7'.)(v=1, ･･･,vo) is subordinate to the family. ' (11) W.(h)--{W,.(k):v----[2Th].･"'･'[2Th] "1}. where ･ '. :t. (12),･ .Wptv(le)=-.;,ehte[m-Z{2,se,(--Z8hi+-le2--/T-hg-)-2,se,(-Z--ge]T-+ 2(rcle"[Tl))h-')} ic 2rchn.. , '(pt==1,･･･,m+n;v=--[2T-h]-,･･･,-[2Th],-1). We remark here that VV2,.(k) are complex-valued Gaussian random variables v. such that, for each k(le = 1, 2, ･･･), (I7Viv(k), ･･', VVnv(k)), Wi+i,v(le), "', Wn+m,v(le)). (v==-[2Th],･･･,-[2Th]-'1)aremutuallyindependentand･ .. ･... '. (13), . '. E W..(k) == O. El7V)v(le) VVpt'v(k) == F62h) gw)( 2("[+Tl])hn )-Fese, ep,( 2["Th]I-). ' (pt,pti!i,･･･,m+n;v---[,T,],'",-[2Th]-i).. Accordingly,wehave '' 'J(ic). == I(((gih,)(7',), ･r ,,gAh,)(1',)), ･･･, (6ite)(7'.,), ･･･,eah,)(]'.,))),. (14) ((rpShk'(]'i),'",rpinhk(]'i)),'",(rpft'(lvo),'",rpts'ek(]'vo)))). !.. .t. ' ;:ill((Wi(k),''',V47n(k)),(VVn÷i(le),''･,VVn-m(le))).

(4) tt 12･･ -',K.YosHI･HA'･RA ' '. '. L[T-]--1 '. P,h == Z I((VViv(le),'",'VVnv(le)),(V'Vn+i,v(le),'",Wn+m,v(k)))'. v=--2h- tT]. .. As is known in [4], for each v(v=-ttT-h],･･･,-[2T-h] ---1), there exist non-singular linear transformations. (15) '. ' ,Uptv(k)=a2,crtq(t2TVZ]-)P'Vqv(k) '(ge==1)''',n) ' `. '. and. '. (16) ' Vptf6(le)=,,￡.,dpt'g'(-t2'li'll'ge-])JVn+g',v(le) (pt'･=1,'",M) N. such that. (i7) '. c,,,'. tt (-[2･l71Z-)i:;ii (pt,q-'i,･･･,n). and. tt. (18) . ･'d,･,･('//4T-]),$'1 (pt',q'=1,･･'･,m). '. and the random variables Ups,(le) and Vpt,(k) are ･all independent Gaussian except for pairs (Uptv(le), Vpt,(le)), pt == 1, ･e･ , e S 'min (n, m) `. Using (13) and the equaldity .. .. Fese) ese)(4) =,-.S..{itpteFe'( 2+h20Z )-Fe,g.･(L2-hO-rr )}. we obtain-''l ' '. '. EI'V]ttv(le)J>V)tt'v(le) =,lllli)..{FeFtgpt･( 2(V[+'Tim)Z + 20hZ )-Fe.:-,,･(L[2-llik!] + 20hZ )}. ' ' ,, [i=,=lili..AFeptgFc･(-t2Ei7Sr+20h'C) .,. and, therefore, for each pt(pt = 1, ･･･, e). s. (19) EiUrtv12==,S.,,lil.i],CFtq(lt[2T"Zi)cbq'(Lt21illS-)EM7qv(k)fi7q･v(E') /-. =,,2,,OO-.. ,El), ,Elwwl)m,cptq(ha[2-17ilSri) cptq'(-[2TVag-)A"Fgqsq･(-i2ge]-+ 20,Z) >o. `. ,t. tt. (20) EIVptvi2==.i.Il],.#.ll].,dptr(-2[-liil!]-'-)dptr'(-2'lilti)EVVn+r,v(fe)VVn+r',v(le). '==e=]ili,., i;l.i,dptr(-2mlliZL])dptr'(-21ii'!]-)AFnrrprt(-e'"V.Z: ･+ 20,Z '). ' ' ' (21) EUptvZtv=tr.,,#,cFtq(-2-li4Z!]L)dptr(L2'miT!l{lr)EWqv(le)tVpti.I::1<Z+rv(le) ,. 't. ttt t -... ' ==,=ZO '. .O. '. .:l`), ,Zmp=,Cptq( Zli'Z] )dptr('38k")AftqTr(T3i' i{r+-2LZ-T)...

(5) '. The Information Rate of Stationary Gaussian Random Processes III. t3. Thus, using (19), (20) and (21) and.the.inequality. oo' 1 Z cj12. .;;i} iElf200s ib,i2 i-".Fg･p,,,,." ial.f;iiZ,i2. y=-co j'=-oo , for the convergent series .]Sl Iajl2, S) lbjl2, S lcjl2, we have. j=-oo j'=-oo j=-oo. s.. -[21i-?--1 . ... -. ￡i.M.. 1(k) $ ,ltLne .=-2-,z,- I(( Wiv(le), "' ,' VVnv(k)), (17Vn'-･i,v(le), "' , PVn"4m,v(le))). 2h ''' '. s･. '. ny !i--1. == 2i.M. ...ll-l,zp..I((Uiv(fe), ':', Unv.(k)), (Viv(le), '" , Vnv(le))). 2h ' -[TL]--J.1. == l/tllg .=2ffZh+[m- ,S,I(Uptv, Vptv). 2h (22) ---l}-.i2flli,It-,' ,;I;liog '(i- .Ebe,ptiv,tti"e2,.i2). 1 , -[2'.]--i. ;:Si-r2",il.}-i.=;-[..,-. ' -. , -2h iQ,g1-s.u,p 'qZn,i.XIti.Cptq([2ii<:)dptv(tillil)AFeqn.([i21TlltiZ+2hOT. ). 12. e=='"'-i'"O'i"'i'{(,S, ,;il.]icptg ([2Tt:) cpl,r(//V-Z]) aFe,/1 ,･ ([21il}]ii + 2hO-'V) ×. × (ri-llii ,tL,dpt'([2T':) dptr" (r2T"mZ]-) A"LFL?r?r･ (r2.'V-Zi + 2ST))}. '. ' ' ' 'i ,S,.￡,Cptq(c21TthZ)dptr([2i<li>aft,n.(3)Vg)i2' --Le oo ' $ 2pt¥iv=2...10g. {<i;iqtZ-iC"q(r2TVZ)'Cptq'(t'ii2Ti-Z])AFe,e,,([2ilil:))×. .. × (.;.ii, .t2,dpt'r (r2TVg) dptr' (t'2TV'i )A]FLtrtp.･ (r2."iili))}. From (17) and (18), if wesuitably choose c.,.(R+a21)(pt･, q== 1, ･･･,n) and d.･,･(R+a2)(pt', q' == 1, ･･･ , m) suchthat for almost all R'. lim c .,,(Z.+aR)=c.*,(Z) (Ft,q=1,･･･,n) d2eo. and･･. '. ' 'S'//II},dpt'q'(Z+A'R) =d."･,t(Z) (pt',q'==1,･･･,,m). where c.",(Z)(Ie.",(R)'･l ;$ 1)(LE. q'= 1, ･･･ , n)･ and d.*t,･(R)(l d."･,r(2)I;.:{ 1)(pt', q' == 1, ･･･, m). are measurable (i, 7' == 1, ･･･,m+n),. then we have.

(6) 14 . b KYosHIHARA (23) Li,M.,'i2'T,￡.,,E.ll-',Cptq(2+12)Cptq'(R+AR){jFl6qsqt(2+a2)-Feq.eq･(R)}. nn = Z 12] Cpt*q(Z)Cpt*q･(1)kqeqr(2). '. q:=1 q' =1. ' (24). >i,M-,,']Iz--.iS.i,,,;l.li,dptr(Z+d2)dper'(2+A2){iFvrvr'(Z+a4)'Frprnr･gZ)}. i mm =ZZdpt"r(2)d,*ar'(2).IC17rnr'(R) '. " .. , r=lri=1 ..'. ' t .ttt. ' (25) Li,M-.,-tr'IT2,#,.S,i-,Cptq(2+aZ)dptr(R+A2){"Fleqopr(R+A2)LFeqvr(Z)}. t.. -. .-' ' ･,'" ==]2)2C.*q(2)dpt"r(2)kqopr(2) '"' '. "pt ･. q=:l r=1 ,. for almost all 2 and for each pt(pt=1, ･･･,e). Since g is regular of so from (23). full. rank,. (26) l$l) SI]cpt"q(R)c--ll"q･(2)kqeq'(R)>O q =1 qt =1 for almost all 2 and for each pt(pt=1, ･･･,e)."''. By construction, it is obvious that. (27) iogdetAd6ZiRl,.d.,e.v?S"A"(R)==iogdettS-gf)k,d.,Ae.(tR)A"N(2). ,, 'r ･nm '2. == ･1...-. q==lr=1. '' e';'"i'. ]E)ZCpt*q(R)d:r(R)kqTr(2) m2 log. st=1 (dS, ,.' ,tt,Cff",qg2)Cpt"q'(Jl)kqeq'(2)). (.#., .tt ,d, pt"r(']l)d pt"r'(R) 117rtpr'(2)). ' (27), the condition of Theorem 1 in [8] are froni (26>' and Thus; we see that satisfied for 'each pt(pt=1, ･･･,e). Accordingly, from the theorem, (5), (6), (22). and (27), we have. 1. 'L t' tt ' .''" liM-J(e6,o75);:;l,,,ill.l:{i-tli;I(6se,?7g'i)+e 1. r.oo T. . ･. '. 1'. ,,S;,i.-M-･･.-.'T, -{I((8(t?),･･･,8(t9,)),(77(t9),･･･,77(tBo)))+ei}+e. s{; 1im-1. {tt. J`ic'}+e -Ti-.oo Ti. kl-oo , ., j. i.tep...log1 q]I'lilir￡iCptq(iTVi:r)dpa(-[2t?iilil])AFg,,.(r2Tvf].)2 ;iiii ,l,lilMIi. ,1,ii?i[. {(,;i ,ii.l],Cpq(t2ii<Z) cpq'(t21i fl)aFeq6qt(r2TVS)) ×. ×(.1.I:, ,S.,dptr(r2T"i.S)dptr' (-3'lit?i)aFnrTr･ (r2TrrV)iS))}. +'E. tt .tt t tu '. ].

(7) The Information RateofStationaryGaussianRandomProcesseslll l5 :Sl) S}cpt*q(R)d`"tr(2)1;qopr(2)2. g -},S, S-OO.i,g i. qSi ,S=' iC"*q(2)Cl* 'i(iiikgeq'(2).S, .S-i.,d:;(2)d.".t(R)h,n.(R) d'Z. +s. .t. tt .t-". = Lerp +e.. ' ' ' ' ' Since E>O is arbitrary, we have (4). Thus we have the theorem, when g is regular of full rank, and li.m.. I.i(6& rp6) is finite. .. .,. v. tt .. that Lerp < oo implies Il/ilm. '; I(8s, rpE),< do. The proof. ･ It remains to prove. ,. of the latter half of Theorem 1 'in [7] can be completely carried over to this case. On the contrary, we assume that L6, < oo and li.m.. ; I(65, rp5)= oo･ Then, .the followi.ng alternatives. would be the possible cases: ,.. I(6,T,rpD<oo forallT>O;. (a). (b) I(e,T,. rpgy==oo forallT>To,. t' ' where To is a positive number. ' In the case (a), for anarbitrarily large number K there exists a sequence of numbers {Ti} such thatTi->oo (l-->oo) and. -"7I(6,Ti,rpgi) (ltl,2,...)..' ' -"-. KS .. By assumption,J(e8t, rp,Ti) < oo (l=1, 2, ･･･). So, proceed as before and we have TI.lll:-MI,.-t: I(6oT`, rpg`)+e;:$ Lg,' +e<t)c),, , Kf{{ 1' which contradicts thearbitrariness of KL (Thus, in the case (a) ･ Pll:s:lli(g"or,rps)=oc)==LE,.) '. e. In the case (b), for an arbitrarily large number K and for any T>To there exist rational numbers t,, ･･･,t. in (O, T] such that' ". 1. t K$ I((e(t,), 'T-. '. ･･･,g(t.)), (rp(t,), ･･･, rp(t.))) < .oo. Accordin' gly, we can use the above method, which leads the contradiction. that KSLen<oo. (Thus, we have '. 1. '. l.'illEl T I(85, rp6)=CX)=Lg"q.. in the case (b), too.). '. th?fl'.e,Mg Hence, we have the (II) Next, we assume if liil:e l i(gs, rps) < oo,. W,,he,",,ti5S, r,efg",i.a.rkOfp(fiUl/ p'a<"kn5. As befor6,. then, for any e>O, there exists a sequence {Tt} of.

(8) 16 ･ K. YosHIHARA positive numbers such that Ti"oo (l.oo) and '. (28). . Ii.m.>1I(g8,rp8)iisl-ti!1,;'I(esL,rp,Ti)±E (l=1,2,･･･) Now, for fixed Ti, we choose rational numbers {1',h, ･･･,1'.h}(O <7'.h ;$ T, v=1, ･･･,v,)(h =,,(S)d, d being an integer) ' such that. '. I(6eTL, rpgt) 5 I((8(1'ih), '" , 6(]'voh)), (rp(1'ih), '" , rp(ivoh))) +E･. .. Since 8 is regular of rank P, so there exists a principal Minor, say, M(2) = det"ft&(R)lli,j--i,.",, of the matrix ll.ICIi,G,.(2)lli,j=,,".. which is almost every.. where different from zero, and the representation. (29) 6pt(t)=Smoo.e"Z,l.ili,gbptt(2)d2et(R) (iLt=P+1,･･･,n)' holds, where. ' - Mlett(R) ￠.t(R)- M(R). (M2,t(R) being the determinant of that matrix which is obtained from, M(R) by replacing its l-th row {fl,(Jl), ･･･,ft,(R)} the row {f2t,(R), ･･:,f}t,(R)} (cf.'[3] and. [5]). From (29), we have (30) 6pt(th) =Smn., eidZ{..Z.oo-.. t?.,gb,i( 2+h2VT )dze,( 2+h2"Z )}. (pt =P+1, ･t･,n;1' ----jp "',7'vo)}. (31) e,(1'h)=S.OO..ei2'hZd2,"(Z)==S-".ei"Z{.=Zoo..dKe.(Z+h29T)}. (pt =:1, ･･･,P, n+1, ･･･,n+m).. ,. , so, if for each 7' (1'=1',, ･･･,]'.,), we put. ･L. k[Tt]. 6hhk'(1')==.=--l¥tiT.,1.ei"2"-i.=];i..{Zspt( Rrc+h2V7V )-z,-pt( Rrc-ih+2VZ )}. !E. (pt = 1, ･･･,P, n+1, ･･･,n+m). (32). 6ZhB(.1)==...Il]:Zi'b,`i,,.f?tjRrc-.--]22il].t?.,gb,.e('2""ii;2V'r){z6,(3Lrc+h2VZ)-ze,('ilrc-ii;2Vn)}. 2h. (pt = p+1, ･･･, n). where 2rchT 2rc=k[T--,]- (rc=O,±1,±2,･･･), then, we have 1. i. m. g2h,, (1") = 6.(.th). ,. Thus, if we put. ic-oo. s,.

(9) '. TheInformationRateofStationaryGaussianRandomProcessesIII 17 ' (33) J(ic'=I(((6R'(]'i),"',8Ahic'(1'i)),"･,(6Shk'(]'vo),'",gspk'(]'vo))), ' '･ -'' th,. ((6￡h"i'k(7')'"''gAh"'m･k(7i)),"',(6Eh+'i･le(lvo),"',8Eh+'m,ic(7.,)))) ''. ,. (34) I((6(7'ih),'",6(7'voh)),(op(1ih),'",rp(7'voh)?):<=-,lleel(ic''.,, .･.. vv..-.2,{exp [i 2hZ,([k.rtt rc) ]K=20-O ..[x,.( 2z￡iet/t ]rc) + 2",T ). '. -z6.( 2z(kler[+Tu,rc]-i) + 2zz )]} v. (pt -i, ･･･,p, n+i, ･･･,n+m;r-= --gg.],-i]-, ...,l-gi].-i).,. Then,' VV..(pt=1, ･･･,p,n+1, ･･･,n+m;r=--[-T2-hL.]-, ･･･, [gA] ttl) are. complex-valuedGaussianrandomsuche'hat ,. -EW.. =O '.･ EVZtrVV2ttr･=O(rlr') ' ,. '. ,. '. p-. '. EWptrfi'itt･rr ==.=il]..{Fetw( 2(ET'S)T + 2V,Z )-JaPe.t(-i-iil'zi-+ 2V,T )}. '. From(32),itisobviousthat ' ' '. {(gsh,)o),･･･,gah,)(]'))} (1'=]',,･･･,]'.,). :r.i' lll :,Ci'le'W P,Oldlnlfe...tP,, ,=-'[;A] i... ,'gi!.i} '. ,'' {(8ah.),,,(1), ,e￡h.).,,(]))} ･(i'L]',,･･･,1'vb)'. are, respectively, subordinate to Th.,i ,,,,,,IV,VK,r,:.g 'mu,', 'Qs ",tY,;rf. T, ['St] ･･･.･･ [SA] T.i}.･ t. (3s)' Jr(ic';;s./I2T-tllllilJl,ii((M/ir･'"･wi)r)･(vpin+i,v･'"･wn+m･v)) '. fi:.".C,e. M9)>O. 2h ''. )7g!,S,t,)all i' i?.z,,f,rOM.(gil),;,llN3,.A5.(1.,./)aSdllh--e ,r,e,S"'t. O` ,(i)' ye p;tf. .O .r. la. where detAe'"v7(2) is the principal minor of order p+p' of th'e determinant. ' '. '. t tt. ' '. '. '. '. '. '. J '.' ,. '. '.

(10) e. 18 K; YosmHARA '. '. '. ' Therefore, det A6rp(2) which c'ontains M(2)=det ll./1?"(2)IIi,j=,,...,, and detAop'v(Z). we have (3), when' g is regular of rank P(1gP< n) and 1.iLm.. I I(g6, rp6) < oo･ In the case where 6 is regular of rank p(1 s. p < n) and 1,i.m. I I(g8, rp6) ± oo,. we can use the same method as the latter half of (I) and pbtain. 1. l/,m... . I(g6･ rp6)= oo == L,,,.. ". Thus, the proof is completed.. PROOF oF THEoREM 2. The proof is completely analogous to the proof of Theorem 1. '. v. ' 5. Thediscrete-parametercase. . .･. ' ' t. tt We can slightly generalize Theorem 10. 4.1, (3) (a) in [4] as follows:. THEoREM 3. SuPPose that the discrete-Parameter Processes g={8(t)}== {(gi(t), ･･･,g.(t))} and rp == {rp(t)}={(rp,(t), ･･･, rp.(t))} form an (n+m)-dimensional. stationary Gaussi.avn Process (6, rp). 11f Ef,e,.(R) are absolutely continuous and 6. is regular, then I(8, rp) and i(6, v) are dofn2d and. I'V(g,rp)==i(6,rp)=Le,. ' " The Proof is obvious from the Proof of Theorem 1. ' / ' ' ' '. .. References i) 8iRsS2M3gR(i,g4･o,).On the theory of stationary ran9om processes, Ann. Math., 41,. 2) GELFAND, L.M. and A.M. YAGLoM, Calculation of the amount of information about a random function contained in another such function, Usp. Mat. Nauk 12, ' 3-52 (1957), English translation･in American Mathema-tical Society Translations,. Providence,R.I.Series2,VoL12(1959),199-246. , '. AL･. 3) MATvEEv, R.F., On multidimensional regular stationary processes, Theory of ' probability qnd its applications, 6, 149-165 (1961).. 4) PiNsKER, M.S., Information and information stability of random'variables and processes, translated and edited by A. Feinstein, Holden-Day Inc. (1964).. 5) RozANov, Yu. A., Spectral properties of multivariate stationary processes and boundary properties of analytic matrices, Theory of probability and its applica-. tions, 5, 362-276 (1960). -. 6) ,Stationa,ryrandom･processes,translatedbyA.Feinstein,Holden-Day Inc., San Francisco, (1967).. 7) YosHiHARA, K., The information rate for continuous.parameter stationary Gaus' sian random processes I, Science reports of the Yokohama National University, Section I, No. 15, 1-8 (1969).. 8) ,Theinforrnationrateforcontinuous-parameterstationaryGaussianranaom processes II, Science reports of the Yokohama National University, Section I, No. 16, 1-8 (1970).. ..

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