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(2) 50 K. YosHIHARA where <4) . ￠,(1')ig.2-e."...x.EIP(a,lxj,i,-i,･･･)-;-P(azixJ,xj--i,''',lXi)i･,･. ･. It is proved in [5] il-that.qnder these assumptiQp.s the series in (3) is absolutely convergent, and the central limit theorem holds, if o2> O. In this note, we shall prove the following three theorems. THEoREM 1. Let a2>O and de.17ne X.(t, (o) as. tt ttt tt <5) Xn(t,w)=.Jli7i7(-logP(xl,''','x[nt])L[pt)H) (Osts1)･. T. t- ttt. Then the distribution of X. converges wealely to a Wiener measure VV on (D, .ED). Here, D is the sPace of functions h on [O, 1] that are right;'contin' u'ozas' and. "･. have left:hand limits, and g the a-Lfield of Borel set foT the Sfeorohod toPology. (cf･[1])･ ,･, ･ .･ .･.･' -,',,i .v/".:':",･,/ ., ,･'r,･.･' .THEoREM2. 11fa2>O,then ･/ .,,.,,/..･..･,.,.,.,.･,,,.i.,･..,;,/ ' l:2,;(fsg`iigX"Sil"i"'=i''i}L''i'. '･- 'i '･ "･' ･''･'. tt tt ttt. `6' , P{ILm..'-. The next is Strassen's version of the law of the iterated logarithm for the. entropyofthestationary,process.. ･ - -･ ･. THEoREM 3. Let C be the sPace of all continzaous junctions oza [O,1]. V(annlllSh/scg za.nt cO'aWsith the usual maximum norm･ Define the junction z.(t, w). t/t tt t. -i:2,;`xei6};X;9-".for t'F,￡･fe=o･ l;･'･･･@; .i, ,.,... (7) Z.(t,(o)== ･ ,. , .. ' tG [ ￡' , igIl ], le = o, 1, ･･･,n-1. linearly interpolated for. Furthermore, let . , (8) :. ,Jic={h(t)GC:S6i{n(t)}2dt;sl} '. ' t with' respect [for he .JT the derivative n(t)' is suPPosed to exist for almost all tO. ･,. L81ZeeSng,UfeorMae7)ilZrsf]bvezy.Eg, lhe seqzaencb of fzanctions ''' {Zn(t, to), n )- .32 }. is PrecomPact and its derived set is the set ,.IC.. 3. Proofs. At first, we define .1{i and gj as. <9) h==-logp(xjlxj-,,･･･)' (1'=O,1,2,･･･). l..
(3) Some Limit Theorems Connectedt with(the.Entropy of a Stationary Process ･51. - tt. ,. (10) v..,.,,,,gj=71ogP(xjixj-i,･･･,xi) (i'=2,3,･･･).. (11) ip(1') =Elfo-E{fo fi EM%'}l2(7' = 1, 2, ･･･).. ,, ,. ,]. Thefollowingfactsareptoved:E.. '. (i) Foranyr>O lf,(r<'oo, . .. (12) Eny. Y. ;1, -/. (cf.Lemma2.lin[5])･ ,,,j-･- ,,. ". ,.(ii)..,Forap,y,/p>O, .. ,.,', .,,{･,,,,,,.. ･･ ･,,.,.,(. (13) ￠(7')SEI.11i-g,l2SK(p){￠,(1')}i-p ;, ･ (cf. the proof of iLemma 2.2 in [5]). Thus, under the assumption (II). iio4r)gom6'si,`l>l6: '''･: Sb(])=O(i-2-63) 'i.'',,1 '/','' '..' (iii) Underthe assumptions of Theorem 2.2 in. [5], the fungtional central limit theorem holds, (cf. Theorem 5 in [7]).. Now, we proceed to prove theoreMs. In what fbllows, by the letter K, we shall denote any quantity (not always the same) which is bounded 1'. p,. absolute value.. THE PRooF oF THEoREM 1. From (13) and Asstimption (II), we have. tt. k. ･-･ ･･. E --IOg,elil.IXil',i'''Xl), //titA. ,;,yli.. t.l)IEi.,{,..,,,i .'.iii 'i'". '''' ''':S-･･''･vfli=iot,ii=l,ll,{Elfl,･-gjt2}-S-.... 1 n .1-a2. S Vff. ,l=,{,￠.o<1)} i2; ' 1' '. ' ･ L''..o(n-S5' ' ''''`:'-'"''. . -l. for all le S n.' Thus, from Theorem 4.1' in [1], it is sUfficient to show that the '. distribution of Y. converges weakly to a Wiener measure on (D, g), where,. Yn(t,tu)= v#. jlllii(L-H) '(Osts1)･･. ' ttthatt the s't' rictly stationary proces.g,. But, using (i) and (ii), we can easily verify {.1`)-H} satisfies all the conditions of Theorem 5ip [,7], or of Theorem 2.2 lli3,[8]l SHO,.t,h,9 S'i,Strhi.b.U,ti?h",Otfh,Y,",,C..9". Verggs weakiy to a wiener measure on. ToproveTheorem2,weneedthefollowinglemmas. ' '.･' ''' .･'･./1.
(4) .. ,. 52 K. YosHIHARA LEMMA 1. There exists a Positive napmber r, such that. t(i5) -g2E,p.co'' p{-'iOgP(Xie7:li'`iXn)L;'=nff<2}-￠(2)=.o(n-ri) ,where. di(z) - v!,i}i'=' Sldab-S' dt''i. PRooF. Let. and 8SS'=E{-1`)IEML- gv}-H. r. rpS･S)=L･-H-6S･S). ,. ' satisfies the stfortg mixifig Then, the strictly stationary }process {6SS) "'`. -,. cortdition. withthefunction ･., .,,.･･.1..,,. ,. exe(n)±-:'1 foritS.2s; =a(it-2s> forn>2s, Furthermore, it follows from (i) and (ii) .tthat the strictly stationary procesges. {L･-H}, {eSS'} and {rpSS'} have the following properties:. 2- ･ , <iv) Forafiy6andarsuehthatO<-sr-2<S'<S, A''' oo ' 6' ''. , , ESSs)th--O, EigSs)I2"6$EIf,-Hl2+6<oo, X {a(n)} i+6' < oQ,. n=i. t(v) ForanyBm>O ' 'r ･:･' ErpS･S) = O, E1rpS-s)l2+6" ;.:s 22-t･antE lfo-Hl2+ant < oo .. B, 6, <vi) IE68S'eSS'lSK{ae(1)}2+6'=K{a(1'-2s)}2+6', , iE(]Cb-H)(L-H)i$ K[<a([-/r -])} 2g'6't +{￠([-ll-])}}], 1Erp6S) rpSS)IK. K[{a([g] )} 2i's') +{ip([g])}S], IErp8s)rpS6)1:.sg:'Elrpss)t2. ' '. For a small positive number P, define P, q and le by. sc. p(n) =[nS"P], q(n)=[nS-P] and le(n) =[peq].. Let i. t'. N-IZI:,(l.;j,'i II ,O;,S//2. :'. and. ' 4ss)=･IiSS" 111 i6el.I',1';iU; c-ss)==-･gs･･s)-･--css.)`,;. '. Furthermore,put .' ''. ,' , ･/y'1.'. i. I I.
(5) SomeLimitTheoremsConnectedwiththeEntropyofaStationaryProcess 53 Sa= .e>ii- tli.l.),(CSS'-ECSS'), S."-='="- .JJ>iT te.l},(4'",(s)-E4-,(s)), '･. tt. .. ' TA= '.l<l)ii- tli.ll,j.//P,(C[S"i)'(p+q)+j':EC[ij"-i)(p+q)+j), 7""--SA-TA.･ Then, it is easily proved from (iv), (v) and (vi) that for some r>O. <16) EIS#,l2=O(n-D., EIT.tii2=O(n-r), IElT.'12-1[=O(n'r). s. Lets==[n-;--2B].. , Then,fOrSOMeh>O .., .. a7) i,(t)= E{expzt(-iOgP(Xiv' t."iX")-"H)}-E{expzt ':'illifiS-.H) } -. :sE exp it( i-.. t;.ll,(gj-X)'))-i ,' ' ;S VIiLi. t.Il,E171i--gji･$ 5-Xi-tY.,{￠,(j)}'l6=9'･･. t ttt 1 $Kltln--ii-, . ',.-･- ,-,,:･･. -t. ' vZ-.to tl.l].SSS')} (18) I2(t)== E{exp( vZ-.t. tA(]{i-H))}-E{exp( 'SE. eXP< .vftfat. tlS.,rpSS')Ll SKItfi2E ../ili=. tli.ll,.PS"'2' '. s. ' ''t -'. tt )Erpss)rpss)} ･- ' '' ,,li -i5,iLt2{Elrpss)l2+2ZIIIIil (1-f. E{;,t2(2[.t-p]t1)￠(,)+?,--.,".z;,idp,({.([-/r-])},.6S'･+<ip([LS-])}S). ' =t2･O(n-ri). ,,. ,.. and. '. ' (19) l3(t)==E{eXP(.Jlitii-tj..,6SS>)}-E{eXP(itTn')} .t .'. , ,. ' $E{exp(-3ii>'ntli.l?sss))}-=E{exp(its.t)} s.. +1E{exP (it S.')' }-E{eXP (it T.')}1 lj. ;:;l E1exp (it S.")-11+Elexp (it T.")-11 ;:ill 1t1 [{E1Sn" 12}'S' + {E1T.n 12}:IY' 1 '. Next,. ;:Elltl･O(n'-ri), ,..' ' ' let 0o,.0i, ･･･,0k be independent. random variables distributed in the. samewayasthecorrespoitdttng '' , oti t?.,(4Eij"-i)(p;-q)+e'-E(IIEg･'-p(,.,,.,i). .,,.
(6) /. 5'4 ';･L"'( ･･ K.Yos'HiHAR'A･ ･':': Sincefromghestrong-niixingcondition , ./,,,,..,., ic . . I,(t)..IEeitf'n.Eeitj-Xie"1;skaE(q)=kev(q-2s)=O(n-2r2) t-.. for some r2>O, so , .i .. (20) ,,･- S-".r,2, (`t(t) dt,:SS,..,,,...':-i,- Ztt(t).4(+S.--tl..i,i...r'. , I`t(t),..d,i''. ' gKS,..t,i...--1-1tldt+ka(q)S.-ts.it:s..r2ldttl ''. T. '''Lo(,i.r2)'' '''''''''i''' -. Finally, from Esseen's lemma there exifits ,a r,> O such that. k (2i) I,(t) ==liEeiS'ie'-eM-ei2i"i;;I KI4,2"ie. Tii' nTT? for all ltlS ･viit (cf. the proof of Theorem 5in [8]).. Hence, from Esseen's theorem and (17)-(21), we have tha,t there. ro'. ii. K,. g KiS -nro i'vi.. ,1n ･'. "･'. exlst a. 2.slukC.2 //' lt i,, ;:,.'i ft ;/I,/t/l' /ii/'r-{',t 2//ilil(11il ..)) -nh}l'Ei't:"'' d'. '. tt t' ''''''. nro. tt. . f, ,9(e,';TO).{...,. :,il tt/tt I/,./,it,tt:. Thus, the proofdf Lem'ma 1 is completed.. LEMMA 2. (22). P{ max I-!og P(x,, ･･･, x.)-mHl ). 8aX(n)} IS-mS-n <2p{l-i6'gb(xl,･･･)'be.)-'ntil>=ax(n)}+o(a,in)3')'i'1'' '-. '.1. ,z77ciently Llarge) Here, a>O and ･.''.,･ ' .'. holds for all n su. ,]. il. '. r. ,,, ･X(n) == (2o･2u log log a2n)r27..1,. PRooF. We remarkfirst･th.a.-t･a..･･. i･:.. ' ' P{max1-logP(xi,･･･,x.)-mHl>=8aX(n)} ,.' IS-mS-n. t. ,,,. ,,...... tt tT.,(L-H) )8ax(n)} ''d['J''/$ P{i=,IL-gjl+,mH..a..x,. n. ' ･e-･ ''''''sb' {'tl.ll=, E'LLg,I>= 2ax(n)}ttP{ ,m.-=.a-.x,. t?.,(lj -"H) )-'- 6aX(n>} i,"'I'1 Since the strictly Stati9,n.,q.ry prg,.cess {7`>. T,￡<.I, ;i satisfies all the conditions of. Theorem 8 in [8], so we have x.
(7) SomeLimitTheoremsConnectedwiththeEntropyofaStationaryProcess 55 P{,M-..a-.X. trv.,(L'-H) l6ax(ap)} ., ,/... , ,. tt 4 '' ". . E2P{'t/.ii=,(hrH) ZaZ(n)}+9( a,gl.)3) '. <cf. the proofs of Theorem 5 in [7] and of Theorem 6 in [8]).,... Ontheotherhand,fromChebychev'sinequality i'･･･'L ' ' '' p{t/;i.l.}il.i{i-gjii2ax(n)}s2'.xi(.)t?.,EI.fr{)･ingjl -g. ''' '"i ''i ':S2axl(n)'t.l),{Elf)'-gjl2}iY.. '' ' ' 'f-{g 2axi(n)' tS{￠,(]')} '"262. '. i , '.. o(n--i.).,. Thus, (15) is obtained.. t. ･ -THE PROoE oF THEoREM 2. Theorem 2 will be proved if we show that for any e>O. tt. :(23) P{I-logP(x,, ･･･,x.)-nHl> (1+e)X(n) i. o.}=O ' ' ' <24) , P{1-log P(x,, ･･･,x.)-nHl> (1-e)X(n) i. o.} =1 .,. and .'･ ,-. 't.. '' Lemmas 1 and 2 (cf. the proof of Theorem 1 in As (23) follows easily from [8]), it is enough to proVe (24). 'Now, we use the method of the proof in [IQ] (cf. the proof of Theorem 6 in [8]). smaP/.efinpeuSSS' and rpSS' aSbefore･ Let ' s==[n-S'-ei] where e,>o is sufficientiy. v ' '. ･[ oo d.2=E166s)12+2,l-m,Eg6S)6SnyS), ,.. n a.2(s)= El2'8S"' 12, ' '. j'=1 'L X'(n)=l(2a2(s)loglogcrfi(s))-2i-. u. Then, from the property of･ 6SS'. gi' =r-i -o(n-r)･. g. for some r>O and so ',.-'' ''-tl--:2(.")) =o(n"r). for somer>O. Put s,=AS-ei and Ek={ t4t,6Ssi) ;!;ll(i-p,)x/(Ai),., i<le; t12.iic,e'ssk) S(i-p,)zi(Ak)},. '. ,.
(8) 56 K. YosHIHARA c==tll.ll,{1ttr(gj-ff)+t4ipsse)l<-:l}'-(p,-p,)xi(Ai)}, tik :P{lt"=t,6Ss,'l> (1-p,)xi(Az) for at least one i, 1;:ili-s le}. k =2P{E,} i=1 where pi and p, are positive numbers such that p2<pi<e. Then, from Chebyshev's inequality and prQper,ties of L･,gj, rpS･Si' P{ c} ! I- p{ ,LMy=, [lt"=t,(g, -.1?)+ tf.lilpsst)1lll t(p,- p,)xi(Az)] }. `. J･. >-i-p{trl.l.i{,[t/S.lltlg,-fl,E+ltrl.lii rpsFi)1lii ± (p,-p,)xi(Az)]}. ). i- P{[ t￡.i¥1g, -LIl {ir(pi-p2)X'(A) ] v(tll.III.,[ltr.l.i,irps-st)Iiiii 2 (p,-p,)xr(Ai)])}. lii1-P{t4¥lg,-Lllli it (p,-p2)X'(A)} . -P{,UM=,[)i.li-Z,rpSsz)l>=t(p,-p,)x!(Az)]} l.,,.-, f,jzll,llE,:;,l'-,,.g,ji -,l.:?, {-}Sl/Elirp,iSi',li.,,}2. . ;-62 4Z{sbo(,i')}T . >=1- j=i -K2(Ai)-ro (Pi-P,)Xi(A) t-i l1-KA-ro(1-A-ro)-i for some ro>O. Thus, for all sufficiently largd e-. A,. Um = P{ tr.Il!, [l -iog p(x,, ･･･ , x.i) - AiHE > (i - e)x(Ai)]} t. >=P{tll.l),[l-logP(x,,･･･,xAi)-AiHl>(1-p,)zi(Az)]}. s l,. '- P{ t/l.l2, [{ltll.liit, 6ssz'I> (i-p,)x!(At)}. v{ltA=t,(g,-n)+tA=T,rpssz)l:il-il-(pi-p2)x!(At)}]}. >--p{t/l.ll!,[lt/I.llikssz'l>(i-p,)z/(Ai)]Ac} ,,,... Illl -1+ Um+P{C} lil U.-KA-ro(1-A-re)-i.. 3.
(9) Some Limit Theorems Connected with the Entropy of a Stationary Process. 57. Finally, let ck =ALI} and 'choose p,>O such that for some E'> O,. '2. +. .v!IZ=+P3+ei < p,.. Then. p{[ tl.t,' gssi) s (1-p,)zi(Ai), i< le]A[ ,.=2,-S.)ic,,.,eSSk) > (1-p3)X'(Aic)]}. Ai .. Ili P{ ,Z=,6S･Si) :.i!{ (1-p,)XtrAi), i< k}. 'P{ j=.ic-S+ic,,+, 6S'Sk' > (1-p3)Xi(Aic)}-a(ck-2s,).'. Since from the proof of Lemma 1 Vk == P{ j--.,mS+k,,+,6S･sk5 > (1-p,)x/(Aic)}. lll{10goltk-Ak-i-,k(sk)}"-(i+p)(i-p4)2 .. for some p>O and p,>O (cf. the proof of Theorem 6 in [8]) a;' (sk) == o2n(1 + Oa)) for all sufficiently large n, so. Vk >= Kk-(i-2i) where 2i is positive and does not depend on k. From the properties of. .i`IJp. and rpS･s). Akml+ck 2IKc,, E Z (L･-H) j･=Aic-1+1 and E Aic,-l=i,+eic rps.s,) 2 ;:$ (Aic-i+c,){E1rps.sic)I2+2Aic1.E)l;ek1Erpssic)rps.sk)l} i. '. $(Aic-i+c,){(2[A}-2ei]+1)￠(sk) +K Aic-2,i'C'C ({a([{l-])}2+66't+{￠([rg-])}-li-)} j'=[ALsi--2el]. s{ KAk-i--kr. for some r (O<r<1). Hence, we obtain. l. Aic-1+ck Aic-1 2' j'=1 j'=1 Aic-1+ck Ak-1+cic rpSFk-i) Ak-1 2 E) =E Z (L･-H)-rpSSk)+]2. E 2 8Ssk)-zgs.sk--p. s'=Aic+1 j'=1 J'=1. ;Sl3[E. ic-1,.11)･2+E " ic-1 ic-1 j=.lll.l)+,C t.m+,Ckrps.sk)2+E j=,rpss,..,)2]i, 1. !.
(10) 58i . K. YosHIHf),.RA S; KAk-i-icr and so, from Chebyshev's inequality. ic-1 le-1 p{ " t.=",Ck6S.sk)-,.zC, gS6k-i) ll eixi(Ak)}:$ KA-i-kr. Hence, as in [10], we have Uic->1 (le-->oo) and consequently Uk->1 (fe-->oo).. So, the proof of Theorem 2 is completed. THE PRooF oF THEoREM 3. Let 6iS' and rpES' be as before. For any integer m and hG C, let IT.h be the piecewise approxirr}ation to h defined by. h(-ilii-) fort==-;7, v=o,i,･･･,m (llmh)(t) =. linearlyinterpolatedfortE[th,V7+tl], v==O,1,･･･,m----1, Let nr=[cr] with some suitably chosen c==c(e)>1. Then the increment' of. 17mz.,(t)over[th,Vm+1]isgivenbythefonowingform: ' ' llmznr( V7+t 1 ) - llmZnr ("). == x(i.,) tr/,(gt-H)+or,. == x(i.,) [te.,(rt-H)+{tr/,(gi-fi)+or,p}] = x(i.,) te.,gS[g])+[x('.,) {t/.ll,(gt-fi)+tl.,rpS[g])+or･v}]･. where i is the smallest integer such that i l th and ]' is the largest integer I !. 'such that 7'. I. < V+1. Let. nr m. yr,. = (2m iog iog nro2)e{ x(in,) tt/,gtetOr,v} i. = 1, Sgi+(2mloglognro2)0r,v. l. ( itzr )-2-a t=i. i. v== O, 1, ･･･,m-1.. ! "-x. l. Put qr =[NbB] with some O<P<1 and M,.,vnr ., and. L. I I l. l !. i. i. '. yi,,- i,ttqr6S[-ll-]). ' (M,.-q,)-lra t==i .. ' 8SS' and rpSS) that Then, it follows easily from the properties of fi, gi,. ' EM2iy;,,-rn2iyi?.-o(nSj. ･. . ..=o v=o.
(11) SomeLimitTheoremsConnectedwiththeEntropyofaStationaryProcess 59 So, the method used in the proofs of Theorems 2, 4 and5in [9] may be applied to this case (cf. [2]) and we have Theorem 3.. References 1) BiLLiNGsLEy, P., ConvergenceofProbability measures, Wiley, New York (1968). 2) CHovER, J,, Ori Strassen's version of the loglog law, Z. Wahrscheinlichkeitstheorie verw. Geb., 8 (1967), 83-90. 3) DyM. H., A note on limit theorems for the entropy of Markov chains, Ann. Math. ,". }.. !･. m. Stat., 37 (1966), 522-524.. 4) EssEEN, G.G., Fourier analysis of distribution functions, A mathematical study of the Laplace-Gaussian law, Acta Math., 77 (1945), 1-125. 5) IBRAGiMov, I.A., Some limit theorems for stationary processes, Theory Prob.. Appli.7(1962), 349-382. .. 6) IBRAGiMov, I.A. and Yu. V. LiNNiK, Independent and stationarily correlated random variables, Iz-vo " Nauka", Moscow 1965 (In Russian). 7) OoDAiRA, H. and K. YosHiHARA, Functional central limit theorems for strictly stationary processes satisfying the strong mixing condition (to appear). 8) OoDAiRA, H. and K. YosHiHARA, The law of the iterated logarithm for stationary. processes satisfying mixing conditiQns (to appear). ･ 9) OoDAiRA, H. and K. YosHiHARA, Note on the law of the iterated logarithm for stationary processes satisfying mixing conditions (to appear), 10) REzNiK, M. KH., The law of the iterated logarithm for some classes of stationary processes. Theory Prob. Appli., 13 (1968), 606-621.. x 1. i.
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