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(1)Functional Central Limit Theorem for some Stationary Process By ,. Hiroshi NEGISHI (Received May 31, 1972). .. 1. Summary. The object of this paper is to prove the functional central limit theorem for the stationary process generated by a stationary process satisfying the strong mixing conditipn, under diffefent assumptions of Oodaira and Yoshi-. 2. Preliminaries.. The following two lemmas are needed to prove our theorem. LEMMA 1. Let 8F and (S; be o-Lfields zvith 85c(S. 11/' El81T< oo (rll. 1), th.en. (1) Elg-E{816} Ir;$ 2r Eig-E{e18i}lr. PROOF･ If rp==6-E{6I8F},then6-E{816}=rp-E{rpl(S3}. Hence it suffices to prove EIrp-E{rp1(S}}1rEil2rE1rp1r. From C.-inequa!ty ([5], p. 155) and Jensen's inequality,. " Elrp-E{rp16}IT;sc.(E"1rplr+ElE{rp1(8$}lr) ;!!l2C.Ei)7ir･, ･ .･. where C. == 2r-i. ,,. We shall concider a process {6.;n =O, ±1, ±2, ･･･} which is strictly stationary and satisfies the strong mixing condition, i.e.. (2) suplP(AB)-P(A)P(B)I=a(n)-->O (n->oo), here the supremum is taken over all AEEMig., BEEM:+., EMZ denote the oalgebra generated by events of the form {(6i,, ･･･,6,i,) E E} with' a';IEI ii < ･･･. .. .,b' ,.an[d,]E,gS, <ikg. afk6d,iM.e"iilLO.",a.',.B,O,r.e',ii2t,',i., f,,. the space of a6tibiy. N. infinite sequences (･･･,a-i, cro, cri, ･･･) of real numbers into the real line. Define. randomvq;i.4P.leS.- L...f(...,e'..,,6j,6j･Fi,'"i. tt. wherg 6j.g.g,uptps. the, O-thn place in. the a.rgument. of f. It is obvious that.

(2) 14 H. NEGIsHI {L･} is a strictly stationary process.. Write. Sn =fi+h+ ''' +fn, and define a random element Xn(t) in D=D[o,i] by. 1 <3) Xn(t)=='--iJ</;=`-S[nt] (O;llt;Sl)･. .. The following lemma is due to Ibragimov, [3], [4]. LEMMA 2. Let the stationaryprocess {e.} satistly the strong mixing con-. s. dition (2), let the random variable f be measurable with resPect to EI]flee., and let. the Process {L} be obtained from f, as described above. Moreover, suPPose the following conditions are satisked:. 1. E(f)=O; lfl<C<oo, with Probabil,it.v 1,. 2. oo ZEIf-E{flg)}ig,}I<oo, le=1. oo 3. Zcr(fe)<c>o. ic=--1. Then. (4) a2=E(fg)+2:iliE(f,f,)<oo, j'=1. and moreover. '(5) P(u-'bEvZ'-.rT<z] ;.;g.r. 'Ju/2=rfi7Sl..e-nveq22-du, ij a# O.. 3.. Theorem. In this section, we shall prove the central limit theorem for the random process defined by (3), under some assumptions. THEOREM. Let the stationaryProcess {6.} satisly the strong mixing condition (2), let the random variable f be ,measzarable with resPect to EMee., and let. the Process {fil be obtained from f, as described above. Moreover, suPPose the following conditions are satisy7ed:. 1, E(f)=O; [fl<C< oo, with Probability 1,. oo 1. .. 2. 2(Elf-E{flEM&k}Ii"S)rFa <oo, forsomeS(O<S<1), ic=1. oo 1 3. Z(cr(le))2+a<oo. Then. .. k==1. o2 == E(fg)+2 co 2 E(foL-) < Qo･ j'--1 if a !O and X. is defined by (3), then the distribution of X. converges weakly to a 17Viener measure W on (D, ES>).. PRooF. The first half of the' theorem is evident from Lemma 2.' To. 1.

(3) Functionalcentrallimittheoremforsomestationaryprocess 15 prove the latter half, it is suthcient to prove that the finite dimensional dis-. tributions of the X. con･verge weakly to those of lli and that the sequence {Xn} is tight. The covergence ofthe finite dimensional distributions is easily obtained by the method in [1] or [2], using Lemma 2. If we prove that for some constant A l. .. (6) EIS.I2"6.SAaZ"" (a2.=E(SZ)), tightness of {X.} will be obtained by Theorem 12.3 in [1]. Now we shall prove (6). Set. S2n= Zsp+Za2'+(Sn-Sn.2p)+(Sn÷2p-Sn)+2ff'+2ff',. where '. ZX) = Sn-2p-E{Sn-2plEMn-boP},. Zfi2) == S2n-Sn÷2p-E{S2nmSn+2p1EMew+p} , ZSi' == E{Snp2p1SMe'oo"} , ZS2) == E{S2n- Sn÷2p1SIJI:+p} ,. and P = [Vit]. From Minkowski's inequality and the stationarity of {fj},. i11. (7) ElS2nl2"6;$[(ElZspl2"6)2+6+(ElZS2'12"6)2'6+2(ElS2pi2"6)2"6. NN 1. +(ElZEi)+Za2)l2+6)2+6]2÷o".. From condition 1 of the the6rem and E(Si)=o2n(1+o(1)),. (8) EIS2p12"6=EiS2plIS2pli"'S ' 1+6. ;:ll 2CpE1S,.1i.6 ;sl 2Cp(ElS,,12) 2. 3+B ' =2(lp(aV2p (1+o(1)))i÷6 = Kbp 2 .. We have '. '. '. 11 (9) E12fii'i2'"'S;IlllEiSn-2pl2'"O".Sl{(ElSn12'"6)2'6+(ElS2pl2"6)'2"a}2'S 1 ,; ;ISEISnl2`"ti(1+(E(liiPsi.21"i',S))-s2.."6]2'i'5. '. SEIS"l2"6[1+..a.,,KlbnL-uPs(i3tiliS6i(1))]'2"6, ''' .. = ElSnl2'"fi(1+Pn)2"6 = ElSnl2"6(1+PA). 3+B where, pn=-.vK-b.P(i(f'aili)) and i+pa=a+p.)2'6. since p=･;+[vn'], '. '. Bn'--"O, PA.O (n-oo). ..

(4) 16 H. NEGIsHI Similarly]. (10) ' El29'l2"6;:IilE]Sn12':'O-(1+Pn)2"6.. FromMinkowski'sinequalityandLemma1,wehave .. In-2p 1. (EiZspli+6)T+6 ;:sl ]z (E1f,-E{f,1EMn-..p}li"6)TFb7. j'--1. tt ' n-2p S2,l.l,v(n-p-]'),. .. where ,. i v(le)==(Eif-E{f1EMig,}Ii･:-o")LiT+-6'L. If. pt(p) ==ooz v(i j'=p. then - ･. (11) (EIZ8i)[i+6)iVa'T ;;s i2pt(p). Moreover, since lfi< C,. (2+B)a+6) 6 - n-2p (2+6)(1+a))6 (2+6)ai6･ --o-' -(12) (EIZS)l 6 )?26)(i+6)s2](Elfli-E{fjiEMe-..p}I. ' J'=1. From (11) and (12), we obtain '. ff{ 2Cn. ,. tt. (13) Elzsu 2÷6= EIzsiI -2L?lz￡bl 2S6 D tt. 2+6 (2+6)(,+6> :;ll(ElZsp1i.6)i(i+6)･(ElZsu 6 -)2(i+6'T)n. 2+6 2+a. S KIn 2 (pt(p)) 2. .. Similarly,. 2+6 2+6 (14) EIZ82)I2+6 ;il Kln 2 (pt(p))'il Since E2S' = E2a2' == O, and 2Si' is EMEH.p-measurable and 2￡2' is EMee+.-measurable.. using Lemma 7 in [2], we find that E(2 si)12s)l62￡2)) -<. (E12spI2+6) S+"6B .(El2ff)l3+s) 3I6 (.(2p)) ,2-.-si,lliili/sJ.. ,. Since E12S2'13"6 :S CnE12S2'12"a, we have . (ls). tt E(2suT2fii)E62a2))i:!lKhn3t6(El2a)i2+6)Slg(E12s2)t2÷6)i;Jl'lj'(af(2p))(2+ok(3+-aT.. Similarly, (16) E(2spl2ff'162fi2)):;$Khn3tB(E12E2)12'"6)2'+'B6 (E12S')12"6)3tS(a(2p))(2.6;(3+arm,.. Moreover, from Lemma 7 in [2],. i. ..

(5) Functional central limit theorem for some stationary process ,. 17. NN. ElZev2lZfi2)1e;SEl2fii)I9El2E2)16. '' +(E12S'l2"6) 2?B (Ei2a2)13"6) 3+B6 (a(2p)) (2+6)6,3+6). o.. '. '. (17) ' :SEI2Ei)l2(El2￡2)12)E- . ,. ' +.Ksn 3+66 (E]2s)l2':-S) 236 (Et2￡2)l2÷S) 3+66 (a(2p)) a+a)93+6) Similarly,. i. NN B. (18) . ElZSi)l61Z￡2)l2m<.(El2Si)12)E-El2￡2)12. a 2' a6. ･ +Ki,n 3+6 (EIZA2)12÷o") 2+a (EIZS)12'i'6) 3+B (cu(2p)) (2+6)(3+B). '. .. From (9), (10) and (15), we obtain. (19) E(2S'l2S'lo"ZN A2');llK>n3#(1+P.)i'6'32+'a6(a(2p))(2+6)i(3+6)(Els.12÷S);'+66". 1. 3+6. '== IEs.l2+6 Klin 3t6 (1+Pn)'+6' 32+'6B(g(2p)) (2+ai3+6). ' (EISn12+O") (2+6)(3+D) ;:S EISnl2"O"rn,. where ･ i...K>n3t6(1+P.)i'6'32+'6a,(a(2p))(2+3)(3+6). (EISn12) 2(3+6) ' Similarly, from (9), (10) and (16),. (20) E(2Ai'l2A2'l62￡2') i:S EiSn12'6rn･'. 1 < oo and a(n) is monotone decreasing, so Since 2(a(n))2+6 a(n) == o(n-(2÷6)) .. ･1 (1+o(1)) and p =[ViT], we find that Moreover, since (EiS.I2)7=aViiH. (21) r.--->O (n->oo).' '' From (9), (10) and (17). NN 2+B. (22) EIZSt'12IZ￡2'l6,<,=(EISp-2pI2)2 L. +Khn 3+66 (1+p.)2+ 6g2+"6a'(a(2p)) (2+6)i3+6) (Els.I2÷6) 2t6 +. 2+6. ;;ll(EISn-2pI2) 2 +ElSnl2"6rA,. '. where , rA, =. Kin 3'66 (i+(Ei(9:)s2ilO,(3)2t.,lla.(,lli(2P)) (2'"'l3'6' .･. Similarly,from(9),(10)and(18), ,. NN 2+6. (23) E[ZS'IfiIZ￡2'[2:l;ll (ElSn.2pl2) 2 +ElSnl2"6rh･. 6. 3+6.

(6) 18 H. NEGIsHI Since (EIS.12)S= aViT(1+o(1)), p==[Viirr] and cr(p)=o(p-(2'6'), we have. (24) rA--;->O' (n->oo). Thus, from (9), (10), (19) and (2-, we have (25) E12Si)+2￡2)12+SsE(2S)+2￡2))2(i2Ai)15+12￡2)l6) ". ==E12S)12-t･S+El2￡2)12÷S+2(E(2su2su62￡2))+E(2su2￡2)162￡2))). +E12Ai)1212￡2)16+Ei2su612fi2)12. 2+B '. ;-:{(2+2PA+4rn+2rA)EiSnl2"6+2(E1Sn-2p12)T. ,. 2+B = (2+Pn*)E1Snl2"5+2(E1Sn-2p12)iM. ' (8), (13), (14) and (25), we heve From (7), '. .. E1S2n1'2"6 ' :$ '[Kbb 22i+"Si +2KlnS(pt(p))S' +{(2+Pn*)E1Sn12'5'i!'2(E1S.-,,12)-2:'llg!u6}ttl'B-]'2÷6. :<=={(2+Pn*)EISn12"O"+2(ElSn-2,12)2J"iB-}(1+6.)2"S. where. 3+a 11 ,.. 6.=KbP2(2'6',+2Kln7(pt(P)li , (2+Pn")LigF'(E1S.12"6)-iiFT, '. 3+a 11･ '''. $ Kbp2(2+6)+,,2Kn7(pt(P),)-2 -->o, ,(n--->'oo)･. (2+Pn*)2'B(E1Sn12)2 It follows that. 2+D ElS2n12"6:lll{(2+Pn")ElSn12"a+2(E1'Sp-2p12)T}(1+6A) ' (n.oo), for any e>O there exist an . Since P.*.O and1+6'.=(1+6.)2÷6. SA,-->O integer N where, and a K (constant) suph that, for all n )- N,. 2+B E1S2n12"6g(2+S)E1Sn12"6+K(E1Sn-2p12)-2-.. L Since (EIS.l2)2==aVii(1+o(1)), we may choose a suitable constant K such. `. ' thatforalln,' '' '' <26) EIS2nl2't'6 ;:Sl (2+e)EISn12"6+KtzZ'B･. We now take r to be any positive integer. From (26) and a. =aVii- (1+o(1)),. EIS,,12"S:Sl(2+s)rEl,S,12"6 , +Kb3."-6i(1+(2+e)( ai,ii2, )2'6+ ''' (2+e)r-'i .:1..i ]. '. ;:$(2+E)rE1Sil2"5+.Kl.T3'.-B.i(i-ILK'.,2.",( 22,IIIil;, )j'-i1. q.

(7) Functionalcentrallimittheoremforsomestationaryprocess 19 Ef we take e such that 2+e<2i+-gU, then '. t(27) El S,.I2"6 :Sl (2+e)rEl S,12"'S'+KZ" a3."i ;l:$ Aoo:."6. -. -where Ao is some constant. Let n be arbitrary positive integer. We may write n= cro2r+cri2r-i+ ･･･ a., ' 'where cro =1, crj=1 or O (]'>1) and 2r ;Sn<2r+T. Let. .. ' '" +fn) Sn =(fi+ '" +fii)+(fii+i+ ''' +fi2)+ '" +(fiT-Fi+ ' v(where, the number of additives of 7'-th bracket is aj2'). From (27) and .Minkowski's inequality, we have. r1. El Sn]2'kO lillll { Z (El S,r-,']2"O") 2+6 }2'O". .･i=o ;ll Ao(j.Z=ro o2r-j･)2"O" = A,oZ+6( tr.o 03r.'J' fi>2'UO". '. j' :;ll Aia;'6(j2=,co2--i)2'6 == AaZ+6. Thus the theorem have been proved.. '. References [1] Billingsley, P., Convergence of probability measures, New York, John Wiley and Sons (1968). [2] Davydov, Yu. A., InvariancA principle for stationary processes, Theory Prob.. - Appl.15(1970),488-509. ･[3] Ibragimov, I.A., Some limit theorems for stationary processes, Theory Prob. Appl. 7 (1962), 349-382. '[4i] Ibragimov, I.A. and Yu. V. Linnik, Independent and stationarily related random variables, Iz-vo "Nauk", Moscow (1965).. [5] Lo6v, M., Probability theory, third edition. Princeton, D. Van Nostrand,. (1962). -. L. .. ,[, 6] Oodaira, H. and K. Yoshihara, Funct.ional central limit theorems for strictly stationary processes satisfying the strong mixing condition, (to appear)..

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