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(1)Functio'nal Central Limit Theorems for ' Some Semi-Stationary Processes. '. '. '. By. ",. Hiroshi NEGISHI v. (Received May 31, 1973). '. '. 1. Summary.. ' '. ' t. processes which In [6], Serfozo introduced a class of stochastic he called semi-stationary processes, and discussed elemehtary properties of these. processes. Moreover, he presented ergodic theorems and functional centra! limit theorems with the uniformly strong mixing condition for them. These processes are a generalization of strictly stationary processes. Hereafter, stationary means strictly stationary. The object of thiS paper is to prove the functional central limit theorems. for the semi-stationary processes with the strong mixing condition. These results contain as special cases the corresponding theorems for the stationary. processes (Theorem 1, Theorem 2 in [4]). 2. Definitions and elementary properties.. Following serfozo [6], we define the semi-stationary pr6cess and stat6 elementary properties of this process.. .. The integers and real numbers are denoted by Zand R respectively. All stochastic processe$ herein are defined on the probability space (9, & P), and ;. real valued and measurable.. s 1. DEFINITIoN. We call a process {8.: nEZ} semi-stationary with respect to a nonconstant integer valued process {Tn: nEZ}, where Wn =Tn+i-Tnl-lrO and TLi:.fO;STo, if the following condition is satisfied:. For any mi, ･･･,mk, ni, ･･･,nk and h in Z, the joint distribution of the ran-. dom variables W(ni+h),･･･,W(nk+h), 8(mi+Tn,+h),"',8(mk+Tnk+h) does not depend on h.. ''. '. ' Note that the process {8.} is statiohary if {6.} is semi-stationary with respect to T.==n. Define random elements 4. and CN. of RZ by ' (1) ･ Cn(m)=g(T.+m)'.

(2) 12 H. NEGIsHI and. n(M)=6(Tn+M)I[o,w.)(M),fOrMEZ, (2) , C"" wherelA(･)istheindicatorfunctign., . Let S" be the 'shift transformation defined by. '. '. Sn('''sX-1, Xo, Xl, ''')=('" 7 Xn-17 Xn: Xn+IJ "'). 'i. In [6], Serfozo proved the following the,Qre.m.. THEOREM A (Serfozo). The followin, g statements are equivalent.. i'. (i)g=={g.}issemi-stationarywithresPectto{T.}. ･' ' (ii) {(Wn,CA..)} is stationary,. (iii) {(W.,C.)} is stationary. . (iv) For any measurable functionf on RZ which talees values in some measurable sPace, the Process 6.i==f(S"6) (nEZ) is semi-stationarN with resPect to {Tn}.. '. t tt. '. tt '. '. ''3.Functionalcentrallimittheorems. ･ ' ,Let {en} be semi-stationary with respect to {T.}, where ToLo for con-. venlence.. . NQte that ' (3) 26,=: ]X Z,+26k .. '. ' n-1 yn-1 n-1 ic=O ic=O k=1'vn. where v.==sup{m: T.;;iln} and. ･ Tn+1-1. '. (4) ' Zn == 26ic k=Tn. /t t. j (we adopt the convention that ]X==O, if i>i i From the definition (2j of 4N'. and Theorem A, we have that the process {Z.} is stationary. Let {M.} be the stationary process defined by. '. m. M.=sup[Zgic1 fornlO,. m<T,n.:.t ic='Tn. and whenever the expressions exist we set. E (Zo). (s.) , A=E(vvssoo ([ Zg'AMZo)(Zk-AJ>v,)] (6) B,,- E(Zo-AWo)2+2,;.ll,EE w(. and. (7) X.(t)=bil7it[n,.#?ili(6ic-A) forof{gts{i.. ,-. ;.

(3) FunctionalCentralLimitTheoremsforSomeSemi-StationaryProcesses 13 THEoREM 1 (cf. Theorem 1 in E4]''and [3]). SuPPose that {M.} and{W.} are bozanded, i.e., M.<C<oo and W.<C<c>o with Probability one. SuPPose further that {(W., g.)} gatisfies the strong mixing. condition with the mixing cooficients a(n) such that Z cr(n)<oo.. n =1 Then A and B2 are 7inite. ILf B2>O and if/X. is defined by (7), then x.--21-i>EIB where E!B is a J7pT//ener' ,. process. ' '. ,. THEOREM2(cf.Theorem2in[4]). ... SuPPose that E(M62"O")<oo and E(VIZ,2"6)<oo for some 6>O. SuPPose further. 'L. that {(W., a.)} satisfies the strong mixing condition with the mixing coe:17Zcients. oo 6a+2 <oo. Then A and B2 are finite. (f B2>Oand if X. a(n) such that Z(a(n)) n=t. D is dofned by (7), then X.-->E!B where E!B is a Wiener Process. To prove the theorems, we need the following lemmas. LEMMA 1. 11f {W.} satisfies the hyPothes2s of Theorem 1 or Theorem 2, then･. (8) .1-i}zg P{ "1"-. n. u2 du {ii;llO) ;!lx}- .v>-. xS..e". ., . ' 'V (E(W,))3 '. where o2==Var(Wo).. PRooF. Using the facts that {Ml.} satisfies the central 1i'mit theorem (cf.. Theorem 1.6 or Theorem 1.7 in [2]) and that P{v.< m} == P{T. > n} -. the same argument as on pp. 269-270 of [5] gives (8).. From Lemma 1, the next lemma is easily obtained.. s.. LEMMA 2. Under the same conditions of Theorem 1 or Theorem 2, (9). Vn P 1 E(Wo). ---l))･ n. REMARK. Serfozo proved that "nn ---> E(lvv,) a.s. (cf.[6],[7]). But, in. our problems, we need only this lemma.. PRooF OF THEoREM 1. Since {Z.} is stationary and since the {M.} are bounded, A is finite. The hypotheses imply that the stationary process {(Z.-AJ7I7.): nlO} satisfies the strong mixing condition with the same mixing. coefficients. Since E(Z.-AW.)==O and {Z.-AW.} are bounded, it follows.

(4) 14 ,.i- .. ,H.NEGIsH･I '. form Theorem 1.6 in [2] that B2 is finite.. From(3)and(7),we'have ･ ' ao) xn(t)='. il7ii-"[2]2)],-i(2k'AWkS+'B'll7f,l'tllllii-,..iSicLkAv-.{["`]T..T"[nt]}'. Let .51.(t) denote the first term on the right of (10). Using Theorem 1 in [4]. and a random time transformation argument as on pp. 143-146 of [1], with rv S. Lemma 2, gives X..E!B. '. i. terms on the right of (10) Gonverge to zero in probability, as n-->oo. But these. t. Thus the assertion will follow upon showing that the second and third. facts may be shown by･the same way as in [6]. '. ' ' ' PROOF OF THEOREM 2. The proof of this theorem differs only 1t slightly from that of Theorem 1, wi,th Theorem 1.7 in [2] and Theorem 2 i,n [4] playing the roles of Theorem 1.6 in [2] and Theorem 1 in [4] respectively.. tt. '. '. References [1] BiLiNGsLEy, P.; Convergence of Probability measures. John Wiley, New York (1968). [12] IBRAGiMov, I.A.: Some limit theorems for stationary processes, Theory of Prob. and Appl. 7 (1962), 349-382. [3] IBRAGiMov, I.A. and Yu, V. LiNNiK: Independent and stationarily related random variables, Iz-vo "Nauk", Moscow (1965). (in Russian) [4] OoDAiRA, H. and K. YosHrHARA: Functional central limit theorems for strictly stationary processes satisfying the strong mixing condition. K6dai Math. Sem. Rep. 24 (1972), 259-269. [5] PARzEN, E.: Stochastic Processes. Holden-Day, San Francisco (1962). [6] SERFozo, R.: Semi-stationary processes. Z. Wahrscheinlichkeits theorie Verw, Geb. 23 (1972), 125-132. [7] SERFozo, R.: Weak convergence of superpositions of randomly selected partial sums, (1971) (Submitted for publication).. .･. ;.

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