Let {Xj1 -∞<j<∞} be a strictly stationary sequence of random variables centered at expectations with finite variance, which satisfies ф-mixing condition (1.1)   supP(A∩B)-P(A)P(B)/P(A)≦ф(n)↓0 (N→∞). ..

Yokoyama Ryozo

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Convergence of moments in the central limittheorem for stationary φ-mixing sequences著者 Yokoyama Ryozo雑誌名 Tsukuba journal of mathematics巻 3号 2ページ 1-6発行年 1979-12URL http://hdl.handle.net/2241/7104TSUKUBA J. MATH.Vol. 3, No. 2 (1979), 1―6CONVERGENCE OF MOMENTS IN THE CENTRAL LIMITTHEOEEM FOR STATIONARY ^-MIXING SEQUENCESByRyozo Yqkoyama1. Introduction and result.Let {Xj, -oo<j<oo} be a strictly stationary sequence of random variablescentered at expectations with finitevariance, which satisfies0-mixing condition(1.1) sup ＼P(Af)B)-P(A)P(B) ＼IP(A)'£4,(n)[ 0 (≪->oo).Here the supremum is taken over ail Ae3t-<x, and BeJM£+n, and 3Ai denotes theafield generated by X, (a^j^b). Let Sn = Xi + -~+Xn and an2= ESn2, ≫= 1, % ･･･.For independent random variables, Brown [1 and 2] has shown that theLindeberg condition of order v'^2 is necessary and sufficientfor the central limittheorem and the convergence of E＼Snjan＼vtowards the corresponding moment ofthe normal distribution. For dependent random variables, such a result seemsless well-known. We study here the convergence of moments for stationary (j>-mixing sequences.Theorem. Let ＼XA satisfy(1.1). // EXfm<oo for some integer m^2, and if(1.2)as w―>oo(a>0), then(1.3)<rna= o*n{l+o(l))E(SnlOn)2m-+fam (≫->Oo)where /3yis the vth absolute moment of N(0, 1).We remark that under the assumptions of the theorem X, satisfiesthe centrallimit theorem (cf.[4, Theorem 18.5.1]). Also remark that'any other conditibeyond (1.1) on the decays of mixing coefficients6(n) are not required.ons2. Preparatory lemmas.Lemma 1 [4, Theorem 17.2.3]. Suppose that (1.1) is satisfiedand that f and7]are measurable with respect to JAk-^ and 3il＼-≪{n'^0) respectively. If E＼g＼p<coand E＼rj＼q<oofor p, q>＼ with {l/p)+ (llq)= l, then(2.1) ＼E(to)-m)Eto)＼ ^ 2{6(n)E＼mi/p{E＼y]＼r/q-Received October 2, 19782Ryozo YokoyamaLemma 2. Let {X,-}satisfy(1.1) and E＼Xi＼v<oofor some v^2. IfGn ―ycoas n-*oo, then there is a constant K, for which(2.2) E＼Sn＼v^Koit w^l,In thislemma, the assumption (1.2) is not necessarily required. If (1.2) holds,the right-hand side of (2.2) can be replaced by Knv/Z.Proof. We apply the method used in the proof of Lemma 7.4 of Doob [3] tothat of Lemma 18.5.1 of Ibragimov-Linnik [4]. Lemma 2 is true for v=2. Weassume therefore that (2.2) holds when v is an integer m^2 and prove that itthen holds for v=m+d, 0<5^1. Let us write2n+kq ― y Y- ft ―F＼^ ＼m+sOn― l_x -^jiU-n―E>＼On＼j=n+k+lWe only prove that, for s>0 there exist K% and k such that(2.3) E＼Sn+ Sn＼m+3^(2+e)an+K1ars-The proof of (2.2) then follows on the same line as in Lemma 18.5.1 of [4].We have(2.4) ElSn + SJ^^EUSn + S^ilSJ' + lSnn^ElSJ^ + ElSn]^TO―1 TO.7=0 7=1Since Sn and Sn have the same distribution,(2.5) E＼Sn＼m+'=E＼§n＼m+'=an.Using (2.1) with p=(m+8)l(j+8),(2.6) E＼Sn＼i+'＼Sn＼m-i^2an[^k)]CJ+nam+n+E＼Sn＼i+'E＼Sn＼m-i,and with p=(m+d/j,(2.7) E＼Sn＼J＼Sn＼m-J+s^2an[&(k)]jnm+d)+E＼Sn＼JE＼Sn＼m-j+s.By Holder'sinequality,E＼Sn＼u^(E＼Sn＼m)u/m,0<u^m.Thus, since(2.2)is assumed to hold for v=m (with some K), for Q^j^m―l,(2.8) £|SBK+a£|S≫|M-'^(£|S≫|*)(w+H/m^/ir<#+<.and for l<j<m,Convergence of Moments in the Central Limit Theorem 3(2.9) E＼Sn＼JE＼Sn＼m-J+8^(E＼Sn＼myM+e>/m^Kaf+e.From (2.4) through (2.9), we obtainE＼Sn+ Sn＼m^^{2+Kz{(j>{k)rm+i))an+KlaT＼for some constants Kx and K2. To prove (2.3) it sufficesto take k so large thatK2[0(k)]s/<m+i≫<e.Wp rpnrpsenf- thf sum .S^in the.form* fc+1i = l i=lwhereip+(.i-tfqft* S *> d^*)(i-i)(P+g)+lVi=iv+nwhere k ―＼n＼{j.+q)], and p―p{n) and q=g(n) are integer-valued functions such that(2.10) p-*x>, q->oo, q=o(p), p=o(n), nq―o(pz) and n<j)(q)=o(p).For such a pair of p and q, see for example [4, Theorem 18.4.1]. Under the re-quirements imposed on p and q, we shall show that Z'k is negligible,and thatconsequently £1(Sn/o>w)2m~£'(2rj;/o'n)2"1.We note that, because of the stationarity,Lemma 2 is applicable to & and -q%. In the following, for convenience' sake theconditions of the theorem are assumed to hold and K denotes generic constant.Lemma 3. As ≫-≫oo(2.11) EZll=ES2nl+o(a2nl), 1=1, 2, -, m,Proof. We firstshow that(2.12) EZ^oisft).Wp havp.(2.13)By Minkowski's inequiality,Lemma 2 and (2.10),(2.14) EZ'% ^ kaiErj?^ K(k2q)l=o(all),4by Lemma 2.(2.15)Ryozo YokoyamaE7$+l^K(n-k(p+q))l = o(ril)and by Holders Inequality,(2.14) and (2.15),(2.16) ＼EZ'ir)V^(EZflly'*HErfAV2l-≫'al= o(oVThen (2.12) follows from (2.13)-(2.16).21-1VL/j k ―J2,＼On £> kit) ―CsOn -f- £j {―1 ) ■'tiOJn £, k+lJi = (＼and by Holder's inequality, Lemma 2 and (2.12),＼K%z'il^mES^y/il(EZfii+lyii-^≫=o(a%1),for j = 0, 1, ■･',21―1. Thus the lemma is proved.Let r＼= EZ＼ for i = l, 2, ■■-,k. Then (2.11) implies that(2.17) rj=ff* (1+0(1)).Since£^?=JB(Si(,+fl-1Z''()≪it follows from the proof of Lemma 3 that(2.18) EZf = ES?(P+q) +<≪,+,),which together with (2.10) implies that(2.19) EZf^K(ip)1,for i = l, 2, ･･-, k, 1=1, 2, ■■■,m. Also (2.10) and (2.18) imply that(2.20) ^=^(1+0(1)), i = l, 2, -, k.3 Proof of Theorem.E(Snl<Jn)2= l, n ―1, 2, ･･■. Assume inductively that as n-^-oo(3.1) EiSnlanf1-^^, 1=1, 2, -, m-1.In view of (2.11) and (2.17), the assumption (3.1) is equivalent to the one that asn―>oo(3.2) E(Zklrk)*l-*fa, 1=1, 2, ･..,m-1.Using (2.11) again, we have only to prove under the assumption (3.2) that as ≪->oo(3.3) E(Zklrkym^>fcm.We haveConvergence of Moments in the Central Limit Theorem(3.4) EZr= £2f!(T)EZuerj,where Z0-0,i=l j=0i=l i=l i=lBy Lemma 2 and (2.20),(3.5) 2 E$T^Kkpm=o(rfl).By Lemmas 1, 2, (2.10),(2.19) and (2.20)(3.6) 2 ＼EZf^l5i＼k 2TO-3jLj Li ＼j )Cl£'i-i.Zii=l j=l^2[0(g)](Zm~1)/2m 2 (^2Ti)(2m~1)/2TO(^r)1/2m^K[0(q)]U2pm 2 (*―l)cam-"/aFor ; = 1, ･■･,2m-3, by Lemma 2, (2.19) and (2.20)(3.7)and so(3.8)Further, by Lemmas 1, 2,(2.19) and (2.20),(3,9)k ki=l i=l^ K[$(q)] {-m-1)/m{kp)m=o{vfl).Consequently, by (3.4)-(3.9), as ≫-≫oo(3.10)By (2.20) and (3.2),(3.11)fckX (DEiZi-tlu-y-iEGDrT!?i=l56 Ryozo Yokoyama~{(r")A,≪-2+o(i)}orh(i-i)m-l+oa)i=l~{(D ft≫-2+o(l)}< (&m/m) +0(1)~{(2m-l)^w-2+o(l)}rr+O(l)~/32rorr.Hence, by (3.10) and (3.11),(3.3) follows, and the proof of the theorem is com-pleted.Remark. If EIX^"'<oo(v'>v>2), then by Lemma 2, {＼Snlon＼v,n^l} is uni-formly integrable. By the central limit theorem we have, without the assump-tion (1.2),E＼SnltJn＼V-*h(≫->oo).References[ 1 ] Brown, B. M., Moments of a stopping rule related to the central limit theorem.Ann. Math. Statist.40 (1969) 1236-1249.[ 2 ] Brown, B. M., Characteristicfunctions, moments, and the central limit theorem.Ann. Math. Statist.41 (1970) 658-664.[ 3 ] Doob, J.L., StochasticProcesses,Wiley, New York, 1953.[ 4 ] Ibragimov, I.A. and Linnik, Yu. V., Independent and Stationary Sequences of Ran-dom Variables,Wolters-Noordhoff,Groninsen. 1971.Department of MathematicsOsaka Kyoiku UniversityIkeda, Osaka, 563

Convergence of moments in the central limit theorem for stationary φ-mixing sequences

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