Let $\{Y_i, -\infty&lt;i&lt;\infty\}$ be a doubly infinite sequence of identically distributed $\rho$-mixing random variables, $\{a_i,-\infty&lt;i&lt; \infty\}$ an absolutely summable sequence of real numbers. In this paper, we prove the complete convergence and Marcinkiewicz-Zygmund strong law of large numbers for the partial sums of the moving average processes $\{\sum\limits^\infty_{i=-\infty}a_i Y_{i+n},n\geq1\}$

Giuliano Antonini, Rita

Hu, Tien-Chung

Volodin, Andrei

English

Università del Salento: ESE - Salento University Publishing

Note di Matematica ISSN 1123-2536, e-ISSN 1590-0932Note Mat. 30 (2010) n. 1, 17–23. doi:10.1285/i15900932v30n1p17Limiting behaviourof moving average processesunder ρ-mixing assumptionPingyan CheniDepartment Mathematics, Jinan University, Guangzhou, 510630, P.R. Chinatchenpy@jnu.edu.cnRita Giuliano AntoniniDepartment of Mathematics, University of Pisa, Largo B. Pontecorvo, 5, 56100 Pisa, Italygiuliano@dm.unipi.itTien-Chung HuDepartment of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwantchu@math.nthu.edu.twAndrei Volodin iiSchool of Mathematics and Statistics, University of Western Australia,35 Stirling HWY, Crawley, WA 6009, Australiaand Department of Mathematics and Statistics, University of Regina,Regina, SK, S4S 0A2, Canadaandrei@maths.uwa.edu.au, andrei@math.uregina.caReceived: 9.5.2008; accepted: 10.11.2009.Abstract. Let {Yi,−∞ < i < ∞} be a doubly infinite sequence of identically distributedρ-mixing random variables, {ai,−∞ < i < ∞} an absolutely summable sequence of realnumbers. In this paper, we prove the complete convergence and Marcinkiewicz-Zygmund stronglaw of large numbers for the partial sums of the moving average processes {∞∑i=−∞aiYi+n, n ≥ 1}.Keywords: moving average, ρ-mixing, complete convergence, Marcinkiewicz-Zygmund stronglaws of large numbers.MSC 2000 classification: primary 60F15Introduction and Main Results.Let {Yi,−∞ < i < +∞} be a doubly infinite sequence of identically distributed randomvariables and {ai,−∞ < i < +∞} be an absolutely summable sequence of real numbers. Next,iThis work is supported by the National Natural Science Foundation of ChinaiiThis work is supported by the National Science and Engineering Council of Canadahttp://siba-ese.unisalento.it/ c© 2010 Universita` del Salento18 Chen, Giuliano, Hu, VolodinletXn =∞∑i=−∞aiYi+n, n ≥ 1be the moving average process based on the sequence {Yi,−∞ < i < +∞}. As usual, we denoteSn =∑nk=1Xk, n ≥ 1, the sequence of partial sums.Under the assumption that {Yi,−∞ < i < +∞} is a sequence of independent identicallydistributed random variables, many limiting results have been obtained for the moving averageprocess {Xn, n ≥ 1}. For example, Ibragimov [3] established the central limit theorem, Burtonand Dehling [4] obtained a large deviation principle assuming E exp{tY1} < ∞ for all t, andLi et al. [5] obtained the complete convergence result for {Xn, n ≥ 1}.Certainly, even if {Yi,−∞ < i < +∞} is the sequence of independent identically dis-tributed random variables, the moving average random variables {Xn, n ≥ 1} are dependent.This kind of dependence is called weak dependence. The partial sums of weakly dependentrandom variables {Xn, n ≥ 1} have similar limiting behaviour properties in comparison withthe limiting properties of independent identically distributed random variables.For example, we could present some the previous results connected with complete conver-gence. The following was proved in Hsu and Robbins [1].Theorem A. Suppose {Xn, n ≥ 1} is a sequence of independent identically distributedrandom variables. If EX1 = 0, E|X1|2 <∞, then∞∑n=1P{|Sn| ≥ εn} <∞ for all ε > 0.Hsu-Robbins result was extended by Li et al. [5] for moving average processes.Theorem B. Suppose {Xn, n ≥ 1} is the moving average processes based on a sequence{Yi,−∞ < i < ∞} of independent identically distributed random variables with EY1 = 0,E|Y1|2 <∞. Then∞∑n=1P{|Sn| ≥ εn} <∞ for all ε > 0.Very few results for a moving average process based on a dependent sequence are known.In this paper, we provide a result on the limiting behavior of a moving average process basedon a ρ-mixing sequence.Let {Zi,−∞ < i <∞} be a sequence of random variables defined on a probability space(Ω,F , P ) and denote σ-algebrasFmn = σ(Zi, n ≤ i ≤ m), −∞ ≤ n ≤ m ≤ +∞.As usual, for a σ-algebra F we denote by L2(F) the class of all F-measurable random variableswith the finite second moment.A sequence of random variables {Zi,−∞ < i < ∞} is called ρ-mixing if the maximalcorrelation coefficientρ(m) = supk≥1sup{∣∣∣∣∣ cov(X,Y )√Var(X)Var(Y )∣∣∣∣∣ , X ∈ L2(Fk−∞), Y ∈ L2(F∞m+k)}→ 0as m→∞.The following maximal inequality can be found in Shao ([7], Theorem 1.1) and plays acrucial role in the proof of our main result. As usual, the notation [·] is used for the integerpart function.Maximal Inequality. Assume that {Zn, n ≥ 1} is a sequence of ρ-mixing random vari-ables with EZn = 0 and E|Zn|q < ∞ for all n ≥ 1 and some q ≥ 2. Then there is a positiveMoving average under ρ-mixing assumption 19constant K = K(q, ρ(·)) depending only on q and ρ(·) such that for any n ≥ 1,E max1≤k≤n|k∑i=1Zi|q ≤ Knq/2 expK[logn]∑i=0ρ(2i) max1≤k≤n(E|Zk|2)q/2+n expK[logn]∑i=0ρq/2(2i) max1≤k≤nE|Zk|q.Recall that a measurable function h is said to be slowly varying if for each λ > 0limx→∞h(λx)h(x)= 1.We refer to Seneta [6] for other equivalent definitions and for detailed and comprehensive studyof properties of such functions.Now we can present the main result of the paper.Theorem 1. Let {Yi,−∞ < i < ∞} be a sequence of identically distributed ρ-mixingrandom variables, and {Xn, n ≥ 1} be the moving average process based on the sequence{Yi,−∞ < i <∞}.Let h(x) be a positive slowly varying function and 1 ≤ p < 2, r ≥ 1. If rp = 1,additionally assume that∑∞i=1 |ai|θ < ∞ for some θ ∈ (0, 1). If rp < 2 take q = 2, and ifrp ≥ 2 take any q > 2p(r−1)2−p. Setφ(t) =[log t]∑i=0ρ2/q(2i).Let K = K(q, ρ(·)) be the constant defined in the Maximal Inequality presented above.If EY1 = 0 and E|Y1|rph(|Y1|p) exp{Kφ(|Y1|p)} <∞ then for all ε > 0∞∑n=1nr−2h(n)P{maxk≤n|Sk| ≥ εn1/p} <∞.In particular, if EY1 = 0 and E|Y1|p exp{Kφ(|Y1|p)} <∞, then the following Marcinkiewicz-Zygmund strong law of large numbersn−1/pSn → 0, a.s.holds.Remark 1. If∑∞n=0 ρ2/q(2n) <∞, thenE|Y1|rph(|Y1|p) exp{Kφ(|Y1|p)} <∞ and E|Y1|rph(|Y1|p) <∞are equivalent. Hence the first statement of Theorem extends and generalizes Theorem 3.1 ofShao [7]. Marcinkiewicz-Zygmund strong law of large number presented in Theorem generalizesTheorem 5.1 of Fazekas and Klesov [2] and Corollary 3.1 of Shao [7].By the following Proposition, the function exp{Kφ(t)} is a slowly varying function. Hencethe assumption about exp{Kφ(t)} in Theorem 5.1 of Fazekas and Klesov [2] is excessiveProposition 1. Let {bn, n ≥ 0} be a sequence of real number with limn→∞ bn = 0 and setΦ(t) =∑[log t]n=0 bn. Then for any constant K > 0 the function exp{KΦ(t)} is a slowly varyingfunction.20 Chen, Giuliano, Hu, VolodinProofsProof of Theorem 1. Note thatn∑k=1Xk =n∑k=1∞∑i=−∞aiYi+k =∞∑i=−∞aii+n∑j=i+1Yjand since∑∞i=−∞ |ai| <∞,n−1/p|E∞∑i=−∞aii+n∑j=i+1YjI{|Yj | ≤ n1/p}|≤ n−1/p∞∑i=−∞|ai|i+n∑j=i+1E|Yj |I{|Yj | ≤ n1/p}≤ n1−1/p(∞∑i=−∞|ai|)E|Y1|I{|Y1| > n1/p}≤ CE(n1/p)p−1|Y1|I{|Y1| > n1/p}≤ CE|Y1|pI{|Y1| > n1/p} → 0, as n→∞.Hence for n large enough we haven1/p|∞∑i=−∞aii+n∑j=i+1YjI{|Yj | ≤ n1/p}| < ε/4.Let Ynj = YjI{|Yj | ≤ n1/p} − EYjI{|Yj | ≤ n1/p}. Then∞∑n=1nr−2h(n)P{ max1≤k≤n|Sk| ≥ εn1/p}≤ C∞∑n=1nr−2h(n)P{ max1≤k≤n|∞∑i=−∞aii+k∑j=i+1YjI{|Yj | > n1/p}| ≥ εn1/p/2}+C∞∑n=1nr−2h(n)P{ max1≤k≤n|∞∑i=−∞aii+k∑j=i+1Ynj | ≥ εn1/p/4}=: I + J, say.For I, if rp > 1, by Markov inequality we haveI ≤ C∞∑n=1nr−2h(n)n−1/pE max1≤k≤n|∞∑i=−∞aii+k∑j=i+1YjI{|Yj | > n1/p}|≤ C∞∑n=1nr−1−1/ph(n)E|Y1|I{|Y1| > n1/p}=∞∑n=1nr−1−1/ph(n)∞∑m=nE|Y1|I{m < |Y1|p ≤ m+ 1}=∞∑m=1E|Y1|I{m < |Y1|p ≤ m+ 1}m∑n=1nr−1−1/ph(n)Moving average under ρ-mixing assumption 21≤ C∞∑m=1mr−1/ph(m)E|Y1|I{m < |Y1|p ≤ m+ 1}≤ CE|Y1|rph(|Y1|p) ≤ CE|Y1|rph(|Y1|p) exp{Kφ(|Y1|p)} <∞.If rp = 1, by Markov inequality and the same argument as the case rp > 1 we haveI ≤ C∞∑n=1nr−2h(n)n−θ/pE max1≤k≤n|∞∑i=−∞aii+k∑j=i+1YjI{|Yj | > n1/p}|θ≤ C∞∑n=1nr−1−θ/ph(n)E|Y1|θI{|Y1| > n1/p} <∞.For J , if rp < 2, by Markov, Ho¨lder, and Maximal inequalities we haveJ ≤ C∞∑n=1nr−2h(n)n−2/pE max1≤k≤n|∞∑i=−∞aii+k∑j=i+1Ynj |2≤ C∞∑n=1nr−2h(n)n−2/pE(∞∑i=−∞(|ai|1− 1/2)(|ai|1/2 max1≤k≤n|i+k∑j=i+1Ynj |))2≤ C∞∑n=1nr−2−2/ph(n)(∞∑i=−∞|ai|)∞∑i=−∞|ai|E max1≤k≤n|i+k∑j=i+1Ynj |2≤ C∞∑n=1nr−2−2/ph(n)n expK[logn]∑i=0ρ(2i)E|Y1|2I{|Y1| ≤ n1/p}= C∞∑n=1nr−1−2/ph(n) exp {Kφ(n)}E|Y1|2I{|Y1| ≤ n1/p}≤ C∞∑n=1nr−1−2/ph(n) exp {Kφ(n)}n∑k=1E|Y1|2I{(k − 1)1/p < |Y1| ≤ k1/p}≤ C∞∑k=1E|Y1|2I{(k − 1)1/p < |Y1| ≤ k1/p}∞∑n=knr−1−2/ph(n) exp {Kφ(n)}≤ C∞∑k=1kr−2/ph(k) exp {Kφ(k)}E|Y1|2I{(k − 1)1/p < |Y1| ≤ k1/p}≤ CE|Y1|rph(|Y1|p) exp{Kφ(|Y1|p)} <∞.If rp ≥ 2, by Markov, Ho¨lder, and Maximal inequalities we haveJ ≤ C∞∑n=1nr−2h(n)n−q/pE max1≤k≤n|∞∑i=−∞aii+k∑j=i+1Ynj |q≤ C∞∑n=1nr−2h(n)n−q/pE(∞∑i=−∞(|ai|1− 1/q)(|ai|1/q max1≤k≤n|i+k∑j=i+1Ynj |))q≤ C∞∑n=1nr−2−q/ph(n)(∞∑i=−∞|ai|)q−1∞∑i=−∞|ai|E max1≤k≤n|i+k∑j=i+1Ynj |q≤ C∞∑n=1nr−2−q/p+q/2h(n) expK[logn]∑i=0ρ(2i) (E|Y1|2I{|Y1| ≤ n1/p})q/222 Chen, Giuliano, Hu, Volodin+C∞∑n=1nr−1−q/ph(n) exp {Kφ(n)}E|Y1|qI{|Y1| ≤ n1/p}=: J1 + J2, say.For J1, since r − 2− q/p+ q/2 < −1, we can take t > 0 small enough such that r − 2− q/p+q/2 + tq/(2p) < −1. ThenJ1 = C∞∑n=1nr−2−q/p+q/2h(n) expK[logn]∑i=0ρ(2i) (E|Y1|2−t+tI{|Y1| ≤ n1/p})q/2≤ C∞∑n=1nr−2−q/p+q/2+tq/(2p)h(n) expK[logn]∑i=0ρ(2i) (E|Y1|2−tI{|Y1| ≤ n1/p})q/2 <∞.For J2, we haveJ2 = C∞∑n=1nr−1−q/ph(n) exp {Kφ(n)}E|Y1|qI{|Y1| ≤ n1/p}≤ C∞∑n=1nr−1−q/ph(n) exp {Kφ(n)}n∑k=1E|Y1|qI{k − 1 < |Y1|p ≤ k}= C∞∑k=1E|Y1|qI{k − 1 < |Y1|p ≤ k}∞∑n=knr−1−q/ph(n) exp {Kφ(n)}≤ C∞∑k=1kr−q/ph(k) exp {Kφ(k)}E|Y1|qI{k − 1 < |Y1|p ≤ k}≤ CE|Y1|rph(|Y1|p) exp {Kφ(|Y1|p)} <∞.Now we show the almost sure convergence. By the first part of Theorem, EY1 = 0 andE|Y1|p exp{Kφ(|Y1|p)} <∞ imply∞∑n=1n−1P{max1≤k≤n|Sn| ≥ εn1/p}<∞, for all ε > 0.Hence∞ >∞∑n=1n−1P{ max1≤m≤n|Sm| > εn1/p}=∞∑k=12k∑n=2k−1n−1P{ max1≤m≤n|Sm| > εn1/p}≥ 1/2∞∑k=1P{ max1≤m≤2k−1|Sm| > ε2k/p}.By Borel-Cantelli lemma,2−k/p max1≤m≤2k|Sm| → 0 almost surelyMoving average under ρ-mixing assumption 23which implies that Sn/n1/p → 0 almost surely.Proof of Proposition 1. Since limn→∞ bn = 0, for all ε > 0 there exists a positive integerN = N(ε) such that−ε < bn < ε for all n > N.For any λ > 1exp{KΦ(λt)}/ exp{KΦ(t)} = exp{K[log(λt)]∑n=[log t]+1bn}.Note that for t > 1[log(λt)]− [log t] ≤ log(λt)− (log t− 1) = log λ+ 1.Hence for t large enoughexp{−K(log λ+ 1)ε} ≤ exp{KΦ(λt)}/ exp{kΦ(t)} ≤ exp{K(log λ+ 1)ε}and by the arbitrariness of ε > 0limt→∞exp{KΦ(λt)}/ exp{KΦ(t)} = 1.References[1] P.L. Hsu, H. Robbins: Complete convergence and the law of large numbers, Proc. Nat.Acad. Sci. U.S.A. 33 (1947) 25–31.[2] I. Fazekas, O. Klesov: A general approach to the strong law of large numbers, TheoryProbab. Appl. 45 (2000) 436–449.[3] I.A. Ibragimov: Some limit theorem for stationary processes, Theory Probab. Appl. 7(1962) 349–382.[4] R.M. Burton, H. Dehling Large deviations for some weakly dependent random pro-cesses, Statist. Probab. Lett. 9 (1990) 397–401.[5] D. Li, M.B. Rao, and X.C. Wang: Complete convergence of moving average processes,Statist. Probab. Lett. 14 (1992) 111–114.[6] E. Seneta: Regularly varying function, Lecture Notes in Math. 508 Springer, Berlin,1976.[7] Q.M. Shao: Maximal inequalities for partial sums of ρ-mixing sequences, Ann. Probab.23 (1995) 948–965.

Limiting behaviour of moving average processes under       $\rho$-mixing assumption

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