Limiting behaviour of moving average processes under ρ-mixing assumption

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1 Note di Matematica ISSN , e-issn Note Mat. 30 (2010) no. 1, doi: /i v30n1p17 Limiting behaviour of moving average processes under ρ-mixing assumption Pingyan Chen i Department Mathematics, Jinan University, Guangzhou, , P.R. China tchenpy@jnu.edu.cn Rita Giuliano Antonini Department of Mathematics, University of Pisa, Largo B. Pontecorvo, 5, Pisa, Italy giuliano@dm.unipi.it Tien-Chung Hu Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan tchu@math.nthu.edu.tw Andrei Volodin ii School of Mathematics and Statistics, University of Western Australia, 35 Stirling HWY, Crawley, WA 6009, Australia andrei@maths.uwa.edu.au Received:...; accepted... Abstract. Let {Y i, < i < } be a doubly infinite sequence of identically distributed ρ-mixing random variables, {, < i < } an absolutely summable sequence of real numbers. In this paper, we prove the complete convergence and Marcinkiewicz-Zygmund strong lawoflargenumbersforthepartialsumsofthemovingaverageprocesses{ Y i+n,n 1}. Keywords: moving average, ρ-mixing, complete convergence, Marcinkiewicz-Zygmund strong laws of large numbers. MSC 2000 classification: primary 60F15 Introduction and Main Results. Let {Y i, < i < + } be a doubly infinite sequence of identically distributed random variables and {, < i < + } be an absolutely summable sequence of real numbers. Next, let X n = Y i+n,n 1 be the moving average process based on the sequence {Y i, < i < + }. As usual, we denote S n = n Xk, n 1, the sequence of partial sums. i This work is supported by the National Natural Science Foundation of China ii This work is supported by the National Science and Engineering Council of Canada c 2010 Università del Salento

2 18 Chen, Giuliano, Hu, Volodin Under the assumption that {Y i, < i < + } is a sequence of independent identically distributed random variables, many limiting results have been obtained for the moving average process {X n,n 1}. For example, Ibragimov [3] established the central limit theorem, Burton and Dehling [4] obtained a large deviation principle assuming Eexp{tY 1} < for all t, and Li et al. [5] obtained the complete convergence result for {X n,n 1}. Certainly, even if {Y i, < i < + } is the sequence of independent identically distributed random variables, the moving average random variables {X n,n 1} are dependent. This kind of dependence is called weak dependence. The partial sums of weakly dependent random variables {X n,n 1} have similar limiting behaviour properties in comparison with the limiting properties of independent identically distributed random variables. For example, we could present some the previous results connected with complete convergence. The following was proved in Hsu and Robbins [1]. Theorem A. Suppose {X n,n 1} is a sequence of independent identically distributed random variables. If EX 1 = 0, E X 1 2 <, then P{ S n εn} < for all ε > 0. Hsu-Robbins result was extended by Li et al. [5] for moving average processes. Theorem B. Suppose {X n,n 1} is the moving average processes based on a sequence {Y i, < i < } of independent identically distributed random variables with EY 1 = 0, E Y 1 2 <. Then P{ S n 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 based on a ρ-mixing sequence. Let {Z i, < i < } be a sequence of random variables defined on a probability space (Ω,F,P) and denote σ-algebras F m n = σ(z i,n i m), n m +. As usual, for a σ-algebra F we denote by L 2 (F) the class of all F-measurable random variables with the finite second moment. A sequence of random variables {Z i, < i < } is called ρ-mixing if the maximal correlation coefficient } ρ(m) = sup k 1 cov(x,y) sup{,x L 2 (F ),Y k L 2 (Fm+k) Var(X)Var(Y) as m. The following maximal inequality can be found in Shao ([7], Theorem 1.1) and plays a crucial role in the proof of our main result. As usual, the notation [ ] is used for the integer part function. Maximal Inequality. Assume that {Z n,n 1} is a sequence of ρ-mixing random variables with EZ n = 0 and E Z n q < for all n 1 and some q 2. Then there is a positive constant K = K(q,ρ( )) depending only on q and ρ( ) such that for any n 1, k E max Z i q K n q/2 exp K ρ(2 i ) max (E Zk 2 ) q/2 i=1 +nexp K ρ q/2 (2 i ) max E Zk q. 0

3 Moving average under ρ-mixing assumption 19 Recall that a measurable function h is said to be slowly varying if for each λ > 0 h(λx) lim x h(x) = 1. We refer to Seneta [6] for other equivalent definitions and for detailed and comprehensive study of properties of such functions. Now we can present the main result of the paper. Theorem 1. Let {Y i, < i < } be a sequence of identically distributed ρ-mixing random variables, and {X n,n 1} be the moving average process based on the sequence {Y i, < 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 if rp 2 take any q > 2p(r 1). Set 2 p [log t] φ(t) = ρ 2/q (2 i ). Let K = K(q,ρ( )) be the constant defined in the Maximal Inequality presented above. If EY 1 = 0 and E Y 1 rp h( Y 1 p )exp{kφ( Y 1 p )} < then for all ε > 0 n r 2 h(n)p{max k n Sk εn1/p } <. In particular, if EY 1 = 0 and E Y 1 p exp{kφ( Y 1 p )} <, then the following Marcinkiewicz- Zygmund strong law of large numbers holds. Remark 1. If n=0 ρ2/q (2 n ) <, then n 1/p S n 0, a.s. E Y 1 rp h( Y 1 p )exp{kφ( Y 1 p )} < and E Y 1 rp h( Y 1 p ) < are equivalent. Hence the first statement of Theorem extends and generalizes Theorem 3.1 of Shao [7]. Marcinkiewicz-Zygmund strong law of large number presented in Theorem generalizes Theorem 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. Hence the assumption about exp{kφ(t)} in Theorem 5.1 of Fazekas and Klesov [2] is excessive Proposition 1. Let {b n,n 0} be a sequence of real number with lim n b n = 0 and set Φ(t) = [logt] n=0 bn. Then for any constant K > 0 the function exp{kφ(t)} is a slowly varying function. Proofs Proof of Theorem 1. Note that n X k = n Y i+k = i+n Y j

4 20 Chen, Giuliano, Hu, Volodin and since ai <, n 1/p E n 1/p Hence for n large enough we have n 1/p n 1 1/p ( i+n i+n ) Y ji{ Y j n 1/p } E Y j I{ Y j n 1/p } E Y 1 I{ Y 1 > n 1/p } E(n 1/p ) p 1 Y 1 I{ Y 1 > n 1/p } E Y 1 p I{ Y 1 > n 1/p } 0, as n. i+n Let Y nj = Y ji{ Y j n 1/p } EY ji{ Y j n 1/p }. Then n r 2 h(n)p{ max Sk εn1/p } Y ji{ Y j n 1/p } < ε/4. n r 2 h(n)p{ max i+k Y ji{ Y j > n 1/p } εn 1/p /2} +C n r 2 h(n)p{ max i+k Y nj εn 1/p /4} =: I +J, say. For I, if rp > 1, by Markov inequality we have I n r 2 h(n)n 1/p E max i+k Y ji{ Y j > n 1/p } n r 1 1/p h(n)e Y 1 I{ Y 1 > n 1/p } = n r 1 1/p h(n) E Y 1 I{m < Y 1 p m+1} m=1 m=n m = E Y 1 I{m < Y 1 p m+1} n r 1 1/p h(n) m r 1/p h(m)e Y 1 I{m < Y 1 p m+1} m=1 E Y 1 rp h( Y 1 p ) E Y 1 rp h( Y 1 p )exp{kφ( Y 1 p )} <.

5 Moving average under ρ-mixing assumption 21 If rp = 1, by Markov inequality and the same argument as the case rp > 1 we have I n r 2 h(n)n θ/p E max i+k Y ji{ Y j > n 1/p } θ n r 1 θ/p h(n)e Y 1 θ I{ Y 1 > n 1/p } <. For J, if rp < 2, by Markov, Hölder, and Maximal inequalities we have J n r 2 h(n)n 2/p E max i+k Y nj 2 ( ( n r 2 h(n)n 2/p E 1 1/2)( 1/2 max i+k = C n r 2 2/p h(n)( ) E max i+k Y nj 2 Y nj n r 2 2/p h(n) nexp K ρ(2 i ) E Y 1 2 I{ Y 1 n 1/p } n r 1 2/p h(n)exp{kφ(n)}e Y 1 2 I{ Y 1 n 1/p } n n r 1 2/p h(n)exp{kφ(n)} E Y 1 2 I{(k 1) 1/p < Y 1 k 1/p } E Y 1 2 I{(k 1) 1/p < Y 1 k 1/p } n r 1 2/p h(n)exp{kφ(n)} n=k k r 2/p h(k)exp{kφ(k)}e Y 1 2 I{(k 1) 1/p < Y 1 k 1/p } E Y 1 rp h( Y 1 p )exp{kφ( Y 1 p )} <. If rp 2, by Markov, Hölder, and Maximal inequalities we have J n r 2 h(n)n q/p E max ( n r 2 h(n)n q/p E n r 2 q/p h(n)( i+k Y nj q ( 1 1/q)( 1/q max i+k ) q 1 E max i+k Y nj q Y nj n r 2 q/p+q/2 h(n)exp K ρ(2 i ) (E Y1 2 I{ Y 1 n 1/p }) q/2 )) 2 )) q

6 22 Chen, Giuliano, Hu, Volodin +C n r 1 q/p h(n)exp{kφ(n)}e Y 1 q I{ Y 1 n 1/p } =: J 1 +J 2, say. For J 1, 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. Then J 1 = C n r 2 q/p+q/2 h(n)exp K ρ(2 i ) (E Y1 2 t+t I{ Y 1 n 1/p }) q/2 n r 2 q/p+q/2+tq/(2p) h(n)exp K ρ(2 i ) (E Y1 2 t I{ Y 1 n 1/p }) q/2 <. For J 2, we have J 2 = C n r 1 q/p h(n)exp{kφ(n)}e Y 1 q I{ Y 1 n 1/p } n n r 1 q/p h(n)exp{kφ(n)} E Y 1 q I{k 1 < Y 1 p k} = C E Y 1 q I{k 1 < Y 1 p k} n r 1 q/p h(n)exp{kφ(n)} n=k k r q/p h(k)exp{kφ(k)}e Y 1 q I{k 1 < Y 1 p k} E Y 1 rp h( Y 1 p )exp{kφ( Y 1 p )} <. Now we show the almost sure convergence. By the first part of Theorem, EY 1 = 0 and E Y 1 p exp{kφ( Y 1 p )} < imply { } n 1 P max Sn εn1/p <, for all ε > 0. Hence > n 1 P{ max 1 m n Sm > εn1/p } 2 k = n 1 P{ max Sm > 1 m n εn1/p } n=2 k 1 1/2 P{ max 1 m 2 k 1 Sm > ε2k/p }. By Borel-Cantelli lemma, 2 k/p max 0 almost surely 1 m 2k Sm which implies that S n/n 1/p 0 almost surely.

7 Moving average under ρ-mixing assumption 23 Proof of Proposition 1. Since lim n b n = 0, for all ε > 0 there exists a positive integer N = N(ε) such that ε < b n < ε for all n > N. For any λ > 1 Note that for t > 1 Hence for t large enough exp{kφ(λt)}/ exp{kφ(t)} = exp{k [log(λt)] n=[log t]+1 b n}. [log(λt)] [logt] log(λt) (logt 1) = logλ+1. exp{ K(logλ+1)ε} exp{kφ(λt)}/exp{kφ(t)} exp{k(logλ+1)ε} and by the arbitrariness of ε > 0 References lim exp{kφ(λt)}/exp{kφ(t)} = 1. t [1] P.L. Hsu, H. Robbins: Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U.S.A. 33 (1947) [2] I. Fazekas, O. Klesov: A general approach to the strong law of large numbers, Theory Probab. Appl. 45 (2000) [3] I.A. Ibragimov: Some limit theorem for stationary processes, Theory Probab. Appl. 7 (1962) [4] R.M. Burton, H. Dehling Large deviations for some weakly dependent random processes, Statist. Probab. Lett. 9 (1990) [5] D. Li, M.B. Rao, and X.C. Wang: Complete convergence of moving average processes, Statist. Probab. Lett. 14 (1992) [6] E. Seneta: Regularly varying function, Lecture Notes in Math. 508 Springer, Berlin, [7] Q.M. Shao: Maximal inequalities for partial sums of ρ-mixing sequences, Ann. Probab. 23 (1995)