Complete Convergence for Moving Average Process of Martingale Differences

Under some simple conditions, by using some techniques such as truncated method for random variables (see e.g., Gut (2005)) and properties of martingale differences, we studied the moving process based on martingale differences and obtained complete convergence and complete moment convergence for this moving process. Our results extend some related ones.


Introduction
Let {Y i , −∞ < i < ∞} be a doubly infinite sequence of random variables.Assume that {a i , −∞ < i < ∞} is an absolutely summable sequence of real numbers and is the moving average process based on the sequence {Y i , −∞ < i < ∞}.As usual, S n n k 1 X k , n ≥ 1, denotes the sequence of partial sums.
For the moving average process {X n , n ≥ 1}, where {Y i , −∞ < i < ∞} is a sequence of independent identically distributed i.i.d.random variables, Ibragimov 1 established the central limit theorem, Burton and Dehling 2 obtained a large deviation principle, and Li et al. 3 gave the complete convergence result for {X n , n ≥ 1}.Zhang 4 and Li and Zhang 5 extended the complete convergence of moving average process for i.i.d.sequence to ϕmixing sequence and NA sequence, respectively.Theorems A and B are due to Zhang 4 and Kim et al. 6 , respectively.
Chen et al. 7 and Zhou 8 also studied the limit behavior of moving average process under ϕ-mixing assumption.Yang et al. 9 investigated the moving average process for AANA sequence.For more works on complete convergence, one can refer to 3-6, 10-13 and the references therein.
Recall that the sequence {X n , n ≥ 1} is stochastically dominated by a nonnegative random variable X if Recently, Chen and Li 14 investigated the limit behavior of moving process under martingale difference sequences.They obtained the following theorems.
Theorem C. Let r ≥ 1, 1 ≤ p < 2 and rp < 2. Assume that {X n , n ≥ 1} is a moving average process defined in 1.1 , where {Y i , F i , −∞ < i < ∞} is a martingale difference related to an increasing sequence of σ-fields F i and stochastically dominated by a nonnegative random variable ∞} is a martingale difference related to an increasing sequence of σ-fields F i and stochastically dominated by a nonnegative random variable Y .If where x x when x > 0 and x 0 when x ≤ 0 and x q x q .
Inspired by Chen and Li 14 , Chen et al. 7 , Sung 13 and other papers above, we go on to investigate the limit behavior of moving process under martingale difference sequence and obtain some similar results of Theorems C and D, but we only need some simple conditions.Our results extend some results of Chen and Li 14 see Remark 3.3 in Section 3 .Two lemmas and two theorems are given in Sections 2 and 3, respectively.The proofs of theorems are presented in Section 4.
For various results of martingales, one can refer to Chow 15 , Hall and Heyde 16 , Yu 17 , Ghosal and Chandra 18 , and so forth.As an application of moving average process based on martingale differences, we can refer to 19-22 and the references therein.Throughout the paper, I A is the indicator function of set A, x max{x, 0} and C, C 1 , C 2 , . . .denote some positive constants not depending on n, which may be different in various places.

Two Lemmas
The following lemmas are our basic techniques to prove our results.Lemma 2.1 cf.Hall and Heyde 16, Theorem 2.11 .If {X i , F i , 1 ≤ i ≤ n} is a martingale difference and p > 0, then there exists a constant C depending only on p such that Lemma 2.2 cf.Wu 23, Lemma 4.1.6 .Let {X n , n ≥ 1} be a sequence of random variables, which is stochastically dominated by a nonnegative random variable X.Then for any a > 0 and b > 0, the following two statements hold: where C 1 and C 2 are positive constants.

Main Results
Theorem 3.1.Let r > 1 and 1 ≤ p < 2. Assume that {X n , n ≥ 1} is a moving average processes defined in 1.1 , where {Y i , F i , −∞ < i < ∞} is a martingale difference related to an increasing sequence of σ-fields F i and stochastically dominated by a nonnegative random variable Y .Let K be a constant.Suppose that EY rp < ∞ for rp > 1 and . .be an increasing family of σ-algebras and { X n , F n , n ≥ 1} be a sequence of martingale differences.Assume that for some p ≥ 2, where K is a constant not depending on n, and other conditions are satisfied, Yu 17 investigated the complete convergence of weighted sums of martingale differences.On the other hand, under the condition sup and other conditions, Ghosal and Chandra 18 obtained the complete convergence of martingale arrays.Thus, if rp ≥ 2, our assumption Chen and Li 14 obtained Theorems C and D for the case 1 ≤ rp < 2. We go on to investigate this moving average process for the case rp > 1, especially for the case rp ≥ 2 and get the results of 3.

The Proofs of Main Results
Proof of Theorem 3.1.First, we show that the moving average process 1.1 converges a.s.under the conditions of Theorem 3.1.Since rp > 1, it has EY < ∞, following from EY rp < ∞.
On the other hand, applying Lemma 2.2 with a 1 and b 1, one has

4.6
Meanwhile, by the martingale property, Lemma 2.2 and the proof of 4.6 , it follows that

4.9
Consequently, we obtain by

4.11
Since q > rp and EY rp < ∞, one has

4.12
By the proof of 4.6 , If rp < 2, then we take q 2. Similar to the proofs of 4.8 , and 4.11 , it has
Inspired by the proof of Theorem 12.1 of Gut 24 , it can be checked that

4.17
Combining 3.1 with these inequalities above, we obtain 3.2 immediately.

4.18
By Theorem 3.1, in order to proof 3.3 , we only have to show that For t > 0, denote By Markov's inequality, Lemma 2.2 and EY rp < ∞,

4.25
following from the fact that q > r − 1 / 1/p − 1/2 .We also have by C r inequality and Lemma 2.2 that

4.26
Since q > rp and EY rp < ∞, it follows that

4.27
From the proof of 4.22 ,

4.28
If rp < 2, then we take q 2. Similar to the proofs of 4.23 and 4.26 , we get that
.4 holds true following from 3.3 .
and rp < 2, result 3.1 follows from Theorem C see Theorem 1.1 of Chen and Li , but we can obtain results 3.1 and 3.2 under weaker condition EY rp < ∞.On the other hand, comparing with the conditions of Theorem D, our conditions of Theorem 3.2 are relatively simple.