Complete Moment Convergence of Weighted Sums for Arrays of Rowwise φ-Mixing Random Variables

The complete moment convergence of weighted sums for arrays of rowwise φ-mixing random variables is investigated. By using moment inequality and truncation method, the sufficient conditions for complete moment convergence of weighted sums for arrays of rowwise φ-mixing random variables are obtained. The results of Ahmed et al. 2002 are complemented. As an application, the complete moment convergence of moving average processes based on a φ-mixing random sequence is obtained, which improves the result of Kim et al. 2008 .


Introduction
Hsu and Robbins 1 introduced the concept of complete convergence of {X n }.A sequence {X n , n 1, 2, . ..} is said to converge completely to a constant C if 1.1 Moreover, they proved that the sequence of arithmetic means of independent identically distributed i.i.d.random variables converge completely to the expected value if the variance of the summands is finite.The converse theorem was proved by Erd ös 2 .This result has been generalized and extended in several directions, see Baum  |a ni | O n α , for some α ∈ 0, r .

1.2
Let β be such that α β / − 1 and fix δ > 0 such that 1 α/r < δ ≤ 2. Denote s max 1 α β 1 /r, δ .If E|X| s < ∞ and S n ∞ i 1 a ni X ni → 0 in probability, then ∞ n 1 n β P S n > < ∞ for all > 0. Chow 4 established the following refinement which is a complete moment convergence result for sums of i.i.d.random variables.
The main purpose of this paper is to discuss again the above results for arrays of rowwise ϕ-mixing random variables.The author takes the inspiration in 8 and discusses the complete moment convergence of weighted sums for arrays of rowwise ϕ-mixing random variables by applying truncation methods.The results of Ahmed et al. 8 are extended to ϕmixing case.As an application, the corresponding results of moving average processes based on a ϕ-mixing random sequence are obtained, which extend and improve the result of Kim and Ko 9 .
For the proof of the main results, we need to restate a few definitions and lemmas for easy reference.Throughout this paper, C will represent positive constants, the value of which may change from one place to another.The symbol I A denotes the indicator function of A; x indicates the maximum integer not larger than x.For a finite set B, the symbol B denotes the number of elements in the set B.
where By the properties of slowly varying function, we can easily prove the following lemma.Here we omit the details of the proof.Lemma 1.5.Let l x > 0 be a slowly varying function as x → ∞, then there exists C (depends only on r) such that i Ck r 1 l k ≤ k n 1 n r l n ≤ Ck r 1 l k for any r > −1 and positive integer k, ii Ck r 1 l k ≤ ∞ n k n r l n ≤ Ck r 1 l k for any r < −1 and positive integer k.
The following lemma will play an important role in the proof of our main results.The proof is due to Shao 10 .Lemma 1.6.Let {X i , 1 ≤ i ≤ n} be a sequence of ϕ-mixing random variables with mean zero.Suppose that there exists a sequence {C n } of positive numbers such that

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Proof.By Lemma 5.4.4 in 11 and H ölder's inequality, we have 1.9 Therefore, 1.8 holds.

Main Results
Now we state our main results.The proofs will be given in Section 3.
Let l x > 0 be a slowing varying function, and {a ni , i ≥ 1, n ≥ 1} be an array of constants such that

2.5
Therefore, from 2.5 , we obtain that the complete moment convergence implies the complete convergence, that is, under the conditions of Theorem 2.1, result 2.2 implies Corollary 2.4.Under the conditions of Theorem 2.1, 2.9 Corollary 2.5.Let {X ni , i ≥ 1, n ≥ 1} be an array of rowwise ϕ-mixing random variables with EX ni 0,{X ni } ≺ X and ∞ m 1 ϕ 1/2 m < ∞.Suppose that l x > 0 is a slowly varying function.
Let l x be a slowly varying function.

2.13
Remark 2.7.Corollary 2.6 obtains the result about the complete moment convergence of moving average processes based on a ϕ-mixing random sequence with different distributions.
We extend the results of Chen et al. 12 from the complete convergence to the complete moment convergence.The result of Kim and Ko 9 is a special case of Corollary 2.6 1 .Moreover, our result covers the case of r t, which was not considered by Kim and Ko.

Proofs of the Main Results
Proof of Theorem 2.1.Without loss of generality, we can assume x for any k ≥ 1, n ≥ 1, and x ≥ 0. First note that E|X| s l |X| 1/r < ∞ implies E|X| t < ∞ for any 0 < t < s.Therefore, for x > n r ,

3.2
Hence, for n large enough we have sup

3.5
By Lemma 1.6, Markov inequality, C r inequality, and 1.5 , for any q ≥ 2, we have

3.6
So, For J 1 , we consider the following two cases.N, where N is the set of positive integers.Note also that for all k ≥ 1, n ≥ 1,

3.10
Hence, we have

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Note that

3.13
International Journal of Mathematics and Mathematical Sciences 11 For J 2 , we obtain In fact, noting 1 a/r < s < s and β qr/2 1 α/r − s < −1, using Markov inequality and 3.1 , we get

3.15
Thus, we complete the proof in a .Next, we prove b .Note that E|X| 1 α/r < ∞ implies that 3.2 holds.Therefore, from the proof in a , to complete the proof of b , we only need to prove

3.18
The proof of Theorem 2.1 is completed.
Let {X ni ; i ≥ 1, n ≥ 1} be an array of rowwise independent random elements in a separable real Banach space B, • .Let P X ni > x ≤ CP |X| > x for some random variable X, constant C and all n, i and x > 0. Suppose that {a ni , i ≥ 1, n ≥ 1} is an array of constants such that sup and Katz 3 , Chow 4 , Gut 5 , Taylor et al. 6 , and Cai and Xu 7 .In particular, Ahmed et al. 8 obtained the following result in Banach space.International Journal of Mathematics and Mathematical Sciences Theorem A. i≥1 |a ni | O n −r , for some r > 0, ∞ i 1 Definition 1.2.A sequence {X n , n ≥ 1} of random variables is said to be stochastically dominated by a random variable X write {X i } ≺ X if there exists a constant C, such that P {|X n | > x} ≤ CP {|X| > x} for all x ≥ 0 and n ≥ 1.