On Complete Moment Convergence of Weighted Sums for Arrays of Rowwise Negatively Associated Random Variables

Abstract

The complete moment convergence of weighted sums for arrays of rowwise negatively associated random variables is investigated. Some sufficient conditions for complete moment convergence of weighted sums for arrays of rowwise negatively associated random variables are established. Moreover, the results of Baek et al. (2008), are complemented. As an application, the complete moment convergence of moving average processes based on a negatively associated random sequences is obtained, which improves the result of Li et al. (2004).

1. Introduction

Let {𝑋_𝑛,𝑛≥1} be a sequence of random variables and, as usual, set 𝑆_𝑛=∑^𝑛_𝑖=1𝑋_𝑖,𝑛≥1. When {𝑋_𝑛,𝑛≥1} are independent and identically distributed (i.i.d.), Baum and Katz [1] proved the following remarkable result concerning the convergence rate of the tail probabilities 𝑃(|𝑆_𝑛|>𝜖𝑛^1/𝑝) for any 𝜖>0.

Theorem A (see [1]). Let 0<𝑝<2 and 𝑟≥𝑝. Then ^∞_𝑛=1𝑛^{(𝑟/𝑝)−2}𝑃𝑆_𝑛>𝜖𝑛^1/𝑝<∞∀𝜖>0, (1.1) if and only if 𝐸|𝑋|^𝑟<∞, where 𝐸𝑋₁=0 whenever 1≤𝑝<2.

There is an interesting and substantial literature of investigation apropos of extending the Baum-Katz theorems along a variety of different paths. One of these extensions is due to Chow [2] who established the following refinement which is a complete moment convergence result for sums of i.i.d. random variables.

Theorem B (see [2]). Let 𝐸𝑋₁=0, 1≤𝑝<2, and 𝑟≥𝑝. Suppose that 𝐸[|𝑋₁|^𝑟+|𝑋₁|log(1+|𝑋₁|)]<∞. Then ^∞_𝑛=1𝑛^{(𝑟/𝑝)−2−(1/𝑝)}𝐸𝑆_𝑛−𝜖𝑛^1/𝑝+<∞∀𝜖>0. (1.2)

Recently, Baum-Katz theorem is extended to the case of dependence random variables. Liang [3] obtained some general results on the complete convergence of weighted sums of negatively associated random variables. Li and Zhang [4] showed complete moment convergence for moving average processes under negative association as follows.

Theorem C (see [4]). Suppose that 𝑋_𝑛=∑^∞_𝑖=−∞𝑎_𝑖+𝑛𝑌_𝑖,𝑛≥1, where {𝑎_𝑖,−∞<𝑖<∞} is a sequence of real numbers with ∑^∞_−∞|𝑎_𝑖|<∞ and {𝑌_𝑖,−∞<𝑖<∞} is a sequence of identically distributed and negatively associated random variables with 𝐸𝑌₁=0,𝐸𝑌²₁<∞. Let 𝑙(𝑥)>0 be a slowly varying function and 1≤𝑝<2,𝑟>1+(𝑝/2). Then 𝐸|𝑌₁|^𝑟𝑙(|𝑌₁|^𝑝)<∞ implies that ^∞_𝑛=1𝑛^{(𝑟/𝑝)−2−(1/𝑝)}𝑙(𝑛)𝐸𝑆_𝑛−𝜖𝑛^1/𝑝+<∞,where𝑆_𝑛=^𝑛_𝑖=1𝑋_𝑖,𝑛≥1. (1.3)

Kuczmaszewska [5] proposed a very general result for complete convergence of rowwise negatively associated arrays of random variables which is stated in Theorem D.

Theorem D (see [5]). Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise negatively associated random variables and let {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of real numbers. Let {𝑏_𝑛,𝑛≥1} be an increasing sequence of positive integers and let {𝑐_𝑛,𝑛≥1} be a sequence of positive real numbers. If for some 𝑞>2,0<𝑡<2 and any 𝜖>0 the following conditions are fulfilled:(a)∑^∞_𝑛=1𝑐_𝑛∑^𝑏𝑛_𝑖=1𝑃{|𝑎_𝑛𝑖𝑋_𝑛𝑖|≥𝜖𝑏_𝑛^1/𝑡}<∞,(b)∑^∞_𝑛=1𝑐_𝑛𝑏_𝑛^−𝑞/𝑡∑^𝑏𝑛_𝑖=1|𝑎_𝑛𝑖|^𝑞𝐸|𝑋_𝑛𝑖|^𝑞𝐼(|𝑎_𝑛𝑖𝑋_𝑛𝑖|<𝜖𝑏_𝑛^1/𝑡)<∞,(c)∑^∞_𝑛=1𝑐_𝑛𝑏_𝑛^−𝑞/𝑡(∑^𝑏𝑛_𝑖=1|𝑎_𝑛𝑖|²𝐸|𝑋_𝑛𝑖|²𝐼(|𝑎_𝑛𝑖𝑋_𝑛𝑖|<𝜖𝑏_𝑛^1/𝑡))^𝑞/2<∞,then ^∞_𝑛=1𝑐_𝑛𝑃max_{1≤𝑘≤𝑏𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝑎_𝑛𝑖𝐸𝑋_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖<𝜖𝑏_𝑛^1/𝑡>𝜖𝑏_𝑛^1/𝑡<∞. (1.4)

Baek et al. [6] discussed complete convergence of weighted sums for arrays of rowwise negatively associated random variables and obtained the following results.

Theorem E (see [6]). Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise negatively associated random variables with 𝐸𝑋_𝑛𝑖=0 and 𝑃{|𝑋_𝑛𝑖|>𝑥}≤𝐶𝑃{|𝑋|>𝑥} for all 𝑛,𝑖 and 𝑥≥0. Suppose that 𝛽≥−1, and that {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} is an array of constants such that sup_𝑖≥1𝑎_𝑛𝑖=𝑂𝑛^−𝑟𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑟>0,_∞𝑖=1𝑎_𝑛𝑖=𝑂𝑛^𝛼𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛼∈[0,𝑟). (1.5)(1.6)(a)If 𝛼+𝛽+1>0 and there exists some 𝛿>0 such that 𝛼/𝑟+1<𝛿≤2, and 𝑠=max(1+((𝛼+𝛽+1)/𝑟),𝛿), then, under 𝐸|𝑋|^𝑠<∞, one has ^∞_𝑛=1𝑛^𝛽𝑃^∞_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜖<∞∀𝜖>0. (1.7)(b)If 𝛼+𝛽+1=0, then, under 𝐸|𝑋|log(1+|𝑋|)<∞, (1.7) remain true.

In this paper, the authors take the inspiration in [5, 6] and discuss the complete moment convergence of weighted sums for arrays of rowwise negatively associated random variables by applying truncation methods, which extend the results of [5, 6]. As an application, the complete moment convergence of moving average processes based on a negatively associated random sequences is obtained, which extend the result of Li and Zhang [4].

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 whom their value may change from one place to another. The symbol 𝐼(𝐴) denotes the indicator function of 𝐴, [𝑥] indicate the maximum integer not larger than 𝑥. For a finite set 𝐵, the symbol #𝐵 denotes the number of elements in the set 𝐵.

Definition 1.1. A finite family of random variables {𝑋_𝑖,1≤𝑖≤𝑛} is said to be negatively associated (abbreviated to NA in the following), if for every pair disjoint subsets 𝐴 and 𝐵 of {1,2,…,𝑛} and any real nondecreasing coordinate-wise functions 𝑓₁ on ℝ^𝐴 and 𝑓₂ on ℝ^𝐵Cov𝑓₁𝑋_𝑖,𝑖∈𝐴,𝑓₂𝑋_𝑖,𝑖∈𝐵≤0 (1.8) whenever the covariance exists.
An infinite family of random variables {𝑋_𝑖,−∞<𝑖<∞} is NA if every finite subfamily is NA.

The definition of negatively associated was introduced by Alam and Saxena [7] and was studied by Joag-Dev and Proschan [8] and Block et al. [9]. As pointed out and proved by Joag-Dev and Proschan, a number of well-known multivariate distributions possess the NA property. Negative association has found important and wide applications in multivariate statistical analysis and reliability. Many investigators discuss applications of negative association to probability, stochastic processes, and statistics.

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.

The following lemma is a well-known result.

Lemma 1.3. Let the sequence {𝑋_𝑛,𝑛≥1} of random variables be stochastically dominated by a random variable 𝑋. Then for any 𝑝>0,𝑥>0𝐸𝑋_𝑛^𝑝𝐼𝑋_𝑛≤𝑥≤𝐶𝐸|𝑋|^𝑝𝐼(|𝑋|≤𝑥)+𝑥^𝑝𝑃𝑋|,𝐸𝑋{|>𝑥}_𝑛𝑝𝐼𝑋_𝑛>𝑥≤𝐶𝐸|𝑋|^𝑝𝐼(|𝑋|>𝑥). (1.9)

Definition 1.4. A real-valued function 𝑙(𝑥), positive and measurable on [𝐴,∞) for some 𝐴>0, is said to be slowly varying if lim_𝑥→∞(𝑙(𝑥𝜆)/𝑙(𝑥))=1 for each 𝜆>0.

By the properties of slowly varying function, we can easily prove the following two lemmas. Here we omit the details of the proof.

Lemma 1.5. Let 𝑙(𝑥)>0 be a slowly varying function as 𝑥→∞.(i)𝐶𝑘^𝑟+1𝑙(𝑘)≤∑^𝑘_𝑛=1𝑛^𝑟𝑙(𝑛)≤𝐶𝑘^𝑟+1𝑙(𝑘) for any 𝑟>−1 and positive integer 𝑘.(ii)𝐶𝑘^𝑟+1𝑙(𝑘)≤∑^∞_𝑛=𝑘𝑛^𝑟𝑙(𝑛)≤𝐶𝑘^𝑟+1𝑙(𝑘) for any 𝑟<−1 and positive integer 𝑘.

Lemma 1.6. Let 𝑋 be a random variable and let 𝑙(𝑥)>0 be a slowly varying function as 𝑥→∞.(i)If 𝛼>0,𝑟>0,𝛽>−1, then 𝐸|𝑋|^{𝛼+((𝛽+1)/𝑟)}𝑙(|𝑋|^1/𝑟)<∞ if and only if ∑^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝐸|𝑋|^𝛼𝐼(|𝑋|>𝑛^𝑟)<∞.(ii)If 𝛼>0,𝑟>0,𝛽<−1, then 𝐸|𝑋|^{𝛼+((𝛽+1)/𝑟)}𝑙(|𝑋|^1/𝑟)<∞ if and only if ∑^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝐸|𝑋|^𝛼𝐼(|𝑋|≤𝑛^𝑟)<∞.

The following lemma will play an important role in the proof of our main results. The proof is according to Shao [10].

Lemma 1.7. Let {𝑋_𝑖,1≤𝑖≤𝑛} be a sequence of NA random variables with mean zero and 𝐸|𝑋_𝑖|^𝑞<∞ for every 1≤𝑖≤𝑛,𝑞≥2. Then 𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑋_𝑖^𝑞≤𝐶^𝑛_𝑖=1𝐸𝑋_𝑖^𝑞+^𝑛_𝑖=1𝐸𝑋²_𝑖^𝑞/2. (1.10)

By monotone convergence and (1.10), we have the following lemma.

Lemma 1.8. Let {𝑋_𝑖,𝑖≥1} be a sequence of NA random variables with mean zero and 𝐸|𝑋_𝑖|^𝑞<∞ for every 𝑖≥1,𝑞≥2. Then 𝐸max_𝑘≥1^𝑘_𝑖=1𝑋_𝑖^𝑞≤𝐶^∞_𝑖=1𝐸𝑋_𝑖^𝑞+^∞_𝑖=1𝐸𝑋²_𝑖^𝑞/2. (1.11)

Using Lemma 1.4, Lemma 1.5, and Theorem 2.11 in Sung [11], we obtain the following lemmas.

Lemma 1.9. Let {𝑋_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} be an array of rowwise NA random variables with 𝐸|𝑋_𝑛𝑖|<∞ for 1≤𝑖≤𝑛,𝑛≥1. Let {𝑏_𝑛,𝑛≥1} be a sequence of real numbers. If for some 𝑞≥2, the following conditions are fulfilled:(a)∑^∞_𝑛=1𝑏_𝑛∑^𝑛_𝑖=1𝐸|𝑋_𝑛𝑖|𝐼(|𝑋_𝑛𝑖|>1)<∞;(b)∑^∞_𝑛=1𝑏_𝑛∑^𝑛_𝑖=1𝐸|𝑋_𝑛𝑖|^𝑞𝐼(|𝑋_𝑛𝑖|≤1)<∞;(c)∑^∞_𝑛=1𝑏_𝑛(∑^𝑛_𝑖=1𝐸|𝑋_𝑛𝑖|²𝐼(|𝑋_𝑛𝑖|≤1))^𝑞/2<∞.
Then ^∞_𝑛=1𝑏_𝑛𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑋_𝑛𝑖−𝐸𝑋_𝑛𝑖−𝜖⁺<∞∀𝜖>0. (1.12)

Lemma 1.10. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise NA random variables with 𝐸|𝑋_𝑛𝑖|<∞ for 𝑖≥1,𝑛≥1. Let {𝑏_𝑛,𝑛≥1} be a sequence of real numbers. If for some 𝑞>2, the following conditions are fulfilled:(a)∑^∞_𝑛=1𝑏_𝑛∑^∞_𝑖=1𝐸|𝑋_𝑛𝑖|𝐼(|𝑋_𝑛𝑖|>1)<∞;(b)∑^∞_𝑛=1𝑏_𝑛∑^∞_𝑖=1𝐸|𝑋_𝑛𝑖|^𝑞𝐼(|𝑋_𝑛𝑖|≤1)<∞;(c)∑^∞_𝑛=1𝑏_𝑛(∑^∞_𝑖=1𝐸|𝑋_𝑛𝑖|²𝐼(|𝑋_𝑛𝑖|≤1))^𝑞/2<∞.
Then ^∞_𝑛=1𝑏_𝑛𝐸sup_𝑘≥1^𝑘_𝑖=1𝑋_𝑛𝑖−𝐸𝑋_𝑛𝑖−𝜖⁺<∞∀𝜖>0. (1.13)

2. Main Results

Now we state our main results. The proofs will be given in Section 3.

Theorem 2.1. Let {𝑋_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} be an array of rowwise NA random variables with 𝐸𝑋_𝑛𝑖=0 and stochastically dominated by a random variable 𝑋. Suppose that 𝑙(𝑥)>0 is a slowing varying function and that {𝑎_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} is an array of constants such that sup_{1≤𝑖≤𝑛}𝑎_𝑛𝑖=𝑂𝑛^−𝑟𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑟>0,_𝑛𝑖=1𝑎_𝑛𝑖=𝑂𝑛^𝛼𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛼∈[0,𝑟). (2.1)(2.2)(i)If 𝛼+𝛽+1>0 and there exists some 𝛿>0 such that (𝛼/𝑟)+1<𝛿≤2, and 𝑠=max(1+((𝛼+𝛽+1)/𝑟),𝛿), then 𝐸|𝑋|^𝑠𝑙(|𝑋|^1/𝑟)<∞ implies ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺<∞∀𝜖>0. (2.3)(ii)If 𝛼+𝛽+1=0, assume also 𝑙(𝑥)≤𝐶𝑙(𝑦) for all 0<𝑥<𝑦. Then 𝐸|𝑋|log(1+|𝑋|)𝑙(|𝑋|^1/𝑟)<∞ implies ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺<∞∀𝜖>0. (2.4)

Theorem 2.2. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise NA random variables with 𝐸𝑋_𝑛𝑖=0 and stochastically dominated by a random variable 𝑋. Suppose that 𝑙(𝑥)>0 is a slowing varying function and that {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} is an array of constants such that sup_𝑖≥1𝑎_𝑛𝑖=𝑂𝑛^−𝑟𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑟>0,_∞𝑖=1𝑎_𝑛𝑖=𝑂𝑛^𝛼𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛼∈[0,𝑟). (2.5)(2.6)(i)If 𝛼+𝛽+1>0 and there exists some 𝛿>0 such that (𝛼/𝑟)+1<𝛿≤2, and 𝑠=max(1+(𝛼+𝛽+1/𝑟),𝛿). Then 𝐸|𝑋|^𝑠𝑙(|𝑋|^1/𝑟)<∞ implies ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝐸sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺<∞∀𝜖>0. (2.7)(ii)If 𝛼+𝛽+1=0, assume also 𝑙(𝑥)≤𝐶𝑙(𝑦) for all 0<𝑥<𝑦. Then 𝐸|𝑋|log(1+|𝑋|)𝑙(|𝑋|^1/𝑟)<∞ implies ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝐸sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺<∞∀𝜖>0. (2.8)

Remark 2.3. If (2.7) and (2.8) hold, then for all 𝜖>0, we have ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝑃sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜖<∞,_∞𝑛=1𝑛⁻¹𝑙(𝑛)𝑃sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜖<∞. (2.9)(2.10) Thus, we improve the results of Baek et al. [6] to supreme value of partial sums.

Remark 2.4. If 𝛼+𝛽+1<0, then 𝐸|𝑋|<∞ implies that (2.10) holds. In fact, ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝐸sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺≤^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^∞_𝑖=1𝑎_𝑛𝑖𝐸𝑋_𝑛𝑖+𝜖^∞_𝑛=1𝑛^𝛽𝑙(𝑛)≤𝐶^∞_𝑛=1𝑛^𝛽+𝛼𝑙(𝑛)𝐸|𝑋|+𝜖^∞_𝑛=1𝑛^𝛽𝑙(𝑛)<∞. (2.11)

Corollary 2.5. Under the conditions of Theorem 2.2, ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝐸^∞_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺<∞∀𝜖>0. (2.12)

Corollary 2.6. Let {𝑋_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} be an array of rowwise NA random variables with 𝐸𝑋_𝑛𝑖=0 and stochastically dominated by a random variable 𝑋. Suppose that 𝑙(𝑥)>0 is a slowing varying function.(1)Let 𝑝>1 and 1≤𝑡<2. If 𝐸|𝑋|^𝑝𝑡𝑙(|𝑋|^𝑡)<∞, then ^∞_𝑛=1𝑛^{𝑝−2−(1/𝑡)}𝑙(𝑛)𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑋_𝑛𝑖−𝜖𝑛^1/𝑡+<∞∀𝜖>0. (2.13)(2)Let 1<𝑡<2. If 𝐸|𝑋|^𝑡𝑙(|𝑋|^𝑡)<∞, then ^∞_𝑛=1𝑛^{−1−(1/𝑡)}𝑙(𝑛)𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑋_𝑛𝑖−𝜖𝑛^1/𝑡+<∞∀𝜖>0. (2.14)

Theorem 2.7. Suppose that 𝑋_𝑛=∑^∞_𝑖=−∞𝑎_𝑖+𝑛𝑌_𝑖,𝑛≥1, where {𝑎_𝑖,−∞<𝑖<∞} is a sequence of real numbers with ∑^∞_−∞|𝑎_𝑖|<∞, and {𝑌_𝑖,−∞<𝑖<∞} is a NA random sequence with 𝐸𝑌_𝑖=0 and is stochastically dominated by a random variable 𝑌. Let 𝑙(𝑥) be a slowly varying function.(1)Let 1≤𝑡<2,𝑟≥1+(𝑡/2). If 𝐸|𝑌|^𝑟𝑙(|𝑌|^𝑡)<∞, then ^∞_𝑛=1𝑛^{(𝑟/𝑡)−2−(1/𝑡)}𝑙(𝑛)𝐸^𝑛_𝑖=1𝑋_𝑖−𝜖𝑛^1/𝑡+<∞∀𝜖>0. (2.15)(2)Let 1<𝑡<2. If 𝐸|𝑌|^𝑡𝑙(|𝑌|^𝑡)<∞, then ^∞_𝑛=1𝑛^{−1−(1/𝑡)}𝑙(𝑛)𝐸^𝑛_𝑖=1𝑋_𝑖−𝜖𝑛^1/𝑡+<∞∀𝜖>0. (2.16)

Remark 2.8. Theorem 2.7 obtains the result about the complete moment convergence of moving average processes based on an NA random sequence with different distributions. The result of Li and Zhang [4] is a special case of Theorem 2.7 (1). Moreover, our result covers the case of 𝑟=𝑡, which was not considered by Li and Zhang.

3. Proofs of the Main Results

Proof of Theorem 2.1. Since 𝑎_𝑛𝑖=𝑎⁺_𝑛𝑖−𝑎⁻_𝑛𝑖, where 𝑎⁺_𝑛𝑖=max(𝑎_𝑛𝑖,0) and 𝑎⁻_𝑛𝑖=max(−𝑎_𝑛𝑖,0), we have 𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺≤𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎⁺_𝑛𝑖𝑋_𝑛𝑖−𝜖2⁺+𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑎⁻_𝑛𝑖𝑋_𝑛𝑖−𝜖2⁺. (3.1) So, without loss of generality, we can assume 𝑎_𝑛𝑖>0. From (2.1) and (2.2), without loss of generality, we assume sup_{1≤𝑖≤𝑛}𝑎_𝑛𝑖≤𝑛^−𝑟,^𝑛_𝑖=1𝑎_𝑛𝑖≤𝑛^𝛼. (3.2) Put 𝑏_𝑛=𝑛^𝛽𝑙(𝑛),𝑛=1,2,… in Lemma 1.9. Noting that 𝛼+𝛽+1>0, by Lemma 1.3 and Lemma 1.7, we have ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖>1≤𝐶^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝐸𝑎_𝑛𝑖|𝑋|𝐼𝑎_𝑛𝑖|𝑋|>1≤𝐶^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝑎_𝑛𝑖𝐸|𝑋|𝐼|𝑋|>𝑛^𝑟≤𝐶^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝑎_𝑛𝑖𝐸|𝑋|𝐼|𝑋|>𝑛^𝑟≤𝐶^∞_𝑛=1𝑛^𝛼+𝛽𝑙(𝑛)𝐸|𝑋|𝐼|𝑋|>𝑛^𝑟≤𝐶𝐸|𝑋|^{1+((𝛼+𝛽+1)/𝑟)}𝑙|𝑋|^1/𝑟<∞. (3.3) Since 𝛼<𝑟, we can take some 𝑡 such that 1+(𝛼/𝑟)<𝑡≤min(𝑠,2). Observe that ^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖²𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤1≤^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖^𝑡𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤1≤𝐶^𝑛_𝑖=1𝑎^𝑡_𝑛𝑖=𝐶^𝑛_𝑖=1𝑎_𝑛𝑖𝑎^𝑡−1_𝑛𝑖≤𝐶𝑛^{𝛼−𝑟(𝑡−1)}. (3.4) Hence, choosing 𝑞 large enough such that 𝛽+(𝑞/2)(𝛼−𝑟(𝑡−1))<−1, we have ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖²𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤1^𝑞/2≤𝐶^∞_𝑛=1𝑛^{𝛽+(𝑞/2)(𝛼−𝑟(𝑡−1))}𝑙(𝑛)<∞. (3.5) By (3.3) and Lemma 1.3, we have ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖^𝑞𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤1≤𝐶^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋^𝑞𝐼𝑎_𝑛𝑖𝑋≤1+𝑃𝑎_𝑛𝑖𝑋>1≤𝐶^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋^𝑞𝐼𝑎_𝑛𝑖𝑋≤1+𝐸𝑎_𝑛𝑖𝑋𝐼𝑎_𝑛𝑖𝑋>1≤𝐶^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋^𝑞𝐼𝑎_𝑛𝑖𝑋≤1+𝐶. (3.6) Set 𝐼_𝑛𝑗={1≤𝑖≤𝑛∣(𝑛(𝑗+1))^−𝑟<𝑎_𝑛𝑖≤(𝑛𝑗)^−𝑟},𝑗=1,2,…. Then ∪_𝑗≥1𝐼_𝑛𝑗={1,2,…,𝑛}. Note also that for all 𝑘≥1,𝑛≥1, 𝑛^𝛼≥^𝑛_𝑖=1𝑎_𝑛𝑖=^∞_{𝑗=1𝑖∈𝐼𝑛𝑗}𝑎_𝑛𝑖≥^∞_𝑗=1#𝐼_𝑛𝑗𝑛𝑗+1^−𝑟≥𝑛^∞−𝑟_𝑗=𝑘#𝐼_𝑛𝑗𝑗+1^−𝑟𝑞(𝑘+1)^{𝑟𝑞−𝑟}. (3.7) Hence, we have ^∞_𝑗=𝑘#𝐼_𝑛𝑗𝑗^−𝑟𝑞≤𝐶𝑛^𝛼+𝑟𝑘^{𝑟−𝑟𝑞}. (3.8) Note that ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋^𝑞𝐼𝑎_𝑛𝑖𝑋=≤1_∞𝑛=1𝑛^𝛽𝑙(𝑛)^∞_{𝑗=1𝑖∈𝐼𝑛𝑗}𝐸𝑎_𝑛𝑖𝑋^𝑞𝐼𝑎_𝑛𝑖𝑋≤≤1_∞𝑛=1𝑛^𝛽𝑙(𝑛)^∞_𝑗=1#𝐼_𝑛𝑗𝑛𝑗^−𝑟𝑞𝐸|𝑋|^𝑞𝐼|𝑋|≤𝑛𝑗+1^𝑟=^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^∞_𝑗=1#𝐼_𝑛𝑗𝑛𝑗^−𝑟𝑞𝐸|𝑋|^𝑞𝐼|𝑋|≤(2𝑛)^𝑟+^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^∞_𝑗=1#𝐼_𝑛𝑗𝑛𝑗^{−𝑟𝑞𝑛(𝑗+1)}_𝑘=2𝑛+1𝐸|𝑋|^𝑞𝐼(𝑘−1)^𝑟<|𝑋|≤𝑘^𝑟=∶𝐽₁+𝐽₂. (3.9) Choosing 𝑞 large enough such that 𝛼+𝛽+𝑟−𝑟𝑞<−1, we obtain by Lemma 1.6 and (3.8) that 𝐽₁=^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^∞_𝑗=1#𝐼_𝑛𝑗𝑛𝑗^−𝑟𝑞𝐸|𝑋|^𝑞𝐼|𝑋|≤(2𝑛)^𝑟≤^∞_𝑛=1𝑛^{𝛼+𝛽+𝑟−𝑟𝑞}𝑙(𝑛)𝐸|𝑋|^𝑞𝐼|𝑋|≤(2𝑛)^𝑟≤𝐶𝐸|𝑋|^{1+((𝛼+𝛽+1)/𝑟)}𝑙|𝑋|^1/𝑟<∞. (3.10) Noting that 𝛼+𝛽>−1, by (3.8) and Lemma 1.5, we see 𝐽₂=^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^∞_𝑗=1#𝐼_𝑛𝑗𝑛𝑗^{−𝑟𝑞𝑛(𝑗+1)}_𝑘=2𝑛+1𝐸|𝑋|^𝑞𝐼(𝑘−1)^𝑟|𝑋|≤𝑘^𝑟≤^∞_𝑛=1𝑛^{𝛽−𝑟𝑞}𝑙(𝑛)^∞_𝑘=2𝑛+1𝐸|𝑋|^𝑞𝐼(𝑘−1)^𝑟<|𝑋|≤𝑘^𝑟∞_{[]𝑗=(𝑘/𝑛)−1}#𝐼_𝑛𝑗𝑗^−𝑟𝑞≤𝐶^∞_𝑛=1𝑛^{𝛽−𝑟𝑞}𝑙(𝑛)^∞_𝑘=2𝑛+1𝑛^𝛼+𝑟𝑘𝑛^{𝑟−𝑟𝑞}𝐸|𝑋|^𝑞𝐼(𝑘−1)^𝑟<|𝑋|≤𝑘^𝑟≤𝐶^∞_𝑘=2𝑘^{𝑟−𝑟𝑞}𝐸|𝑋|^𝑞𝐼(𝑘−1)^𝑟<|𝑋|≤𝑘^𝑟[𝑘/2]_𝑛=1𝑛^𝛼+𝛽𝑙(𝑛)≤𝐶^∞_𝑘=2𝑘^{𝛼+𝛽+1+𝑟−𝑟𝑞}𝑙(𝑘)𝐸|𝑋|^𝑞𝐼(𝑘−1)^𝑟<|𝑋|≤𝑘^𝑟≤𝐶𝐸|𝑋|^{1+((𝛼+𝛽+1)/𝑟)}𝑙|𝑋|^1/𝑟<∞. (3.11) From (3.6), (3.9), (3.10), and (3.11), we know that ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖^𝑞𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤1<∞. (3.12) By (3.3), (3.5), and (3.12), we see that (a), (b), and (c) in Lemma 1.9 with 𝑋_𝑛𝑖 replaced by 𝑎_𝑛𝑖𝑋_𝑛𝑖 are fulfilled. Since {𝑎_𝑛𝑖𝑋_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} is also an array of rowwise NA random variables, by Lemma 1.9, we complete the proof of (2.3).
Next, we prove (2.4). If 𝛼+𝛽+1=0, then ∑^𝑘_𝑛=1𝑛^𝛼+𝛽≤𝐶log(1+𝑛). Similarly for the proof of (3.3), noting that 𝑙(𝑥)≤𝐶𝑙(𝑦),0<𝑥<𝑦, we have ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖>1≤𝐶^∞_𝑛=1𝑛^𝛼+𝛽𝑙(𝑛)𝐸|𝑋|𝐼|𝑋|>𝑛^𝑟=𝐶^∞_𝑛=1𝑛⁻¹𝑙(𝑛)𝐸|𝑋|𝐼|𝑋|>𝑛^𝑟=𝐶^∞_𝑛=1𝑛⁻¹𝑙(𝑛)^∞_𝑘=𝑛𝐸|𝑋|𝐼𝑘^𝑟<|𝑋|≤(𝑘+1)^𝑟=𝐶^∞_𝑘=1𝐸|𝑋|𝐼𝑘^𝑟<|𝑋|≤(𝑘+1)^𝑟𝑘_𝑛=1𝑛⁻¹𝑙(𝑛)≤𝐶^∞_𝑘=1log(1+𝑘)𝑙(𝑘)𝐸|𝑋|𝐼𝑘^𝑟<|𝑋|≤(𝑘+1)^𝑟≤𝐶𝐸|𝑋|log(1+|𝑋|)𝑙|𝑋|^1/𝑟<∞. (3.13) Taking 𝑞=2, from the proof of (3.9), (3.10), and (3.11), we obtain ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋²𝐼𝑎_𝑛𝑖𝑋≤1≤𝐶^∞_𝑛=1𝑛^{𝛼+𝛽+𝑟−2𝑟}𝑙(𝑛)𝐸|𝑋|²𝐼|𝑋|≤(2𝑛)^𝑟+𝐶^∞_𝑛=1𝑛^𝛼+𝛽𝑙(𝑛)^∞_𝑘=2𝑛+1𝑘^𝑟−2𝑟𝐸|𝑋|²𝐼(𝑘−1)^𝑟<|𝑋|≤𝑘^𝑟=𝐶^∞_𝑛=1𝑛^−1−𝑟𝑙(𝑛)𝐸|𝑋|²𝐼|𝑋|≤(2𝑛)^𝑟+𝐶^∞_𝑛=1𝑛⁻¹𝑙(𝑛)^∞_𝑘=2𝑛+1𝑘^−𝑟𝐸|𝑋|²𝐼(𝑘−1)^𝑟<|𝑋|≤𝑘^𝑟≤𝐶𝐸|𝑋|𝑙|𝑋|^1/𝑟+𝐶𝐸|𝑋|log(1+|𝑋|)𝑙|𝑋|^1/𝑟<∞. (3.14) Thus, for 𝑞=2, (a), (b), and (c) in Lemma 1.9 with 𝑋_𝑛𝑖 replaced by 𝑎_𝑛𝑖𝑋_𝑛𝑖 are fulfilled. So (2.4) holds.

Proof of Theorem 2.2. By Lemma 1.10, the rest of the proof is similar to that of Theorem 2.1 and is omitted.

Proof of Corollary 2.5. Note that ^∞_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺≤sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺. (3.15) Therefore, by (2.7) and (2.8), we prove that (2.12) holds.

Proof of Corollary 2.6. By applying Theorem 2.1, taking 𝛽=𝑝−2,𝑎_𝑛𝑖=𝑛^−1/𝑡 for 1≤𝑖≤𝑛, and 𝑎_𝑛𝑖=0 for 𝑖>𝑛, then we obtain (2.13). Similarly, taking 𝛽=−1,𝑎_𝑛𝑖=𝑛^−1/𝑡 for 1≤𝑖≤𝑛, and 𝑎_𝑛𝑖=0 for 𝑖>𝑛, we obtain (2.14) by Theorem 2.1.

Proof of Theorem 2.7. Let 𝑋_𝑛𝑖=𝑌_𝑖 and 𝑎_𝑛𝑖=𝑛^−1/𝑡∑^𝑛_𝑗=1𝑎_𝑖+𝑗 for all 𝑛≥1,−∞<𝑖<∞. Since ∑^∞_−∞|𝑎_𝑖|<∞, we have sup_𝑖|𝑎_𝑛𝑖|=𝑂(𝑛^−1/𝑡) and ∑^∞_𝑖=−∞|𝑎_𝑛𝑖|=𝑂(𝑛^1−1/𝑡). By applying Corollary 2.5, taking 𝛽=(𝑟/𝑡)−2,𝑟=1/𝑡,𝛼=1−(1/𝑡), we obtain ^∞_𝑛=1𝑛^{(𝑟/𝑡)−2−(1/𝑡)}𝑙(𝑛)𝐸^𝑛_𝑖=1𝑋_𝑖−𝜖𝑛^1/𝑡+=^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝐸^∞_𝑖=−∞𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺<∞∀𝜖>0. (3.16) Therefore, (2.15) and (2.16) hold.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Grant no. 10901003), the Key Project of Chinese Ministry of Education (Grant no. 211077), and the Anhui Provincial Natural Science Foundation (Grant no. 10040606Q30).