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

Abstract

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).

1. 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 ^∞_𝑛=1𝑃𝑋_𝑛−𝐶>𝜖<∞,∀𝜖>0. (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 Erdös [2]. This result has been generalized and extended in several directions, see Baum 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.

Theorem A. 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_𝑖≥1𝑎_𝑛𝑖=𝑂𝑛^−𝑟,forsome𝑟>0,_∞𝑖=1𝑎_𝑛𝑖=𝑂𝑛^𝛼,forsome𝛼∈[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=1𝑎_𝑛𝑖𝑋_𝑛𝑖→0 in probability, then ∑^∞_𝑛=1𝑛^𝛽𝑃(‖𝑆_𝑛‖>𝜖)<∞ for all 𝜖>0.

Chow [4] established the following refinement which is a complete moment convergence result for sums of (i.i.d.) random variables.

Theorem B. 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)}𝐸^𝑛_𝑖=1𝑋_𝑖−𝜖𝑛^1/𝑝+<∞,∀𝜖>0. (1.3)

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 𝜑-mixing 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.

Definition 1.1. A sequence of random variables {𝑋_𝑖,1≤𝑖≤𝑛} is said to be a sequence of 𝜑-mixing random variables, if 𝜑(𝑚)=sup_𝑘≥1|𝑃(𝐵∣𝐴)−𝑃(𝐵)|;𝐴∈ℑ^𝑘₁,𝐵∈ℑ^∞_𝑘+𝑚,𝑃(𝐴)>0→0,as𝑚→∞, (1.4) where ℑ^𝑘_𝑗=𝜎{𝑋_𝑖;𝑗≤𝑖≤𝑘}, 1≤𝑗≤𝑘≤∞.

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≤𝐶𝐸𝑋^𝑝𝐼(|𝑋|≤𝑥)+𝑥^𝑝𝑃{|𝑋|>𝑥},𝐸𝑋_𝑛^𝑝𝐼𝑋_𝑛>𝑥≤𝐶𝐸|𝑋|^𝑝𝐼(|𝑋|>𝑥). (1.5)(1.6)

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 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+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.

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 𝐸(∑^𝑘+𝑚_𝑖=𝑘+1𝑋_𝑖)²≤𝐶_𝑛 for any 𝑘≥0,𝑛≥1,𝑚≤𝑛. Then for any 𝑞≥2, there exists 𝐶=𝐶(𝑞,𝜑(⋅)) such that 𝐸max_{1≤𝑗≤𝑛}^𝑘+𝑗_𝑖=𝑘+1𝑋_𝑖^𝑞≤𝐶𝐶_𝑛^𝑞/2+𝐸max_{𝑘+1≤𝑖≤𝑘+𝑛}𝑋_𝑖^𝑞. (1.7)

Lemma 1.7. Let {𝑋_𝑖,1≤𝑖≤𝑛} be a sequence of 𝜑-mixing random variables with ∑^∞_𝑖=1𝜑^1/2(𝑖)<∞, then there exists 𝐶 such that for any 𝑘≥0 and 𝑛≥1𝐸^𝑘+𝑛_𝑖=𝑘+1𝑋_𝑖²≤𝐶^𝑘+𝑛_𝑖=𝑘+1𝐸𝑋²_𝑖. (1.8)

Proof. By Lemma 5.4.4 in [11] and Hölder's inequality, we have 𝐸^𝑘+𝑛_𝑖=𝑘+1𝑋_𝑖²=^𝑘+𝑛_𝑖=𝑘+1𝐸𝑋²_𝑖+2_{𝑘+1≤𝑖<𝑗≤𝑘+𝑛}𝐸𝑋_𝑖𝑋_𝑗≤^𝑘+𝑛_𝑖=𝑘+1𝐸𝑋²_𝑖+4_{𝑘+1≤𝑖<𝑗≤𝑘+𝑛}𝜑^1/2𝑗−𝑖𝐸𝑋²_𝑖^1/2𝐸𝑋²_𝑗^1/2≤^𝑘+𝑛_𝑖=𝑘+1𝐸𝑋²_𝑖+2^{𝑘+𝑛−1}_𝑖=𝑘+1^𝑘+𝑛_𝑗=𝑖+1𝜑^1/2𝑗−𝑖𝐸𝑋²_𝑖+𝐸𝑋²_𝑗≤1+4^𝑛_𝑖=1𝜑^1/2(𝑖)^𝑘+𝑛_𝑖=𝑘+1𝐸𝑋²_𝑖. (1.9) Therefore, (1.8) holds.

2. Main Results

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

Theorem 2.1. 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=1𝜑^1/2(𝑚)<∞. Let 𝑙(𝑥)>0 be a slowing varying function, and {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of constants such that sup_𝑖≥1𝑎_𝑛𝑖=𝑂𝑛^−𝑟,𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑟>0,_∞𝑖=1𝑎_𝑛𝑖=𝑂𝑛^𝛼,𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛼∈[0,𝑟). (2.1)(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 𝐸|𝑋|^𝑠𝑙(|𝑋|^1/𝑟)<∞ implies ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝐸sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺<∞,∀𝜖>0. (2.2)(b)If 𝛽=−1,𝛼>0, then 𝐸|𝑋|^{1+(𝑎/𝑟)}(1+𝑙(|𝑋|^1/𝑟))<∞ implies ^∞_𝑛=1𝑛⁻¹𝑙(𝑛)𝐸sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺<∞,∀𝜖>0. (2.3)

Remark 2.2. If 𝛼+𝛽+1<0, then 𝐸|𝑋|<∞ implies that (2.2) 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)<∞. (2.4)

Remark 2.3. Note that ∞>^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝐸sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺=^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^∞₀𝑃sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖=−𝜖>𝑥𝑑𝑥_{∞0∞𝑛=1}𝑛^𝛽𝑙(𝑛)𝑃sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>𝑥+𝜖𝑑𝑥. (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 ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝑃sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜖<∞, (2.6) and (2.3) implies ^∞_𝑛=1𝑛⁻¹𝑙(𝑛)𝑃sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜖<∞. (2.7)

Corollary 2.4. Under the conditions of Theorem 2.1, (1)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)𝐸^∞_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺<∞,∀𝜖>0, (2.8)(2)if 𝛽=−1,𝛼>0, then 𝐸|𝑋|^{1+(𝛼/𝑟)}(1+𝑙(|𝑋|^1/𝑟))<∞ implies ^∞_𝑛=1𝑛⁻¹𝑙(𝑛)𝐸^∞_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺<∞,∀𝜖>0. (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=1𝜑^1/2(𝑚)<∞. Suppose that 𝑙(𝑥)>0 is a slowly 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. (2.10)(2)Let 1<𝑡<2. If 𝐸|𝑋|^𝑡[1+𝑙(|𝑋|^𝑡)]<∞, then ^∞_𝑛=1𝑛^{−1−(1/𝑡)}𝑙(𝑛)𝐸max_{1≤𝑘≤𝑛}^𝑘_𝑖=1𝑋_𝑛𝑖−𝜖𝑛^1/𝑡+<∞,∀𝜖>0. (2.11)

Corollary 2.6. 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 𝜑-mixing random variables with 𝐸𝑌_𝑖=0, {𝑌_𝑖}≺𝑌 and ∑^∞_𝑚=1𝜑^1/2(𝑚)<∞. 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. (2.12)(2)Let 1<𝑡<2. If 𝐸|𝑌|^𝑡[1+𝑙(|𝑌|^𝑡)]<∞, then ^∞_𝑛=1𝑛^{−1−(1/𝑡)}𝑙(𝑛)𝐸^𝑛_𝑖=1𝑋_𝑖−𝜖𝑛^1/𝑡+<∞,∀𝜖>0. (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.

3. Proofs of the Main Results

Proof of Theorem 2.1. Without loss of generality, we can assume sup_𝑖≥1𝑎_𝑛𝑖≤𝑛^−𝑟,^∞_𝑖=1𝑎_𝑛𝑖≤𝑛^𝛼. (3.1) Let 𝑆_𝑛𝑘(𝑥)=∑^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖𝐼(|𝑎_𝑛𝑖𝑋_𝑛𝑖|≤𝑛^−𝑟𝑥) for any 𝑘≥1, 𝑛≥1, and 𝑥≥0. First note that 𝐸|𝑋|^𝑠𝑙(|𝑋|^1/𝑟)<∞ implies 𝐸|𝑋|^𝑡<∞ for any 0<𝑡<𝑠. Therefore, for 𝑥>𝑛^𝑟, 𝑥⁻¹𝑛^𝑟sup_𝑘≥1𝐸𝑆_𝑛𝑘(𝑥)=𝑥⁻¹𝑛^𝑟sup_𝑘≥1𝐸^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖>𝑛^−𝑟𝑥𝐸𝑋_𝑛𝑖≤=0_∞𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖>𝑛^−𝑟𝑥≤^∞_𝑖=1𝐸𝑎_𝑛𝑖𝑋𝐼𝑎_𝑛𝑖𝑋>𝑛^−𝑟𝑥≤^∞_𝑖=1𝑎_𝑛𝑖𝐸|𝑋|𝐼(|𝑋|>𝑥)≤𝑛^𝛼𝐸|𝑋|𝐼(|𝑋|>𝑥)≤𝑥^𝛼/𝑟𝐸|𝑋|𝐼(|𝑋|>𝑥)≤𝐸|𝑋|^{1+(𝛼/𝑟)}𝐼|𝑋|>𝑛^𝑟→0as𝑛→∞. (3.2) Hence, for 𝑛 large enough we have sup_𝑘≥1𝐸|𝑆_𝑛𝑘(𝑥)|<(𝜖/2)𝑛^−𝑟𝑥. Then ^∞_𝑛=1𝑛^𝛽𝑙(𝑛)𝐸sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺=^∞_𝑛=1𝑛^𝛽𝑙(𝑛)^∞_𝜖𝑃sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖=≥𝑥𝑑𝑥_∞𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)𝜖^∞_𝑛𝑟𝑃sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖≥𝜖𝑛^−𝑟𝑥𝑑𝑥≤𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑛𝑟𝑃sup_𝑖𝑎_𝑛𝑖𝑋_𝑛𝑖>𝑛^−𝑟𝑥𝑑𝑥+𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑛𝑟𝑃sup_𝑘≥1𝑆_𝑛𝑘(𝑥)−𝐸𝑆_𝑛𝑘(𝑥)≥𝑛^−𝑟𝑥𝜖2𝑑𝑥∶=𝐼₁+𝐼₂. (3.3) Noting that 𝛼+𝛽>−1, by Lemma 1.5, Markov inequality, (1.6), and (3.1), we have 𝐼₁≤𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑛𝑟^∞_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑛𝑖>𝑛^−𝑟𝑥𝑑𝑥≤𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑛𝑟𝑛^𝑟𝑥^∞−1_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖>𝑛^−𝑟𝑥𝑑𝑥≤𝐶^∞_𝑛=1𝑛^𝛽+𝛼𝑙(𝑛)^∞_𝑛𝑟𝑥⁻¹𝐸|𝑋|𝐼(|𝑋|>𝑥)𝑑𝑥≤𝐶^∞_𝑛=1𝑛^𝛽+𝛼𝑙(𝑛)^∞_𝑘=𝑛^𝑘𝑟+1_𝑘𝑟𝑥⁻¹𝐸|𝑋|𝐼(|𝑋|>𝑥)𝑑𝑥≤𝐶^∞_𝑛=1𝑛^𝛽+𝛼𝑙(𝑛)^∞_𝑘=𝑛𝑘⁻¹𝐸|𝑋|𝐼|𝑋|>𝑘^𝑟≤𝐶^∞_𝑘=1𝑘⁻¹𝐸|𝑋|𝐼|𝑋|>𝑘^𝑟𝑘_𝑛=1𝑛^𝛽+𝛼𝑙(𝑛)≤𝐶^∞_𝑘=1𝑘^𝛽+𝛼𝑙(𝑘)𝐸|𝑋|𝐼|𝑋|>𝑘^𝑟≤𝐶𝐸|𝑋|^{1+((1+𝛼+𝛽)/𝑟)}𝑙|𝑋|^1/𝑟<∞. (3.4) Now we estimate 𝐼₂, noting that ∑^∞_𝑚=1𝜑^1/2(𝑚)<∞, by Lemma 1.7, we have sup_1≤𝑚<∞𝐸^𝑚_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤𝑛^−𝑟𝑥−𝐸^𝑚_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤𝑛^−𝑟𝑥²≤𝐶^∞_𝑖=1𝐸𝑎²_𝑛𝑖𝑋²_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤𝑛^−𝑟𝑥. (3.5) By Lemma 1.6, Markov inequality, 𝐶_𝑟 inequality, and (1.5), for any 𝑞≥2, we have 𝑃sup_𝑘≥1𝑆_𝑛𝑘(𝑥)−𝐸𝑆_𝑛𝑘(𝑥)≥𝑛^−𝑟𝑥𝜖2≤𝐶𝑛^𝑟𝑞𝑥^−𝑞𝐸sup_𝑘≥1𝑆_𝑛𝑘(𝑥)−𝐸𝑆_𝑛𝑘(𝑥)^𝑞≤𝐶𝑛^𝑟𝑞𝑥^−𝑞∞_𝑖=1𝐸𝑎²_𝑛𝑖𝑋²_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤𝑛^−𝑟𝑥^𝑞/2+^∞_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖^𝑞𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤𝑛^−𝑟𝑥≤𝐶𝑛^𝑟𝑞𝑥^−𝑞∞_𝑖=1𝐸𝑎²_𝑛𝑖𝑋²𝐼𝑎_𝑛𝑖𝑋≤𝑛^−𝑟𝑥^𝑞/2+𝐶𝑛^𝑟𝑞𝑥^∞−𝑞_𝑖=1𝐸𝑎_𝑛𝑖𝑋^𝑞𝐼𝑎_𝑛𝑖𝑋≤𝑛^−𝑟𝑥+𝐶^∞_𝑖=1𝑃𝑎_𝑛𝑖𝑋>𝑛^−𝑟𝑥^𝑞/2+𝐶^∞_𝑖=1𝑃𝑎_𝑛𝑖𝑋>𝑛^−𝑟𝑥∶=𝐽₁+𝐽₂+𝐽₃+𝐽₄. (3.6) So, 𝐼₂≤^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑛𝑟𝐽₁+𝐽₂+𝐽₃+𝐽₄𝑑𝑥. (3.7) From (3.4), we have ∑^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)∫^∞_𝑛𝑟𝐽₄𝑑𝑥<∞.
For 𝐽₁, we consider the following two cases.
If 𝑠>2, then 𝐸𝑋²<∞. Taking 𝑞≥2 such that 𝛽+(𝑞(𝛼−𝑟)/2)<−1, we have ^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑛𝑟𝐽₁𝑑𝑥≤𝐶^∞_𝑛=1𝑛^{𝛽−𝑟+𝑟𝑞}𝑙(𝑛)^∞_𝑛𝑟𝑥^−𝑞∞_𝑖=1𝑎²_𝑛𝑖^𝑞/2𝑑𝑥≤𝐶^∞_𝑛=1𝑛^{𝛽−𝑟+𝑟𝑞}𝑙(𝑛)𝑛^{𝑞(𝛼−𝑟)/2}𝑛^{𝑟(−𝑞+1)}≤𝐶^∞_𝑛=1𝑛^{𝛽+(𝑞(𝛼−𝑟)/2)}𝑙(𝑛)<∞. (3.8) If 𝑠≤2, we choose 𝑠^′ such that 1+(𝛼/𝑟)<𝑠^′<𝑠. Taking 𝑞≥2 such that 𝛽+(𝑞𝑟/2)(1+(𝛼/𝑟)−𝑠^′)<−1, we have ^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑛𝑟𝐽₁𝑑𝑥≤𝐶^∞_𝑛=1𝑛^{𝛽−𝑟+𝑟𝑞}𝑙(𝑛)^∞_𝑛𝑟𝑥^−𝑞∞_𝑖=1𝑎_𝑛𝑖𝑎_𝑛𝑖^𝑠′−1𝐸𝑎_𝑛𝑖𝑋^2−𝑠′|𝑋|^𝑠′𝐼𝑎_𝑛𝑖𝑋≤𝑛^−𝑟𝑥^𝑞/2𝑑𝑥≤𝐶^∞_𝑛=1𝑛^{𝛽−𝑟+𝑟𝑞}𝑙(𝑛)𝑛^𝑞𝛼/2𝑛^{−(𝑞𝑟/2)(𝑠′−1)∞}_𝑛𝑟𝑥^−𝑞𝑛^−𝑟𝑥^{(𝑞/2)(2−𝑠′)}𝑑𝑥≤𝐶^∞_𝑛=1𝑛^{𝛽+(𝑞𝑟/2)(1+(𝛼/𝑟)−𝑠′)}𝑙(𝑛)<∞. (3.9) So, ∑^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)∫^∞_𝑛𝑟𝐽₁𝑑𝑥<∞.
Now, we estimate 𝐽₂. Set 𝐼_𝑛𝑗={𝑖≥1∣(𝑛(𝑗+1))^−𝑟<|𝑎_𝑛𝑖|≤(𝑛𝑗)^−𝑟},𝑗=1,2,…. Then ∪_𝑗≥1𝐼_𝑛𝑗=𝑁, where 𝑁 is the set of positive integers. Note also that for all 𝑘≥1,𝑛≥1, 𝑛^𝛼≥^∞_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}. (3.10) Hence, we have ^∞_𝑗=𝑘♯𝐼_𝑛𝑗𝑗^−𝑟𝑞≤𝐶𝑛^𝛼+𝑟𝑘^{𝑟−𝑟𝑞}. (3.11) Note that ^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑛𝑟𝐽₂𝑑𝑥=𝐶^∞_𝑛=1𝑛^{𝛽−𝑟+𝑟𝑞}𝑙(𝑛)^∞_𝑛𝑟𝑥^∞−𝑞_{𝑗=1𝑖∈𝐼𝑛𝑗}𝐸𝑎_𝑛𝑖𝑋^𝑞𝐼𝑎_𝑛𝑖𝑋≤𝑛^−𝑟𝑥𝑑𝑥=𝐶^∞_𝑛=1𝑛^{𝛽−𝑟+𝑟𝑞}𝑙(𝑛)^∞_𝑗=1♯𝐼_𝑛𝑗𝑛𝑗^{∞−𝑟𝑞}_𝑘=𝑛^(𝑘+1)𝑟_𝑘𝑟𝑥^−𝑞𝐸|𝑋|^𝑞𝐼|𝑋|≤𝑥𝑗+1^𝑟𝑑𝑥≤𝐶^∞_𝑛=1𝑛^{𝛽−𝑟+𝑟𝑞}𝑙(𝑛)^∞_𝑗=1♯𝐼_𝑛𝑗𝑛𝑗^{∞−𝑟𝑞}_𝑘=𝑛𝑘^{𝑟(−𝑞+1)−1}𝐸|𝑋|^𝑞𝐼|𝑋|≤(𝑘+1)^𝑟𝑗+1^𝑟=𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑘=𝑛𝑘^{∞𝑟(−𝑞+1)−1}_𝑗=1♯𝐼_𝑛𝑗𝑗^{−𝑟𝑞(𝑘+1)(𝑗+1)−1}_𝑖=0𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟≤𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑘=𝑛𝑘^{∞𝑟(−𝑞+1)−1}_𝑗=1♯𝐼_𝑛𝑗𝑗^{−𝑟𝑞2(𝑘+1)−1}_𝑖=0𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟+𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑘=𝑛𝑘^{∞𝑟(−𝑞+1)−1}_𝑗=1♯𝐼_𝑛𝑗𝑗^{−𝑟𝑞(𝑘+1)(𝑗+1)}_{()𝑖=2𝑘+1}𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟∶=𝐽^′₂+𝐽₂^′′. (3.12) Taking 𝑞≥2 large enough such that 𝛽+𝛼−𝑟𝑞+𝑟<−1, for 𝐽^′₂, by Lemma 1.6 and (3.11), we get 𝐽^′₂≤𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑘=𝑛𝑘^{𝑟(−𝑞+1)−1}𝑛^{𝛼+𝑟2(𝑘+1)−1}_𝑖=0𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟=𝐶^∞_𝑘=1𝑘^{𝑟(−𝑞+1)−12(𝑘+1)−1}_𝑖=0𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟𝑘_𝑛=1𝑛^𝛽+𝛼𝑙(𝑛)≤𝐶^∞_𝑘=1𝑘^{𝛽+𝛼−𝑟𝑞+𝑟}𝑙(𝑘)^{2(𝑘+1)−1}_𝑖=0𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟≤𝐶+𝐶^∞_𝑖=3𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟∞_{[]𝑘=𝑖/2}𝑘^{𝛽+𝛼−𝑟𝑞+𝑟}𝑙(𝑘)≤𝐶+𝐶^∞_𝑖=3𝑖^{𝛽+𝛼−𝑟𝑞+𝑟+1}𝑙(𝑖)𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟≤𝐶+𝐶𝐸|𝑋|^{1+((𝛽+𝛼+1)/𝑟)}𝑙|𝑋|^1/𝑟<∞. (3.13) For 𝐽₂^′′, we obtain 𝐽₂^′′≤𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑘=𝑛𝑘^{∞𝑟(−𝑞+1)−1}_𝑗=1♯𝐼_𝑛𝑗𝑗^{−𝑟𝑞(𝑗+1)(𝑘+1)}_{()𝑖=2𝑘+1}𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟≤𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑘=𝑛𝑘^{∞𝑟(−𝑞+1)−1}_{()𝑖=2𝑘+1}𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟∞_{𝑖()𝑗=𝑘+1−1−1}♯𝐼_𝑛𝑗𝑗^−𝑟𝑞≤𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑘=𝑛𝑘^{∞𝑟(−𝑞+1)−1}_{()𝑖=2𝑘+1}𝑛^𝑟+𝛼𝑖^{𝑟(1−𝑞)}𝑘^{−𝑟(1−𝑞)}𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟=𝐶^∞_𝑘=1𝑘^∞−1_{()𝑖=2𝑘+1}𝑖^{𝑟(1−𝑞)}𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟𝑘_𝑛=1𝑛^𝛽+𝛼𝑙(𝑛)≤𝐶^∞_𝑘=1𝑘^𝛽+𝛼𝑙(𝑘)^∞_{()𝑖=2𝑘+1}𝑖^{𝑟(1−𝑞)}𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟≤𝐶^∞_𝑖=4𝑖^{𝛽+𝛼+1+𝑟−𝑟𝑞}𝐸|𝑋|^𝑞𝐼𝑖^𝑟<|𝑋|≤(𝑖+1)^𝑟≤𝐶𝐸|X|^{1+((𝛽+𝛼+1)/𝑟)}𝑙|𝑋|^1/𝑟<∞. (3.14) So ∑^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)∫^∞_𝑛𝑟𝐽₂𝑑𝑥<∞. Finally, we prove ∑^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)∫^∞_𝑛𝑟𝐽₃𝑑𝑥<∞. 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 ^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑛𝑟𝐽₃𝑑𝑥≤𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)^∞_𝑛𝑟^∞_𝑖=1𝑛^𝑟𝑠′𝑥^−𝑠′𝐸𝑎_𝑛𝑖𝑋^{𝑠′𝑞/2}𝑑𝑥≤𝐶^∞_𝑛=1𝑛^𝛽−𝑟𝑙(𝑛)𝑛^{𝑞𝑟𝑠′/2}𝑛^{−𝑟(𝑠′−1)(𝑞/2)}𝑛^{𝛼(𝑞/2)∞}_𝑛𝑟𝑥^{−𝑠′(𝑞/2)}𝑑𝑥≤𝐶^∞_𝑛=1𝑛^{𝛽−𝑟+𝑟(𝑞/2)+𝛼(𝑞/2)}𝑙(𝑛)𝑛^{𝑟(−𝑠′(𝑞/2)+1)}≤𝐶^∞_𝑛=1𝑛^{𝛽+(𝑞𝑟/2)(1+(𝛼/2)−𝑠′)}𝑙(𝑛)<∞. (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 𝐼₂=𝐶^∞_𝑛=1𝑛^−1−𝑟𝑙(𝑛)^∞_𝑛𝑟𝑃sup_𝑘≥1𝑆_𝑛𝑘(𝑥)−𝐸𝑆_𝑛𝑘(𝑥)≥𝑛^−𝑟𝑥𝜖2𝑑𝑥<∞. (3.16) In fact, noting 𝛽=−1, 𝛼+𝛽+1>0, 𝛼+𝛽−𝑟<−1 and 𝐸|𝑋|^1+𝛼/𝑟𝑙(|𝑋|^1/r)<∞. By taking 𝑞=2 in the proof of (3.12), (3.13), and (3.14), we get 𝐶^∞_𝑛=1𝑛^−1+𝑟𝑙(𝑛)^∞_𝑛𝑟𝑥^∞−2_𝑖=1𝐸𝑎²_𝑛𝑖𝑋²𝐼𝑎_𝑛𝑖𝑋≤𝑛^−𝑟𝑥𝑑𝑥≤𝐶+𝐶𝐸|𝑋|^{1+(𝛼/𝑟)}𝑙|𝑋|^1/𝑟<∞. (3.17) Then, by (3.17), we have 𝐼₂≤𝐶^∞_𝑛=1𝑛^−1−𝑟𝑙(𝑛)^∞_𝑛𝑟𝑛^2𝑟𝑥⁻²𝐸𝑆_𝑥𝑛−𝐸𝑆_𝑥𝑛²𝑑𝑥≤𝐶^∞_𝑛=1𝑛^−1+𝑟𝑙(𝑛)^∞_𝑛𝑟𝑥^∞−2_𝑖=1𝐸𝑎²_𝑛𝑖𝑋²_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤𝑛^−𝑟𝑥𝑑𝑥≤𝐶^∞_𝑛=1𝑛^−1+𝑟𝑙(𝑛)^∞_𝑛𝑟𝑥^∞−2_𝑖=1𝐸𝑎²_𝑛𝑖𝑋²𝐼𝑎_𝑛𝑖𝑋≤𝑛^−𝑟𝑥𝑑𝑥+𝐶^∞_𝑛=1𝑛^−1−𝑟𝑙(𝑛)^∞_𝑛𝑟^∞_𝑖=1𝑃𝑎_𝑛𝑖𝑋>𝑛^−𝑟𝑥𝑑𝑥≤𝐶^∞_𝑛=1𝑛^−1+𝑟𝑙(𝑛)^∞_𝑛𝑟𝑥^∞−2_𝑖=1𝐸𝑎²_𝑛𝑖𝑋²𝐼𝑎_𝑛𝑖𝑋≤𝑛^−𝑟𝑥𝑑𝑥+𝐶<∞. (3.18) The proof of Theorem 2.1 is completed.

Proof of Corollary 2.4. Note that ^∞_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺≤sup_𝑘≥1^𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖−𝜖⁺. (3.19) Therefore, (2.8) and (2.9) hold by Theorem 2.1.

Proof of Corollary 2.5. 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 (2.10). Similarly, taking 𝛽=−1, 𝑎_𝑛𝑖=𝑛^−1/𝑡 for 1≤𝑖≤𝑛, and 𝑎_𝑛𝑖=0 for 𝑖>𝑛, we obtain (2.11) by Theorem 2.1.

Proof of Corollary 2.6. 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 2.4, 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. (3.20) Therefore, (2.12) and (2.13) hold.

Acknowledgment

The paper is supported by the National Natural Science Foundation of China (no. 11271020 and 11201004), the Key Project of Chinese Ministry of Education (no. 211077), the Natural Science Foundation of Education Department of Anhui Province (KJ2012ZD01), and the Anhui Provincial Natural Science Foundation (no. 10040606Q30 and 1208085MA11).