Complete Convergence of the Maximum Partial Sums for Arrays of Rowwise of AANA Random Variables

Abstract

The limiting behavior of the maximum partial sums (1/𝑎_𝑛)max_{1≤𝑗≤𝑛}|∑^𝑗_𝑖=1𝑋_𝑛𝑖| is investigated, and some new results are obtained, where {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} is an array of rowwise AANA random variables and {𝑎_𝑛,𝑛≥1} is a sequence of positive real numbers. As an application, the Chung-type strong law of large numbers for arrays of rowwise AANA random variables is obtained. The results extend and improve the corresponding ones of Hu and Taylor (1997) for arrays of rowwise independent random variables.

1. Introduction

Let {𝑋_𝑛,𝑛≥1} be a sequence of random variables defined on a fixed probability space (Ω,ℱ,𝑃) with value in a real space ℝ. We say that the sequence {𝑋_𝑛,𝑛≥1} satisfies the general strong law of large numbers if there exist some increasing sequence {𝑏_𝑛,𝑛≥1} and some sequence {𝑎_𝑛,𝑛≥1} such that 1𝑏_𝑛^𝑛_𝑖=1𝑋_𝑖−𝑎_𝑖⟶0a.s.as𝑛⟶∞. (1) For the definition of general strong law of large numbers, one can refer to Chow and Teicher [1] or Kuczmaszewska [2]. Many authors have extended the strong law of large numbers for sequences of random variables to the case of triangular array of rowwise random variables and arrays of rowwise random variables. In the case of independence, Hu and Taylor [3] proved the following strong law of large numbers.

Theorem A. Let {𝑋_𝑛𝑖:1≤𝑖≤𝑛,𝑛≥1} be a triangular array of rowwise independent random variables. Let {𝑎_𝑛,𝑛≥1} be a sequence of positive real numbers such that 0<𝑎_𝑛↑∞. Let 𝑔(𝑡) be a positive, even function such that 𝑔(|𝑡|)/|𝑡|^𝑝 is an increasing function of |𝑡| and 𝑔(|𝑡|)/|𝑡|^𝑝+1 is a decreasing function of |𝑡|, respectively, that is, 𝑔(|𝑡|)|𝑡|^𝑝𝑔𝑡↑,(||)|𝑡|^𝑝+1↓,𝑎𝑠|𝑡|↑ (2) for some nonnegative integer 𝑝. If 𝑝≥2 and 𝐸𝑋_𝑛𝑖=0,_{∞𝑛𝑛=1𝑖=1}𝐸𝑔𝑋_𝑛𝑖𝑔𝑎_𝑛<∞,_{∞𝑛=1𝑛𝑖=1}𝐸𝑋_𝑛𝑖𝑎_𝑛^22𝑘<∞, (3) where 𝑘 is a positive integer, then 1𝑎_𝑛^𝑛_𝑖=1𝑋_𝑛𝑖⟶0𝑎.𝑠. (4)

Gan and Chen [4] extended and improved the result of Theorem A for rowwise independent random variable arrays to the case of rowwise negatively associated (NA, in short) random variable arrays. For the details about negatively associated random variables, one can refer to Joag-Dev and Proschan [5], Wu and Jiang [6, 7], Wu [8], Wang et al. [9], and so forth. In this paper, we will further study the strong law of large numbers for arrays of rowwise asymptotically almost negatively associated random variables based on some different conditions.

The concept of asymptotically almost negatively associated random variables was introduced by Chandra and Ghosal [10] as follows.

Definition 1. A sequence {𝑋_𝑛,𝑛≥1} of random variables is called asymptotically almost negatively associated (AANA, in short) if there exists a nonnegative sequence 𝑢(𝑛)→0 as 𝑛→∞ such that Cov𝑓𝑋_𝑛,𝑔𝑋_𝑛+1,𝑋_𝑛+2,…,𝑋_𝑛+𝑘≤𝑢(𝑛)Var𝑓𝑋_𝑛Var𝑔𝑋_𝑛+1,𝑋_𝑛+2,…,𝑋_𝑛+𝑘^1/2 (5) for all 𝑛, 𝑘≥1 and for all coordinatewise nondecreasing continuous functions 𝑓 and 𝑔 whenever the variances exist.
An array of random variables {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} is called rowwise AANA random variables if for every 𝑛≥1, {𝑋_𝑛𝑖,𝑖≥1} is a sequence of AANA random variables.

It is easily seen that the family of AANA sequence contains NA (in particular, independent) sequences (with 𝑢(𝑛)=0, 𝑛≥1) and some more sequences of random variables which are not much deviated from being negatively associated. An example of an AANA sequence which is not NA was constructed by Chandra and Ghosal [10]. Hence, extending the limit properties of independent or NA random variables to the case of AANA random variables is highly desirable in the theory and application.

Since the concept of AANA sequence was introduced by Chandra and Ghosal [10], many applications have been found. See, for example, Chandra and Ghosal [10] who derived the Kolmogorov-type inequality and the strong law of large numbers of Marcinkiewicz-Zygmund; Chandra and Ghosal [11] obtained the almost sure convergence of weighted averages; Wang et al. [12] established the law of the iterated logarithm for product sums; Ko et al. [13] studied the Hájek-Rényi-type inequality; Yuan and An [14] established some Rosenthal-type inequalities for maximum partial sums of AANA sequence; Wang et al. [15] obtained some strong growth rate and the integrability of supremum for the partial sums of AANA random variables; Wang et al. [16] established some maximal inequalities and strong law of large numbers for AANA sequences; Wang et al. [17, 18] studied complete convergence for arrays of rowwise AANA random variables and weighted sums of arrays of rowwise AANA random variables, respectively; Hu et al. [19] studied the strong convergence properties for AANA sequence; Yang et al. [20] investigated the complete convergence, complete moment convergence, and the existence of the moment of supremum of normed partial sums for the moving average process for AANA sequence; and so forth.

The main purpose of this paper is to study the limiting behavior of the maximum partial sums for arrays of rowwise AANA random variables. As an application, the Chung-type strong law of large numbers for arrays of rowwise AANA random variables is obtained. We will give some sufficient conditions for the complete convergence for an array of rowwise AANA random variables without assumptions of identical distribution and stochastic domination. The results presented in this paper are obtained by using the truncated method and the maximal Rosenthal-type inequality of AANA random variables.

Throughout this paper, let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise of AANA random variables with the mixing coefficients {𝑢(𝑖),𝑖≥1} in each row. The symbol 𝐶 denotes a positive constant which may be different in various places.

2. Preliminaries

To prove the main results of the paper, we need the following two lemmas.

Lemma 2 (see [18, Lemma 2.2]). Let {𝑋_𝑛,𝑛≥1} be a sequence of AANA random variables with mixing coefficients {𝑢(𝑛),𝑛≥1}, and let 𝑓₁,𝑓₂,… be all nondecreasing (or all nonincreasing) continuous functions, and then {𝑓_𝑛(𝑋_𝑛),𝑛≥1} is still a sequence of AANA random variables with mixing coefficients {𝑢(𝑛),𝑛≥1}.

Lemma 3 (see [14, Theorem 2.1]). Let 𝑝>1 and let {𝑋_𝑛,𝑛≥1} be a sequence of zero mean random variables with mixing coefficients {𝑢(𝑛),𝑛≥1}.
If ∑^∞_𝑛=1𝑢²(𝑛)<∞, then there exists a positive constant 𝐶_𝑝 depending only on 𝑝 such that for all 𝑛≥1 and 1<𝑝≤2, 𝐸max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑖^𝑝≤𝐶_𝑝^𝑛_𝑖=1𝐸𝑋_𝑖^𝑝. (6)
If   ∑^∞_𝑛=1𝑢^1/(𝑝−1)(𝑛)<∞ for some 𝑝∈(3⋅2^𝑘−1,4⋅2^𝑘−1], where the integer number 𝑘≥1, then there exists a positive constant 𝐷_𝑝 depending only on 𝑝 such that for all 𝑛≥1, 𝐸max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑖^𝑝≤𝐷_𝑝^𝑛_𝑖=1𝐸𝑋_𝑖^𝑝+^𝑛_𝑖=1𝐸𝑋²_𝑖^𝑝/2. (7)

3. Main Results and Proofs

In this section, we will investigate the limiting behavior of the maximum partial sums for arrays of rowwise of AANA random variables. The first three theorems consider different conditions from Hu and Taylor’s [3].

Theorem 4. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise AANA random variables with ∑^∞_𝑛=1𝑢²(𝑛)<∞ and let {𝑎_𝑛,𝑛≥1} be a sequence of positive real numbers. Let {𝑔_𝑛(𝑥),𝑛≥1} be a sequence of nonnegative, even functions such that 𝑔_𝑛(|𝑥|) is an increasing function of |𝑥| for every 𝑛≥1. Assume that there exists a constant 𝛿>0 such that 𝑔_𝑛(𝑥)≥𝛿𝑥 for 0<𝑥≤1. If ^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛<∞, (8) then for any 𝜀>0, ^∞_𝑛=1𝑃max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝑋_𝑛𝑖>𝜀<∞. (9)

Proof. For fixed 𝑛≥1, define 𝑋_𝑖^(𝑛)=−𝑎_𝑛𝐼𝑋_𝑛𝑖<−𝑎_𝑛+𝑋_𝑛𝑖𝐼𝑋_𝑛𝑖≤𝑎_𝑛+𝑎_𝑛𝐼𝑋_𝑛𝑖>𝑎_𝑛𝑇,𝑖≥1,_𝑗(𝑛)=1𝑎_𝑛^𝑗_𝑖=1𝑋_𝑖^(𝑛)−𝐸𝑋_𝑖^(𝑛),𝑗=1,2,…,𝑛. (10) By Lemma 2, we can see that for fixed 𝑛≥1, {𝑋_𝑖^(𝑛),𝑖≥1} is still a sequence of AANA random variables. It is easy to check that for any 𝜀>0, max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝑋_𝑛𝑖>𝜀⊂max_{1≤𝑖≤𝑛}𝑋_𝑛𝑖>𝑎_𝑛max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝑋_𝑖^(𝑛)>𝜀, (11) which implies that 𝑃max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝑋_𝑛𝑖>𝜀≤𝑃max_{1≤𝑖≤𝑛}𝑋_𝑛𝑖>𝑎_𝑛+𝑃max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝑋_𝑖^(𝑛)≤>𝜀_𝑛𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛+𝑃max_{1≤𝑗≤𝑛}𝑇_𝑗^(𝑛)>𝜀−max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝐸𝑋_𝑖^(𝑛). (12)
Firstly, we will show that max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝐸𝑋_𝑖^(𝑛)⟶0,as𝑛⟶∞. (13) In fact, by the conditions 𝑔_𝑛(𝑥)≥𝛿𝑥 for 0<𝑥≤1 and (8), we have max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝐸𝑋_𝑖^(𝑛)≤^𝑛_𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛+^𝑛_𝑖=1𝐸𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖≤𝑎_𝑛≤1𝛿^𝑛_𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛+1𝛿^𝑛_𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖≤𝑎_𝑛≤2𝛿^𝑛_𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛⟶0,as𝑛⟶∞, (14) which implies (13).
It follows from (12) and (13) that for 𝑛 large enough, 𝑃max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝑋_𝑛𝑖≤>𝜀_𝑛𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛+𝑃max_{1≤𝑗≤𝑛}𝑇_𝑗^(𝑛)>𝜀2. (15) Hence, to prove (9), we only need to show that ^∞𝑛_𝑛=1𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛<∞,_∞𝑛=1𝑃max_{1≤𝑗≤𝑛}𝑇_𝑗^(𝑛)>𝜀2<∞. (16)(17)
When |𝑋_𝑛𝑖|>𝑎_𝑛>0, we have 𝑔_𝑖(𝑋_𝑛𝑖/𝑎_𝑛)≥𝑔_𝑖(1)≥𝛿, which yields that 𝑃𝑋_𝑛𝑖>𝑎_𝑛=𝐸𝐼𝑋_𝑛𝑖>𝑎_𝑛≤1𝛿𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛. (18) Hence, ^∞𝑛_𝑛=1𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛≤1𝛿^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛<∞, (19) which implies (16).
By Markov’s inequality, Lemma 3 (for 𝑝=2), 𝑔_𝑛(𝑥)≥𝛿𝑥 for 0<𝑥≤1, and (8), we can get that ^∞_𝑛=1𝑃max_{1≤𝑗≤𝑛}𝑇_𝑗^(𝑛)>𝜀2≤𝐶^∞_𝑛=1𝐸max_{1≤𝑗≤𝑛}𝑇_𝑗^(𝑛)2≤𝐶^∞_𝑛=11𝑎²_𝑛^𝑛_𝑖=1𝐸𝑋_𝑖^(𝑛)−𝐸𝑋_𝑖^(𝑛)2≤𝐶^∞_𝑛=11𝑎²_𝑛^𝑛_𝑖=1𝐸𝑋_𝑖^(𝑛)2≤𝐶^∞𝑛_𝑛=1𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖²𝐼𝑋_𝑛𝑖≤𝑎_𝑛𝑎²_𝑛≤𝐶+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖𝐼𝑋_𝑛𝑖≤𝑎_𝑛𝑎_𝑛≤𝐶+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖≤𝑎_𝑛≤𝐶+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛<∞, (20) which implies (17). This completes the proof of the theorem.

Corollary 5. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise AANA random variables with ∑^∞_𝑛=1𝑢²(𝑛)<∞ and let {𝑎_𝑛,𝑛≥1} be a sequence of positive real numbers. If there exists a constant 𝛽∈(0,1] such that^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖^𝛽𝑎_𝑛^𝛽+𝑋_𝑛𝑖^𝛽<∞, (21) then (9) holds for any 𝜀>0.

Proof. In Theorem 4, we take 𝑔_𝑛(𝑥)≡|𝑥|^𝛽1+|𝑥|^𝛽,0<𝛽≤1,𝑛≥1. (22) It is easy to check that {𝑔_𝑛(𝑥),𝑛≥1} is a sequence of nonnegative, even functions such that 𝑔_𝑛(|𝑥|) is an increasing function of |𝑥| for every 𝑛≥1. And 𝑔_𝑛(𝑥)≥12𝑥^𝛽≥12𝑥,0<𝑥≤1,0<𝛽≤1. (23) Therefore, by Theorem 4, we can easily get (9).

Corollary 6. Under the conditions of Theorem 4 or Corollary 5, 1𝑎_𝑛^𝑛_𝑖=1𝑋_𝑛𝑖⟶0𝑎.𝑠. (24)

Theorem 7. Let {𝑋_𝑛𝑖:𝑖≥1,𝑛≥1} be an array of rowwise AANA random variables with ∑^∞_𝑛=1𝑢²(𝑛)<∞ and let {𝑎_𝑛,𝑛≥1} be a sequence of positive real numbers. 𝐸𝑋_𝑛𝑖=0, 𝑖≥1, 𝑛≥1. Let {𝑔_𝑛(𝑥),𝑛≥1} be a sequence of nonnegative, even functions. Assume that there exist 𝛽∈(1,2] and 𝛿>0 such that 𝑔_𝑛(𝑥)≥𝛿𝑥^𝛽 for 0<𝑥≤1 and there exists a 𝛿>0 such that 𝑔_𝑛(𝑥)≥𝛿𝑥 for 𝑥>1. If (8) satisfies, then (9) holds for any 𝜀>0.

Proof. We use the same notations as those in Theorem 4. The proof is similar to that of Theorem 4.
Firstly, we will show that (13) holds true. Actually, by the conditions 𝐸𝑋_𝑛𝑖=0, 𝑔_𝑛(𝑥)≥𝛿𝑥 for 𝑥>1, and (8), we have max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝐸𝑋_𝑖^(𝑛)≤^𝑛_𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛+max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝐸𝑋_𝑛𝑖𝐼𝑋_𝑛𝑖>𝑎_𝑛≤2^𝑛_𝑖=1𝐸𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖>𝑎_𝑛≤2𝛿^𝑛_𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖>𝑎_𝑛≤2𝛿^𝑛_𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛⟶0,as𝑛⟶∞, (25) which implies (13). Hence, to prove (9), we only need to show that (16) and (17) hold true.
The conditions 𝑔_𝑛(𝑥)≥𝛿𝑥 for 𝑥>1 and (8) yield that ^∞𝑛_𝑛=1𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛=^∞𝑛_𝑛=1𝑖=1𝐸𝐼𝑋_𝑛𝑖>𝑎_𝑛≤^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖>𝑎_𝑛≤1𝛿^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖>𝑎_𝑛≤1𝛿^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛<∞, (26) which implies (16).
By Markov’s inequality, Lemma 3 (for 𝑝=2), 𝑔_𝑛(𝑥)≥𝛿𝑥^𝛽 for 1<𝛽≤2, 0<𝑥≤1, and (8), we can get that ^∞_𝑛=1𝑃max_{1≤𝑗≤𝑛}𝑇_𝑗^(𝑛)>𝜀2≤𝐶^∞𝑛_𝑛=1𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖²𝐼𝑋_𝑛𝑖≤𝑎_𝑛𝑎²_𝑛≤𝐶+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖^𝛽𝐼𝑋_𝑛𝑖≤𝑎_𝑛𝑎^𝛽_𝑛≤𝐶+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖≤𝑎_𝑛≤𝐶+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛<∞, (27) which implies (17). This completes the proof of the theorem.

Corollary 8. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise AANA random variables with ∑^∞_𝑛=1𝑢²(𝑛)<∞ and let {𝑎_𝑛,𝑛≥1} be a sequence of positive real numbers. 𝐸𝑋_𝑛𝑖=0, 𝑖≥1, 𝑛≥1. If there exists a constant 𝛽∈(1,2] such that^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖^𝛽𝑎_𝑛𝑋_𝑛𝑖^𝛽−1+𝑎^𝛽_𝑛<∞, (28) then (9) holds for any 𝜀>0.

Proof. In Theorem 7, we take 𝑔_𝑛(𝑥)≡|𝑥|^𝛽1+|𝑥|^𝛽−1,1<𝛽≤2,𝑛≥1. (29) It is easy to check that {𝑔_𝑛(𝑥),𝑛≥1} is a sequence of nonnegative, even functions satisfying 𝑔_𝑛(𝑥)≥12𝑥^𝛽𝑔,0<𝑥≤1,1<𝛽≤2,_𝑛(𝑥)≥12𝑥,𝑥>1. (30) Therefore, by Theorem 7, we can easily get (9).

Furthermore, by Corollaries 5 and 8, we can get the following important Chung-type strong law of large numbers for arrays of rowwise AANA random variables.

Corollary 9. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise AANA random variables with ∑^∞_𝑛=1𝑢²(𝑛)<∞ and let {𝑎_𝑛,𝑛≥1} be a sequence of positive real numbers. If there exists some 𝛽∈(0,2] such that^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖^𝛽𝑎^𝛽_𝑛<∞, (31) and 𝐸𝑋_𝑛𝑖=0, 𝑖≥1, 𝑛≥1 if 𝛽∈(1,2], then (9) holds for any 𝜀>0 and (1/𝑎_𝑛)∑^𝑛_𝑖=1𝑋_𝑛𝑖→0𝑎.𝑠.

For 𝛽≥2, we have the following result.

Theorem 10. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise AANA random variables with ∑^∞_𝑛=1𝑢²(𝑛)<∞ and let {𝑎_𝑛,𝑛≥1} be a sequence of positive real numbers. Let {𝑔_𝑛(𝑥),𝑛≥1} be a sequence of nonnegative, even functions. Assume that there exists some 𝛽≥2 such that 𝑔_𝑛(𝑥)≥𝛿𝑥^𝛽 for 𝑥>0. If ^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛^1/𝛽<∞, (32) then (9) holds for any 𝜀>0 and (1/𝑎_𝑛)∑^𝑛_𝑖=1𝑋_𝑛𝑖→0𝑎.𝑠.

Proof. We use the same notations as those in Theorem 4. The proof is similar to that of Theorem 4. It is easily seen that (32) implies that ^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛<∞,_{∞𝑛𝑛=1𝑖=1}𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛^2/𝛽<∞. (33)(34)
Firstly, we will show that (13) holds true. In fact, by Hölder’s inequality, 𝑔_𝑛(𝑥)≥𝛿𝑥^𝛽 for 𝑥>0, (32), and (33), we have max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝐸𝑋_𝑖^(𝑛)≤^𝑛_𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛+^𝑛_𝑖=1𝐸𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖≤𝑎_𝑛≤^𝑛_𝑖=1𝐸𝑋_𝑛𝑖^𝛽𝑎^𝛽_𝑛𝐼𝑋_𝑛𝑖>𝑎_𝑛+^𝑛_𝑖=1𝐸|𝑋_𝑛𝑖|^𝛽𝑎^𝛽_𝑛𝐼𝑋_𝑛𝑖≤𝑎_𝑛^1/𝛽≤𝐶^𝑛_𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛+𝐶^𝑛_𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖≤𝑎_𝑛^1/𝛽≤𝐶^𝑛_𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛+𝐶^𝑛_𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛^1/𝛽⟶0,as𝑛⟶∞, (35) which implies (13). To prove (9), we only need to show that (16) and (17) hold true.
By the condition 𝑔_𝑛(𝑥)≥𝛿𝑥^𝛽 for 𝑥>0 again and (33), we have ^∞𝑛_𝑛=1𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛=^∞𝑛_𝑛=1𝑖=1𝐸𝐼𝑋_𝑛𝑖>𝑎_𝑛≤^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖^𝛽𝑎^𝛽_𝑛𝐼𝑋_𝑛𝑖>𝑎_𝑛≤1𝛿^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛<∞, (36) which implies (16).
By Markov’s inequality, Lemma 3 (for 𝑝=2), 𝑔_𝑛(𝑥)≥𝛿𝑥^𝛽 for 𝑥>0, and (34), we can get that ^∞_𝑛=1𝑃max_{1≤𝑗≤𝑛}𝑇_𝑗^(𝑛)>𝜀2≤𝐶^∞𝑛_𝑛=1𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖²𝐼𝑋_𝑛𝑖≤𝑎_𝑛𝑎²_𝑛≤𝐶+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖^𝛽𝑎^𝛽_𝑛𝐼𝑋_𝑛𝑖≤𝑎_𝑛^2/𝛽≤𝐶+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖≤𝑎_𝑛^2/𝛽≤𝐶+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑎_𝑛^2/𝛽<∞, (37) which implies (17). This completes the proof of the theorem.

The next two theorems extend and improve Theorem A for arrays of rowwise of independent random variables to the case of AANA random variables.

Theorem 11. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise AANA random variables with ∑^∞_𝑛=1𝑢²(𝑛)<∞ and let {𝑎_𝑛,𝑛≥1} be a sequence of positive real numbers. Let {𝑔_𝑛(𝑥),𝑛≥1} be a sequence of nonnegative and even functions such that 𝑔_𝑛(𝑥)>0 for 𝑥>0 and every 𝑛≥1, and 𝑔_𝑛(|𝑥|)|𝑥|𝑔↑,^𝑛(|𝑥|)|𝑥|^𝑝↓𝑎𝑠|𝑥|↑, (38) where 1<𝑝≤2. Assume further that 𝐸𝑋_𝑛𝑖=0,𝑖≥1,𝑛≥1,_{∞𝑛𝑛=1𝑖=1}𝐸𝑔_𝑖𝑋_𝑛𝑖𝑔_𝑖𝑎_𝑛<∞. (39)(40) Then (9) holds for any 𝜀>0 and (1/𝑎_𝑛)∑^𝑛_𝑖=1𝑋_𝑛𝑖→0   𝑎.𝑠.

Proof. We use the same notations as those in Theorem 4. The proof is similar to that of Theorem 4.
Firstly, we will show that (13) holds true. Actually, by the conditions 𝐸𝑋_𝑛𝑖=0 and (38)–(40), we have max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝐸𝑋_𝑖^(𝑛)≤^𝑛_𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛+max_{1≤𝑗≤𝑛}1𝑎_𝑛^𝑗_𝑖=1𝐸𝑋_𝑛𝑖𝐼𝑋_𝑛𝑖>𝑎_𝑛≤2^𝑛_𝑖=1𝐸𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖>𝑎_𝑛≤2^𝑛_𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝐼𝑋_𝑛𝑖>𝑎_𝑛𝑔_𝑖𝑎_𝑛≤2^𝑛_𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑔_𝑖𝑎_𝑛⟶0,as𝑛⟶∞, (41) which implies (13). Hence, to prove (9), it suffices to show that (16) and (17) hold true.
The conditions (38) and (40) yield that ^∞𝑛_𝑛=1𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛≤^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖𝑎_𝑛𝐼𝑋_𝑛𝑖>𝑎_𝑛≤2^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑔_𝑖𝑎_𝑛<∞, (42) which implies (16).
By Markov’s inequality, Lemma 3, (38), and (40), we can get that ^∞_𝑛=1𝑃max_{1≤𝑗≤𝑛}𝑇_𝑗^(𝑛)>𝜀2≤𝐶^∞𝑛_𝑛=1𝑖=1𝑃𝑋_𝑛𝑖>𝑎_𝑛+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖^𝑝𝐼𝑋_𝑛𝑖≤𝑎_𝑛𝑎^𝑝_𝑛≤𝐶+𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑔_𝑖𝑎_𝑛×𝐼𝑋_𝑛𝑖≤𝑎_𝑛<∞, (43) which implies (17). This completes the proof of the theorem.

Theorem 12. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise AANA random variables and let {𝑎_𝑛,𝑛≥1} be a sequence of positive real numbers. Let 𝑝>2, 0<𝑟≤2, 𝑠>0, and 𝛿≐max{𝑝,2𝑠}∈(3⋅2^𝑘−1,4⋅2^𝑘−1], where the integer number 𝑘≥1. Let {𝑔_𝑛(𝑥),𝑛≥1} be a sequence of nonnegative and even functions such that 𝑔_𝑛(𝑥)>0 for 𝑥>0 and every 𝑛≥1. Assume that conditions (38)–(40) are satisfied. If ∑^∞_𝑛=1𝑢^1/(𝛿−1)(𝑛)<∞ and ^∞_𝑛=1^𝑛_𝑖=1𝐸𝑋_𝑛𝑖^𝑟𝑎^𝑟_𝑛^𝑠<∞, (44) then (9) holds for any 𝜀>0 and (1/𝑎_𝑛)∑^𝑛_𝑖=1𝑋_𝑛𝑖→0𝑎.𝑠.

Proof. We use the same notations as those in Theorem 4. The proof is similar to that of Theorems 4 and 11.
Following the methods of the proof in Theorems 4 and 11, (13) and (16) hold. So we need only to show (17). Denote 𝛿=max{𝑝,2𝑠}. We have by Markov’s inequality, Lemma 3, and 𝐶_𝑟’s inequality that ^∞_𝑛=1𝑃max_{1≤𝑗≤𝑛}𝑇_𝑗^(𝑛)>𝜀2≤𝐶^∞_𝑛=1𝐸max_{1≤𝑗≤𝑛}𝑇_𝑗^(𝑛)𝛿≤𝐶^∞_𝑛=11𝑎^𝛿_𝑛^𝑛_𝑖=1𝐸𝑋_𝑖^(𝑛)−𝐸𝑋_𝑖^(𝑛)𝛿+^𝑛_𝑖=1𝐸𝑋_𝑖^(𝑛)−𝐸𝑋_𝑖^{(𝑛)2𝛿/2}≤𝐶^∞_𝑛=11𝑎^𝛿_𝑛^𝑛_𝑖=1𝐸𝑋_𝑖^(𝑛)𝛿+^𝑛_𝑖=1𝐸𝑋_𝑖^{(𝑛)2𝛿/2}. (45) Since 𝛿≥𝑝 and (𝑔_𝑛(|𝑥|)/|𝑥|)↑, (𝑔_𝑛(|𝑥|)/|𝑥|^𝑝)↓ as |𝑥|↑, it follows that 𝑔_𝑛(|𝑥|)↑ and (𝑔_𝑛(|𝑥|)/|𝑥|^𝛿)↓ as |𝑥|↑. Hence, ^∞_𝑛=11𝑎^𝛿_𝑛^𝑛_𝑖=1𝐸𝑋_𝑖^(𝑛)𝛿≤^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑖^(𝑛)𝑔_𝑖𝑎_𝑛≤^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑔_𝑖𝑎_𝑛=^∞𝑛_𝑛=1𝑖=1𝐸𝑔_𝑖𝑋_𝑛𝑖𝑔_𝑖𝑎_𝑛<∞. (46) It follows by 0<𝑟≤2, 𝑠>0, 𝛿≥2𝑠, and (44) that ^∞_𝑛=11𝑎^𝛿_𝑛^𝑛_𝑖=1𝐸𝑋_𝑖^{(𝑛)2𝛿/2}=^∞_𝑛=1^𝑛_𝑖=1𝐸𝑋_𝑖^(𝑛)2𝑎²_𝑛^{𝑠𝛿/2𝑠}≤^∞_𝑛=1^𝑛_𝑖=1𝐸𝑋_𝑖^(𝑛)2𝑎²_𝑛^{𝑠𝛿/2𝑠}≤^∞_𝑛=1^𝑛_𝑖=1𝐸𝑋_𝑖^(𝑛)𝑟𝑎^𝑟_𝑛^{𝑠𝛿/2𝑠}≤^∞_𝑛=1^𝑛_𝑖=1𝐸𝑋_𝑛𝑖^𝑟𝑎^𝑟_𝑛^{𝑠𝛿/2𝑠}<∞. (47) Therefore, (17) follows from (45)–(47) immediately. This completes the proof of the theorem.

Acknowledgments

The authors are most grateful to the Editor and anonymous referee for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11201001, 11171001, and 11226207), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20093401120001), the Natural Science Foundation of Anhui Province (1308085QA03, 11040606M12, and 1208085QA03), the 211 project of Anhui University, the Youth Science Research Fund of Anhui University, Applied Teaching Model Curriculum of Anhui University (XJYYXKC04), and the Students Science Research Training Program of Anhui University (KYXL2012007).