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

Abstract

Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise asymptotically almost negatively associated (AANA, in short) random variables. The complete convergence for weighted sums of arrays of rowwise AANA random variables is studied, which complements and improves the corresponding result of Baek et al. (2008). As applications, the Baum and Katz type result for arrays of rowwise AANA random variables and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of AANA random variables are obtained.

1. Introduction

The concept of complete convergence was introduced by Hsu and Robbins [1] as follows. A sequence of random variables {𝑈_𝑛,𝑛≥1} is said to converge completely to a constant 𝐶 if ∑^∞_𝑛=1𝑃(|𝑈_𝑛−𝐶|>𝜀)<∞, for all 𝜀>0. In view of the Borel-Cantelli lemma, this implies that 𝑈_𝑛→𝐶 almost surely (a.s.). The converse is true if the {𝑈_𝑛,𝑛≥1} are independent. Hsu and Robbins [1] proved that the sequence of arithmetic means of independent and identically distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite. Erdös [2] proved the converse. The result of Hsu-Robbins-Erdös is a fundamental theorem in probability theory and has been generalized and extended in several directions by many authors. One of the most important generalizations is Baum and Katz [3] for the strong law of large numbers.

Recently, Baek et al. [4] discussed the complete convergence of weighted sums for arrays of rowwise negatively associated random variables and obtained the following result.

Theorem 1.1. 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 𝑖≥1, 𝑛≥1 and 𝑥≥0. Suppose that 𝛽≥−1, and that {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of constants such that sup_𝑖≥1𝑎_𝑛𝑖=𝑂𝑛^−𝑟forsome𝑟>0,_∞𝑖=1𝑎_𝑛𝑖=𝑂𝑛^𝛼forsome𝛼∈[0,𝑟). (1.1)
(i) If 1+𝛼+𝛽>0 and there exists some 𝛿>0 such that 𝛼/𝑟+1<𝛿≤2 and 𝑠=max(1+((1+𝛼+𝛽)/𝑟),𝛿), then under 𝐸|𝑋|^𝑠<∞, we have ^∞_𝑛=1𝑛^𝛽𝑃^∞_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜀<∞∀𝜀>0. (1.2)
(ii) If 1+𝛼+𝛽=0, then under 𝐸(|𝑋|log(1+|𝑋|))<∞, (1.2) remains true.

The main purpose of this paper is to generalize and improve the above results for arrays of rowwise negatively associated random variables to the case of asymptotically almost negatively associated random variables. In addition, we will also consider the case 1+𝛼+𝛽<0, which complements the result of Baek et al. [4] and Wu [5].

Definition 1.2. A finite collection of random variables 𝑋₁,𝑋₂,…,𝑋_𝑛 is said to be negatively associated (NA, in short) if for every pair of disjoint subsets 𝐴₁,𝐴₂ of {1,2,…,𝑛}, Cov𝑓𝑋_𝑖∶𝑖∈𝐴₁,𝑔𝑋_𝑗∶𝑗∈𝐴₂≤0, (1.3) whenever 𝑓 and 𝑔 are coordinatewise nondecreasing such that this covariance exists. An infinite sequence {𝑋_𝑛,𝑛≥1} is NA if every finite subcollection is NA.
An array of random variables {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} is called rowwise NA random variables if for every 𝑛≥1, {𝑋_𝑛𝑖,𝑖≥1} is a sequence of NA random variables.

The concept of negative association was introduced by Block et al. [6] and carefully studied by Joag-Dev and Proschan [7]. By inspecting the proof of maximal inequality for the NA random variables in Matula [8], one also can allow negative correlations provided they are small. Primarily motivated by this, Chandra and Ghosal [9, 10] introduced the following dependence.

Definition 1.3. A sequence {𝑋_𝑛,𝑛≥1} of random variables is called asymptotically almost negatively associated (AANA, in short) if there exists a nonnegative sequence 𝑞(𝑛)→0 as 𝑛→∞ such that Cov𝑓𝑋_𝑛,𝑔𝑋_𝑛+1,𝑋_𝑛+2,…,𝑋_𝑛+𝑘≤𝑞(𝑛)Var𝑓𝑋_𝑛Var𝑔𝑋_𝑛+1,𝑋_𝑛+2,…,𝑋_𝑛+𝑘^1/2, (1.4) 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.

The family of AANA sequence contains NA (in particular, independent) sequences (with 𝑞(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 [9].

Since the concept of AANA sequence was introduced by Chandra and Ghosal [9], many applications have been found. See for example, Chandra and Ghosal [9] derived the Kolmogorov type inequality and the strong law of large numbers of Marcinkiewicz-Zygmund, Chandra and Ghosal [10] obtained the almost sure convergence of weighted averages, Ko et al. [11] studied the Hájek-Rényi type inequality, Wang et al. [12] established the law of the iterated logarithm for product sums, Yuan and An [13] established some Rosenthal type inequalities for maximum partial sums of AANA sequence, and Wang et al. [14] obtained some strong growth rate and the integrability of supremum for the partial sums of AANA random variables, and so forth. Our aim is to further study the complete convergence of weighted sums for arrays of rowwise AANA random variables.

Throughout this paper, let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise AANA random variables with the mixing coefficients {𝑞(𝑖),𝑖≥1} in each row. For 𝑝>1, let 𝑞≐𝑝/(𝑝−1) be the dual number of 𝑝. The symbol 𝐶 denotes a positive constant which is not necessarily the same one in each appearance and ⌈𝑥⌉ denotes the integer part of 𝑥. For a finite set 𝐴, the symbol ♯𝐴 denotes the number of elements in the set 𝐴. Let 𝐼(𝐴) be the indicator function of the set 𝐴. 𝑎_𝑛=𝑂(𝑏_𝑛) stands for 𝑎_𝑛≤𝐶𝑏_𝑛. Denote log𝑥=lnmax(𝑥,𝑒), 𝑋⁺=max(𝑋,0) and 𝑋⁻=max(−𝑋,0).

The paper is organized as follows. Three important lemmas are provided in Section 2. The main results and their proofs are presented in Section 3. We will provide some sufficient conditions for complete convergence for arrays of rowwise AANA random variables which are stochastically dominated by a random variable 𝑋.

2. Preliminaries

Firstly, we will give the definition of stochastic domination.

Definition 2.1. A sequence {𝑋_𝑛,𝑛≥1} of random variables is said to be stochastically dominated by a random variable 𝑋 if there exists a positive constant 𝐶 such that 𝑃𝑋_𝑛>𝑥≤𝐶𝑃(|𝑋|>𝑥), (2.1) for all 𝑥≥0 and 𝑛≥1.
An array {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} of rowwise random variables is said to be stochastically dominated by a random variable 𝑋 if there exists a positive constant 𝐶 such that 𝑃𝑋_𝑛𝑖>𝑥≤𝐶𝑃(|𝑋|>𝑥), (2.2) for all 𝑥≥0, 𝑖≥1 and 𝑛≥1.

The proofs of the main results of the paper are based on the following three lemmas.

Lemma 2.2 2.2(cf. Yuan and An [13, Lemma 2.1]). Let {𝑋_𝑛,𝑛≥1} be a sequence of AANA random variables with mixing coefficients {𝑞(𝑛),𝑛≥1}, let 𝑓₁,𝑓₂,… be all nondecreasing (or all nonincreasing) and continuous functions, then {𝑓_𝑛(𝑋_𝑛),𝑛≥1} is still a sequence of AANA random variables with mixing coefficients {𝑞(𝑛),𝑛≥1}.

Lemma 2.3 2.3(cf. Yuan and An [13, Theorem 2.1]). Let 𝑝>1 and {𝑋_𝑛,𝑛≥1} be a sequence of zero mean random variables with mixing coefficients {𝑞(𝑛),𝑛≥1}.
If ∑^∞_𝑛=1𝑞²(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. (2.3)
If ∑^∞_𝑛=1𝑞^𝑞/𝑝(𝑛)<∞ for some 𝑝∈(3⋅2^𝑘−1,4⋅2^𝑘−1], where 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. (2.4)

Lemma 2.4. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise random variables which is stochastically dominated by a random variable 𝑋. For any 𝛼>0 and 𝑏>0, the following two statements hold: 𝐸𝑋_𝑛𝑖^𝛼𝐼𝑋_𝑛𝑖≤𝑏≤𝐶₁𝐸|𝑋|^𝛼𝐼(|𝑋|≤𝑏)+𝑏^𝛼𝑃𝑋|,𝐸𝑋(|>𝑏)_𝑛𝑖𝛼𝐼𝑋_𝑛𝑖>𝑏≤𝐶₂𝐸|𝑋|^𝛼𝐼(|𝑋|>𝑏), (2.5) where 𝐶₁ and 𝐶₂ are positive constants.

3. Main Results

In this section, we will study the complete convergence for weighted sums of arrays of rowwise AANA random variables. As applications, the Baum and Katz type result for arrays of rowwise AANA random variables and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of AANA random variables are obtained. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise AANA random variables with the mixing coefficients {𝑞(𝑖),𝑖≥1} in each row and {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of real numbers. Let {𝑋_𝑖,𝑖≥1} be a sequence of AANA random variables with the mixing coefficients {𝑞(𝑖),𝑖≥1}. Our main results are as follows.

Theorem 3.1. Suppose that 𝛽≥−1. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise AANA random variables, which is stochastically dominated by a random variable 𝑋, and {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of constants such that sup_𝑖≥1𝑎_𝑛𝑖=𝑂𝑛^−𝑟forsome𝑟>0,_∞𝑖=1𝑎_𝑛𝑖^𝜃=𝑂𝑛^𝛼𝛼forsome0<𝜃<2andsome𝛼suchthat𝜃+𝑟<2. (3.1)(3.2)
(i) Assume that ∑^∞_𝑛=1𝑞²(𝑛)<∞ when 1<𝜃<2. If 1+𝛼+𝛽<0 and 𝐸|𝑋|^𝜃<∞, then ^∞_𝑛=1𝑛^𝛽𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜀<∞∀𝜀>0 (3.3) and (1.2) holds.
(ii) If 1+𝛼+𝛽>0, 𝛽>−1 and 𝐸|𝑋|^𝑠<∞,where𝑠=𝜃+1+𝛼+𝛽𝑟, (3.4) and assume further that 𝐸𝑋_𝑛𝑖=0 and ∑^∞_𝑛=1𝑞^𝑞/𝑝(𝑛)<∞ for some 𝑝∈(3⋅2^𝑘−1,4⋅2^𝑘−1] and 2𝑝>max2,1+𝛽𝑟(2−𝜃)−𝛼,𝑠, (3.5) when 𝑠≥1, where integer number 𝑘≥1, then (1.2) and (3.3) hold.
(iii) If 1+𝛼+𝛽=0 and 𝐸|𝑋|^𝜃log|𝑋|<∞, (3.6) and assume further that 𝐸𝑋_𝑛𝑖=0 and ∑^∞_𝑛=1𝑞²(𝑛)<∞ when 1≤𝜃<2, then (1.2) and (3.3) hold.

Proof. The proof of (1.2) is similar to that of (3.3), so we only prove (3.3). Without loss of generality, we assume that 𝑎_𝑛𝑖>0 for all 𝑖≥1 and 𝑛≥1 (Otherwise, we use 𝑎⁺_𝑛𝑖 and 𝑎⁻_𝑛𝑖 instead of 𝑎_𝑛𝑖, resp., and note that 𝑎_𝑛𝑖=𝑎⁺_𝑛𝑖−𝑎⁻_𝑛𝑖). From the conditions (3.1) and (3.2), we assume that sup_𝑖≥1𝑎_𝑛𝑖=𝑛^−𝑟,^∞_𝑖=1𝑎^𝜃_𝑛𝑖=𝑛^𝛼,𝑛≥1. (3.7)
(i) If 1+𝛼+𝛽<0, then the result can be easily proved by the following: ^∞_𝑛=1𝑛^𝛽𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜀≤𝐶^∞_𝑛=1𝑛^𝛽𝐸max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖^𝜃≤𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖^𝜃≤𝐶^∞_𝑛=1𝑛^𝛼+𝛽𝐸|𝑋|^𝜃<∞. (3.8)
In the following, we will prove the result when 1+𝛼+𝛽≥0. Denote 𝑋^′_𝑛𝑖=−𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖<−1+𝑎_𝑛𝑖𝑋_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤1+𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖>1,𝑖≥1,𝑛≥1. (3.9) Thus, {𝑋^′_𝑛𝑖,𝑖≥1,𝑛≥1} is still an array of rowwise AANA random variables with the mixing coefficients {𝑞(𝑖),𝑖≥1} in each row by Lemma 2.2. It is easy to check that for any 𝜀>0, max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜀⊂^𝑛_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>1max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋^′_𝑛𝑖>𝜀, (3.10) which implies that ^∞_𝑛=1𝑛^𝛽𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖>𝜀≤^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑛𝑖+>1_∞𝑛=1𝑛^𝛽𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋^′_𝑛𝑖>𝜀≐𝐼+𝐽. (3.11) Hence, in order to prove (3.3), it suffices to prove that 𝐼<∞ and 𝐽<∞.
(ii) If 1+𝛼+𝛽>0, then by Markov's inequality, (3.7) and 𝐸|𝑋|^𝑠<∞, we can get that ^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑛𝑖>1≤𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋>1≤𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝑎^𝜃_𝑛𝑖𝐸|𝑋|^𝜃𝐼|𝑋|>1𝑎_𝑛𝑖≤𝐶^∞_𝑛=1𝑛^𝛼+𝛽𝐸|𝑋|^𝜃𝐼|𝑋|>𝑛^𝑟≤𝐶^∞_𝑛=1𝑛^∞𝛼+𝛽_𝑘=𝑛𝐸|𝑋|^𝜃𝐼𝑘^𝑟≤|𝑋|<(𝑘+1)^𝑟=𝐶^∞𝑘_𝑘=1𝑛=1𝑛^𝛼+𝛽𝐸|𝑋|^𝜃𝐼𝑘^𝑟≤|𝑋|<(𝑘+1)^𝑟≤𝐶^∞_𝑘=1𝑘^1+𝛼+𝛽𝐸|𝑋|^𝜃𝐼𝑘^𝑟≤|𝑋|<(𝑘+1)^𝑟≤𝐶^∞_𝑘=1𝐸|𝑋|^{𝜃+(1+𝛼+𝛽)/𝑟}𝐼𝑘^𝑟≤|𝑋|<(𝑘+1)^𝑟≤𝐶𝐸|𝑋|^{𝜃+(1+𝛼+𝛽)/𝑟}<∞, (3.12) which implies that 𝐼<∞.
Next, we will prove that 𝐽<∞ for 𝑠≥1 and 𝑠<1, respectively.
Case  1  (𝑠≥1). Firstly, we will show that max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝐸𝑋^′_𝑛𝑖⟶0,as𝑛⟶∞. (3.13) Actually, by the conditions 𝐸𝑋_𝑛𝑖=0, Lemma 2.4, (3.7), and 𝐸|𝑋|^𝑠<∞, we have that max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝐸𝑋^′_𝑛𝑖≤max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤1+^𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑛𝑖>1=max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖>1+^𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑛𝑖>1≤𝐶^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖^𝑠𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖>1≤𝐶^𝑛_𝑖=1𝑎^𝑠_𝑛𝑖𝐸|𝑋|^𝑠𝐼|𝑋|>1𝑎_𝑛𝑖≤𝐶sup_𝑖≥1𝑎_𝑛𝑖^{𝑛𝑠−𝜃}_𝑖=1𝑎^𝜃_𝑛𝑖𝐸|𝑋|^𝑠𝐼|𝑋|>𝑛^𝑟≤𝐶𝑛^{−𝑟𝑠−𝜃}𝑛^𝛼𝐸|𝑋|^𝑠𝐼|𝑋|>𝑛^𝑟=𝐶𝑛^−(1+𝛽)𝐸|𝑋|^𝑠𝐼|𝑋|>𝑛^𝑟⟶0,as𝑛⟶∞, (3.14) which implies (3.13). Hence, to prove 𝐽<∞, we only need to show that for all 𝜀>0, 𝐽^∗≐^∞_𝑛=1𝑛^𝛽𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋^′_𝑛𝑖−𝐸𝑋^′_𝑛𝑖>𝜀2<∞. (3.15) By Markov's inequality, Lemma 2.3, 𝐶_𝑟's inequality, and Jensen's inequality, we have for 𝑝≥2 that 𝐽^∗≤𝐶^∞_𝑛=1𝑛^𝛽𝐸max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋^′_𝑛𝑖−𝐸𝑋^′_𝑛𝑖^𝑝≤𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝐸𝑋^′_𝑛𝑖^2𝑝/2+^𝑛_𝑖=1𝐸𝑋^′_𝑛𝑖^𝑝≐𝐽₁+𝐽₂. (3.16) Take 2𝑝>max2,1+𝛽𝑟(2−𝜃)−𝛼,𝜃+1+𝛼+𝛽𝑟, (3.17) which implies that 𝛽−[𝑟(2−𝜃)−𝛼]𝑝/2<−1 and 𝛼+𝛽−𝑟(𝑝−𝜃)<−1. By 𝐶_𝑟's inequality and Lemma 2.4, we can get 𝐽₁≐𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝐸𝑋^′_𝑛𝑖^2𝑝/2≤𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋>1+^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋²𝐼𝑎_𝑛𝑖𝑋≤1^𝑝/2. (3.18) If 1≤𝑠<2, then by Markov's inequality, 𝐸|𝑋|^𝑠<∞, and (3.7), we have 𝐽₁≤𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝑎^𝑠_𝑛𝑖𝐸|𝑋|^𝑠𝑝/2≤𝐶^∞_𝑛=1𝑛^𝛽sup_𝑖≥1𝑎_𝑛𝑖^{𝑛𝑠−𝜃}_𝑖=1𝑎^𝜃_𝑛𝑖^𝑝/2≤𝐶^∞_𝑛=1𝑛^𝛽𝑛^{−𝑟(𝑠−𝜃)}⋅𝑛^𝛼𝑝/2=𝐶^∞_𝑛=1𝑛^{𝛽−(((1+𝛽)𝑝)/2)}<∞since1+𝛽>0. (3.19) If 𝑠≥2, then by Markov's inequality, 𝐸|𝑋|^𝑠<∞, and (3.7) again, we have 𝐽₁≤𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝑎²_𝑛𝑖𝐸|𝑋|^2𝑝/2≤𝐶^∞_𝑛=1𝑛^𝛽sup_𝑖≥1𝑎_𝑛𝑖^𝑛2−𝜃_𝑖=1𝑎^𝜃_𝑛𝑖^𝑝/2≤𝐶^∞_𝑛=1𝑛^𝛽𝑛^{−𝑟(2−𝜃)}⋅𝑛^𝛼𝑝/2=𝐶^∞_𝑛=1𝑛^{𝛽−[𝑟(2−𝜃)−𝛼]𝑝/2}<∞. (3.20) From (3.18)–(3.20), we have proved that 𝐽₁<∞.
By Lemma 2.4 again and the definition of stochastic domination, we can see that 𝐽₂≐𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝐸𝑋^′_𝑛𝑖^𝑝≤𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑛𝑖^𝑝𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖≤1+𝑃𝑎_𝑛𝑖𝑋_𝑛𝑖>1≤𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋>1+𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋^𝑝𝐼𝑎_𝑛𝑖𝑋≤1≐𝐽₃+𝐽₄. (3.21)𝐽₃<∞ has been proved by (3.12). In the following, we will show that 𝐽₄<∞. Denote 𝐼_𝑛𝑗=𝑖∶𝑛𝑗^𝑟≤1/𝑎_𝑛𝑖<𝑛𝑗+1^𝑟,𝑛≥1,𝑗≥1. (3.22) It is easily seen that 𝐼_𝑛𝑘⋂𝐼_𝑛𝑗=∅ for 𝑘≠𝑗 and ⋃^∞_𝑗=1𝐼_𝑛𝑗=ℕ for all 𝑛≥1. Hence, 𝐽₄≐𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋^𝑝𝐼𝑎_𝑛𝑖𝑋≤1≤𝐶^∞_𝑛=1𝑛^𝛽∞_{𝑗=1𝑖∈𝐼𝑛𝑗}𝐸𝑎_𝑛𝑖𝑋^𝑝𝐼𝑎_𝑛𝑖𝑋≤1≤𝐶^∞_𝑛=1𝑛^𝛽∞_𝑗=1♯𝐼_𝑛𝑗𝑛𝑗^−𝑟𝑝𝐸|𝑋|^𝑝𝐼|𝑋|≤𝑛𝑗+1^𝑟≤𝐶^∞_𝑛=1𝑛^𝛽∞_𝑗=1♯𝐼_𝑛𝑗𝑛𝑗^{−𝑟𝑝𝑛(𝑗+1)}_𝑘=0𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1=𝐶^∞_𝑛=1𝑛^𝛽∞_𝑗=1♯𝐼_𝑛𝑗𝑛𝑗^{−𝑟𝑝2𝑛}_𝑘=0𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1+𝐶^∞_𝑛=1𝑛^𝛽∞_𝑗=1♯𝐼_𝑛𝑗𝑛𝑗^{−𝑟𝑝𝑛(𝑗+1)}_𝑘=2𝑛+1𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1≐𝐽₅+𝐽₆. (3.23) It is easily seen that for all 𝑚≥1, we have that 𝑛^𝛼=^∞_𝑖=1𝑎^𝜃_𝑛𝑖=^∞_{𝑗=1𝑖∈𝐼𝑛𝑗}𝑎^𝜃_𝑛𝑖≥^∞_𝑗=1♯𝐼_𝑛𝑗𝑛𝑗+1^−𝑟𝜃≥^∞_𝑗=𝑚♯𝐼_𝑛𝑗𝑛𝑗+1^−𝑟𝜃≥^∞_𝑗=𝑚♯𝐼_𝑛𝑗𝑛𝑗+1^−𝑟𝜃𝑛(𝑚+1)𝑛𝑗+1^{𝑟(𝑝−𝜃)}=^∞_𝑗=𝑚♯𝐼_𝑛𝑗𝑛𝑗+1^−𝑟𝑝[𝑛(𝑚+1)]^{𝑟(𝑝−𝜃)}, (3.24) which implies that for all 𝑚≥1, ^∞_𝑗=𝑚♯𝐼_𝑛𝑗𝑛𝑗^−𝑟𝑝≤𝐶𝑛^𝛼⋅𝑛^{−𝑟(𝑝−𝜃)}⋅𝑚^{−𝑟(𝑝−𝜃)}=𝐶𝑛^{𝛼−𝑟(𝑝−𝜃)}⋅𝑚^{−𝑟(𝑝−𝜃)}. (3.25) Therefore, 𝐽₅≐𝐶^∞_𝑛=1𝑛^𝛽∞_𝑗=1♯𝐼_𝑛𝑗𝑛𝑗^{−𝑟𝑝2𝑛}_𝑘=0𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1≤𝐶^∞_𝑛=1𝑛^𝛽⋅𝑛^{𝛼−𝑟(𝑝−𝜃)2𝑛}_𝑘=0𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1≤𝐶^2∞_𝑘=0𝑛=1𝑛^{𝛼+𝛽−𝑟(𝑝−𝜃)}𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1+𝐶^∞∞_{𝑘=2⌈⌉𝑛=𝑘/2}𝑛^{𝛼+𝛽−𝑟(𝑝−𝜃)}𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1≤𝐶+𝐶^∞_𝑘=2𝑘^{1+𝛼+𝛽−𝑟(𝑝−𝜃)}𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1≤𝐶+𝐶^∞_𝑘=2𝐸|𝑋|^{𝑝+((1+𝛼+𝛽)/𝑟)−(𝑝−𝜃)}𝐼𝑘≤|𝑋|^1/𝑟|𝑋|<𝑘+1≤𝐶+𝐶𝐸^{𝜃+((1+𝛼+𝛽)/𝑟)}𝐽<∞,₆≐𝐶^∞_𝑛=1𝑛^𝛽∞_𝑗=1♯𝐼_𝑛𝑗𝑛𝑗^{−𝑟𝑝𝑛(𝑗+1)}_𝑘=2𝑛+1𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1≤𝐶^∞_𝑛=1𝑛^𝛽∞_{𝑘=2𝑛+1()𝑗≥𝑘/𝑛−1}♯𝐼_𝑛𝑗𝑛𝑗^−𝑟𝑝𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1≤𝐶^∞_𝑛=1𝑛^𝛽∞_𝑘=2𝑛+1𝑛^{𝛼−𝑟(𝑝−𝜃)}𝑘𝑛^{−𝑟(𝑝−𝜃)}𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1≤𝐶^∞_𝑘=2^⌈𝑘/2⌉_𝑛=1𝑛^𝛼+𝛽⋅𝑘^{−𝑟(𝑝−𝜃)}𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1≤𝐶^∞_𝑘=2𝑘^{1+𝛼+𝛽−𝑟(𝑝−𝜃)}𝐸|𝑋|^𝑝𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1≤𝐶^∞_𝑘=2𝐸|𝑋|^{𝑝+((1+𝛼+𝛽)/𝑟)−(𝑝−𝜃)}𝐼𝑘≤|𝑋|^1/𝑟|𝑋|<𝑘+1≤𝐶𝐸^{𝜃+((1+𝛼+𝛽)/𝑟)}<∞. (3.26)(3.27) Thus, the inequality (3.15) follows from (3.16)–(3.21), (3.23), (3.26), and (3.27). The desired result (3.3) follows from (3.11), (3.12), and (3.15), immediately.
Case  2  (𝑠<1). We take 𝑝>0 such that 𝜃+(1+𝛼+𝛽)/𝑟=𝑠<𝑝<1, which implies that 𝛼+𝛽−𝑟(𝑝−𝜃)<−1. By Markov's inequality and 𝐶_𝑟's inequality, we have 𝐽≐^∞_𝑛=1𝑛^𝛽𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋^′_𝑛𝑖>𝜀≤𝐶^∞_𝑛=1𝑛^𝛽𝐸max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋^′_𝑛𝑖^𝑝≤𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝐸𝑋^′_𝑛𝑖^𝑝. (3.28) The rest proof is similar to the process of 𝐽₂<∞ in Case 1, so we omit the details.
(iii) If 1+𝛼+𝛽=0, then by Markov's inequality, (3.6), and similar to the process of (3.12), we can get that 𝐼≐^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋_𝑛𝑖>1≤𝐶^∞𝑘_𝑘=1𝑛=1𝑛⁻¹𝐸|𝑋|^𝜃𝐼𝑘^𝑟≤|𝑋|<(𝑘+1)^𝑟≤𝐶^∞_𝑘=1log𝑘𝐸|𝑋|^𝜃𝐼𝑘^𝑟≤|𝑋|<(𝑘+1)^𝑟≤𝐶^∞_𝑘=1𝐸|𝑋|^𝜃log|𝑋|𝐼𝑘^𝑟≤|𝑋|<(𝑘+1)^𝑟≤𝐶𝐸|𝑋|^𝜃log|𝑋|<∞. (3.29) Hence, to prove (3.3), we only need to show 𝐽<∞. We will still consider the Cases 𝑠≥1 and 𝑠<1. Here, 𝑠=𝜃.
Case  1  (𝑠≥1). Since 1+𝛼+𝛽=0, 1+𝛽=−𝛼≥0 and 𝑠=𝜃, it follows that (3.14) still holds. Thus, it suffices to show that 𝐽^∗<∞.
By Markov's inequality, 𝐶_𝑟's inequality, Lemma 2.3, Lemma 2.4, and (3.29), we have 𝐽^∗≐^∞_𝑛=1𝑛^𝛽𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋^′_𝑛𝑖−𝐸𝑋^′_𝑛𝑖>𝜀2≤𝐶^∞_𝑛=1𝑛^𝛽𝐸max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋^′_𝑛𝑖−𝐸𝑋^′_𝑛𝑖²≤𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝐸𝑋^′_𝑛𝑖²≤𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝑃𝑎_𝑛𝑖𝑋>1+𝐶^∞_𝑛=1𝑛^𝛽𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋²𝐼𝑎_𝑛𝑖𝑋≤1≤𝐶+𝐽^∗₅+𝐽^∗₆. (3.30) Here, 𝐽^∗₅ and 𝐽^∗₆ are 𝐽₅ and 𝐽₆ when 𝑝=2 in (ii), respectively. Notice that 𝛼+𝛽=−1 and 𝛼+𝛽−𝑟(2−𝜃)<−1, similar to the proof of 𝐽₅<∞, we have 𝐽^∗₅≤𝐶+𝐶𝐸|𝑋|^𝜃<∞, (3.31) and similar to the proof of 𝐽₆<∞, we have 𝐽^∗₆≤𝐶^∞_𝑘=2^⌈𝑘/2⌉_𝑛=1𝑛⁻¹⋅𝑘^{−𝑟(2−𝜃)}𝐸|𝑋|²𝐼𝑘≤|𝑋|^1/𝑟<𝑘+1≤𝐶^∞_𝑘=2log𝑘⋅𝑘^{−𝑟(2−𝜃)}𝐸|𝑋|²𝐼𝑘≤|𝑋|^1/𝑟|𝑋|<𝑘+1≤𝐶𝐸^𝜃log|𝑋|<∞. (3.32) Thus, 𝐽^∗<∞ follows from (3.30)–(3.32), immediately.
Case  2  (𝑠<1). The process of the proof is similar to that of Case 2 in (ii). We only need to show that 𝐽₅<∞ and 𝐽₆<∞. Actually, similar to the proof of (3.26), we have 𝐽₅≤𝐶+𝐶𝐸|𝑋|^𝜃<∞, (3.33) and similar to the proof of (3.27), we have 𝐽₆≤𝐶𝐸|𝑋|^𝜃log|𝑋|<∞. (3.34) This completes the proof of the theorem.

Remark 3.2. It is easily seen that the conditions (3.2), (3.4), and (3.6) in Theorem 3.1 are more general than the corresponding ones in Theorem 1.1. So Theorem 3.1 generalizes and improves the corresponding results of Theorem 3.1 in Baek et al. [4]. In addition, we not only consider the cases 1+𝛼+𝛽>0 and 1+𝛼+𝛽=0, we also consider the case 1+𝛼+𝛽<0. This complements the corresponding result of Baek et al. [4] and Wu [5].
By Theorems 3.1, we can extend the results of Baum and Katz [3] for independent and identically distributed random variables to the case of arrays of rowwise AANA random variables as follows.

Corollary 3.3. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise AANA random variables which is stochastically dominated by a random variable 𝑋 and 𝐸𝑋_𝑛𝑖=0 for all 𝑖≥1, 𝑛≥1.
(i) Let 𝛾>1 and 1≤𝑡<2. If 𝐸|𝑋|^𝛾𝑡<∞ and ∑^∞_𝑛=1𝑞^𝑞/𝑝(𝑛)<∞ for some 𝑝∈(3⋅2^𝑘−1,4⋅2^𝑘−1] and 𝑝>max2,2𝑡𝛾−12−𝑡,𝛾𝑡, (3.35) where integer number 𝑘≥1, then for all 𝜀>0, ^∞_𝑛=1𝑛^𝛾−2𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑛𝑖>𝜀𝑛^1/𝑡<∞. (3.36)
(ii) If 𝐸|𝑋|log|𝑋|<∞ and ∑^∞_𝑛=1𝑞²(𝑛)<∞, then for all 𝜀>0, ^∞_𝑛=11𝑛𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑛𝑖>𝜀𝑛<∞. (3.37)

Proof . (i) Let 𝑎_𝑛𝑖=0 if 𝑖>𝑛 and 𝑎_𝑛𝑖=𝑛^−1/𝑡 if 𝑖≤𝑛. Hence, conditions (3.1) and (3.2) hold for 𝜃=1, 𝑟=1/𝑡 and 𝛼=1−1/𝑡<𝑟. 𝛽≐𝑝−2>−1. It is easy to check that 11+𝛼+𝛽=𝑝−𝑡>0,1+1+𝛼+𝛽𝑟=𝑝𝑡≐𝑠. (3.38) Therefore, the desired result (3.36) follows from Theorem 3.1(ii), immediately.
(ii) Let 𝑎_𝑛𝑖=0 if 𝑖>𝑛 and 𝑎_𝑛𝑖=𝑛⁻¹ if 𝑖≤𝑛. Hence, conditions (3.1) and (3.2) hold for 𝑟=−1, 𝜃=1 and 𝛼=0. Therefore, the desired result (3.37) follows from Theorem 3.1(iii), immediately. This completes the proof of the corollary.

Similar to the proofs of Theorem 3.1 and Corollary 3.3, we can get the Baum and Katz type result for sequences of AANA random variables as follows.

Theorem 3.4. Let {𝑋_𝑛,𝑛≥1} be a sequence of AANA random variables which is stochastically dominated by a random variable 𝑋 and 𝐸𝑋_𝑛=0 for 𝑛≥1.
(i) Let 𝛾>1 and 1≤𝑡<2. If 𝐸|𝑋|^𝛾𝑡<∞ and ∑^∞_𝑛=1𝑞^𝑞/𝑝(𝑛)<∞ for some 𝑝∈(3⋅2^𝑘−1,4⋅2^𝑘−1] and 𝑝>max2,2𝑡𝛾−12−𝑡,𝛾𝑡, (3.39) where integer number 𝑘≥1, then for all 𝜀>0, ^∞_𝑛=1𝑛^𝛾−2𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑖>𝜀𝑛^1/𝑡<∞. (3.40)
(ii) If 𝐸|𝑋|log|𝑋|<∞ and ∑^∞_𝑛=1𝑞²(𝑛)<∞, then for all 𝜀>0, ^∞_𝑛=11𝑛𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑖>𝜀𝑛<∞. (3.41)

By Theorem 3.4, we can get the Marcinkiewicz-Zygmund type strong law of large numbers for AANA random variables as follows.

Corollary 3.5. Let {𝑋_𝑛,𝑛≥1} be a sequence of AANA random variables which is stochastically dominated by a random variable 𝑋 and 𝐸𝑋_𝑛=0 for 𝑛≥1.
(i) Let 𝛾>1 and 1≤𝑡<2. If 𝐸|𝑋|^𝛾𝑡<∞ and ∑^∞_𝑛=1𝑞^𝑞/𝑝(𝑛)<∞ for some 𝑝∈(3⋅2^𝑘−1,4⋅2^𝑘−1] and 𝑝>max2,2𝑡𝛾−12−𝑡,𝛾𝑡, (3.42) where integer number 𝑘≥1, then 𝑛^{𝑛−1/𝑡}_𝑖=1𝑋_𝑖⟶0a.s.𝑛⟶∞. (3.43)
(ii) If 𝐸|𝑋|log|𝑋|<∞ and ∑^∞_𝑛=1𝑞²(𝑛)<∞, then 1𝑛^𝑛_𝑖=1𝑋_𝑖⟶0a.s.𝑛⟶∞. (3.44)

Proof. (i) By (3.40), we can get that for all 𝜀>0, ∞>^∞_𝑛=1𝑛^𝛾−2𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑖>𝜀𝑛^1/𝑡=^∞2_𝑘=0^𝑘+1−1_𝑛=2𝑘𝑛^𝛾−2𝑃max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑖>𝜀𝑛^1/𝑡≥^∞𝑘=02^𝑘𝛾−22^𝑘𝑃max_{1≤𝑗≤2𝑘}^𝑗_𝑖=1𝑋_𝑖>𝜀2^{(𝑘+1)/𝑡},if𝛾≥2,_∞𝑘=0(2^𝑘+1)^𝛾−22^𝑘𝑃max_{1≤𝑗≤2𝑘}^𝑗_𝑖=1𝑋_𝑖>𝜀2^{(𝑘+1)/𝑡}≥,if1<𝛾<2,_∞𝑘=0𝑃max_{1≤𝑗≤2𝑘}^𝑗_𝑖=1𝑋_𝑖>𝜀2^{(𝑘+1)/𝑡}1,if𝛾≥2,2^∞_𝑘=0𝑃max_{1≤𝑗≤2𝑘}^𝑗_𝑖=1𝑋_𝑖>𝜀2^{(𝑘+1)/𝑡},if1<𝛾<2. (3.45) By Borel-Cantelli Lemma, we obtain that max_{1≤𝑗≤2𝑘}∑^𝑗_𝑖=1𝑋_𝑖2^{(𝑘+1)/𝑡}⟶0a.s.𝑘⟶∞. (3.46) For all positive integers 𝑛, there exists a positive integer 𝑘₀ such that 2^𝑘0−1≤𝑛<2^𝑘0. We have by (3.46) that ∑^𝑛_𝑖=1𝑋_𝑖𝑛^1/𝑡≤max_{2𝑘0−1≤𝑛<2𝑘0}∑^𝑛_𝑖=1𝑋_𝑖𝑛^1/𝑡≤2^2/𝑡max_{1≤𝑗≤2𝑘0}∑^𝑗_𝑖=1𝑋_𝑖2^𝑘0+1/𝑡⟶0a.s.𝑘₀⟶∞, (3.47) which implies (3.43).
(ii) Similar to the proof of (i), we can get (ii), immediately. The details are omitted. This completes the proof of the corollary.

Acknowledgments

The authors are most grateful to the editor Irena Rachunková and anonymous referees for careful reading of the manuscript and valuable suggestions which helped in significantly improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11171001 and 11126176), Natural Science Foundation of Anhui Province (1208085QA03), Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), the Academic Innovation Team of Anhui University (KJTD001B), Doctoral Research Start-up Funds Projects of Anhui University, and the Talents Youth Fund of Anhui Province Universities (2011SQRL012ZD).