Sufficient and necessary conditions of complete convergence for asymptotically negatively associated random variables

In this investigation, some sufficient and necessary conditions of the complete convergence for weighted sums of asymptotically negatively associated (ANA, in short) random variables are presented without the assumption of identical distribution. As an application of the main results, the Marcinkiewicz–Zygmund type strong law of large numbers based on weighted sums of ANA cases is obtained. The results of this paper extend and generalize some well-known corresponding ones.


Introduction
The complete convergence is a very important research field in probability limit theory of summation of random variables as well as weighted sums of random variables, which was first introduced by Hsu and Robbins [1] as follows: A sequence {X n ; n ≥ 1} of random variables converges completely to a constant λ if ∞ n=1 P(|X n -λ| > ε) < ∞ for all ε > 0. In view of the Borel-Cantelli lemma, this implies that X n → λ almost surely (a.s., in short). Hsu and Robbins [1] proved that the arithmetic means of independent and identically distributed (i.i.d., in short) random variables converges completely to the expected value of the summands, provided the variance is finite. Erdös [2] showed the converse. The Hsu-Robbins-Erdös theorem was generalized in different approaches. One of the most important generalizations was given by Baum and Katz [3] for the following strong law of large numbers. Theorem 1.1 Let 1 2 < α ≤ 1 and αp ≥ 1. Suppose that {X, X n ; n ≥ 1} is a sequence of i.i.d. random variables with EX n = 0. Then the following statements are equivalent: (1) E|X| p < ∞; (2) ∞ n=1 n αp-2 P max 1≤j≤n j i=1 X i > εn α < ∞ for all ε > 0. Peligrad and Gut [4] extended the result of Baum and Katz [3] for i.i.d. random variables toρ-mixing cases. Theorem 1.2 Let 1 2 < α ≤ 1 and αp > 1. Suppose that {X, X n ; n ≥ 1} is a sequence of identically distributedρ-mixing random variables with EX n = 0. Then the above equations (1) and (2) are also equivalent.
However, Peligrad and Gut [4] did not prove whether the result of Baum and Katz [3] for the case αp = 1 holds forρ-mixing random variables. Recently, Cai [5] complemented the result of Peligrad and Gut [4] for the case αp = 1. For more details about this type of complete convergence theorem, one can refer to Huang et al. [6], Wang and Hu [7], Deng et al. [8], Ding et al. [9], Wu et al. [10] among others.
In the following, some concepts of dependent structures are restated.
whenever this covariance exists. A sequence {X n ; n ≥ 1} of random variables is NA if every finite subfamily is NA.
The notion of NA random variables was introduced by Alam and Saxena [11] and carefully studied by Joag-Dev and Proschan [12]. As pointed out and proved by Joag-Dev and Proschan [12], a number of well-known multivariate distributions possess the NA property. Definition 1.2 A sequence {X n ; n ≥ 1} of random variables is calledρ-mixing if, for some integer n ≥ 1, the mixing coefficient where the outside sup is taken over all pairs of nonempty finite sets S and T of integers such that min{|s -t|, s ∈ S, t ∈ T} ≥ n and σ (S) = σ {X i ; i ∈ S}. Definition 1.3 A sequence {X n ; n ≥ 1} of random variables is called asymptotically negatively associated (ANA, in short) if and C is the set of nondecreasing for every variable functions.
It is obvious that ρ -(n) ≤ρ(n), and a sequence of ANA random variables is NA if and only if ρ -(1) = 0. Compared with NA andρ-mixing, ANA cases define a strictly larger class of random variables (for detailed examples, see [13]). Consequently, extending and improving the convergence theorems for NA andρ-mixing random variables to the wider ANA cases is highly desirable in the theory and applications.
In the past decade, many probabilists and statisticians studied and established a series of important results for ANA random variables. For example, see Zhang and Wang [13], Zhang [14,15] for some moment inequalities of partial sums, the central limit theorems, and the complete convergence, Kim et al. [16] for the strong law of large numbers, Wang and Lu [17] for some moment inequalities of the maximum of partial sums, Wang and Zhang [18] for a Berry-Esséen theorem and the law of the iterated logarithm, Liu and Liu [19] for the moments of the maximum of normed partial sums, Budsaba et al. [20] for the complete convergence for moving average process based on a sequence of ANA and NA random variables, Yuan and Wu [21] for the limiting behavior for ANA random variables under residual Cesàro alpha-integrability assumption, Huang et al. [22] for the complete convergence and the complete moment convergence, Wu and Jiang [23] for the almost sure convergence, and so forth.
Let {X n ; n ≥ 1} be a sequence of random variables defined on a fixed probability space (Ω, F, P), and let {a n ; n ≥ 1} be a sequence of real numbers. The probability limit behavior of the maximum weighted sum max 1≤j≤n j i=1 a i X i is very useful in applied probability theory and mathematical statistics. In the theoretical statistical frameworks, many useful linear statistics are based on weighted sums of random samples. For example, least-squares estimators, nonparametric regression function estimators, jackknife estimators, and so on. For that reason, studying the convergence properties for weighted sums of random variables is of much interest.
In this paper, the authors discuss the strong convergence of ANA random variables without identical distributions, and provide some equivalent conditions of Baum-Katz type complete convergence theorem for weighted sums of ANA cases. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of ANA random variables is also obtained. The main results of this paper extend and improve the known corresponding ones of Peligrad and Gut [4], Cai [5], and Wu and Jiang [23], respectively.
The definition of stochastic domination, which is used frequently throughout this paper, is as follows. Definition 1.4 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 for all x ≥ 0 and n ≥ 1.
Throughout this paper, the symbols C, C 1 , C 2 , . . . will represent generic positive constants which may be different in various places, and a n = O(b n ) will mean a n ≤ Cb n for all n ≥ 1. I(A) is the indicator function on the set A. [x] denotes the integer part of x.

Main results and proofs
In this section, we will first restate some preliminary lemmas which are useful to proving the main results of this paper.

Lemma 2.1
Increasing or decreasing functions defined on disjoint subsets of a sequence of {X n ; n ≥ 1} of ANA random variables with the mixing coefficients ρ -(n) are also ANA random variables with the mixing coefficients not greater than ρ -(n). [17]) For some positive integers n ∈ N and 0 ≤ s < 1 12 , suppose that {X n ; n ≥ 1} is a sequence of ANA random variables with ρ -(n) ≤ s, EX n = 0, and E|X n | 2 < ∞. Then there exists a positive constant C = C(2, n, s) for all n ≥ 1 such that

Lemma 2.3
For some positive integers n ∈ N and 0 ≤ s < 1 12 , suppose that {X n ; n ≥ 1} is a sequence of ANA random variables with ρ -(n) ≤ s. Then there exists a positive constant C such that, for all x > 0 and n ≥ 1, . Without loss of generality, assume that α n > 0. It follows that {I( are two sequences of ANA random variables with the mixing coefficients not greater than ρ -(n) ≤ s by Lemma 2.1. Hence, by the C r inequality and Lemma 2.2, we can have Hence, by Hölder's inequality and (2.3), we also have that By reorganizing the above inequality, the desired result (2.2) follows immediately.

Lemma 2.4
Let {X n ; n ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable X. Then, for all α > 0, b > 0, and n ≥ 1, the following statements hold: where C 1 and C 2 represent different positive constants.
Now we state and prove the main results of this paper.
Theorem 2.1 Let 0 < p < 2, α > 1 2 , αp > 1, and 0 ≤ s < 1 12 . Suppose that {X n ; n ≥ 1} is a sequence of ANA random variables with the mixing coefficients ρ -(n) ≤ s, which is stochastically dominated by a random variable X. Assume further that EX n = 0 if 1 ≤ p < 2 for all n ≥ 1. Let {a n ; n ≥ 1} be a sequence of real numbers such that n i=1 (2.7) Proof of Theorem 2.1 The proof is primarily inspired by Wang and Wu [24]. Without loss of generality, assume that a n ≥ 0 for all n ≥ 1. For all 0 < γ ≤ 2, For all i ≥ 1 and n ≥ 1, define Therefore, for fixed n ≥ 1, {X ni -EX ni ; i ≥ 1} is still a sequence of ANA random variables by Lemma 2.1. For all ε > 0, it easily follows that In the following, we will proceed with three cases. Case 1: For α > 1 2 , αp > 1, and 1 < p < 2. Firstly, we will show that Ea i X ni → 0 as n → ∞. (2.9) Note that |Y ni | ≤ |X i |I(|X i | > n α ) and EX n = 0 for all n ≥ 1, Hence, for n large enough and all ε > 0, (2.11) To prove (2.7), it suffices to show that By some standard computations, we can easily have that (2.14) For I 1 , it follows from the Markov inequality, Lemma 2.2, (2.5) of Lemma 2.4 that (2.16) Hence, by an argument similar to those in the proofs of (2.14) and (2.15), we also have I 1 < CE|X| < ∞ and I 2 < CE|X| < ∞.
The following theorem provides the necessary condition of complete convergence for weighted sums of ANA random variables. Theorem 2.2 Let 0 < p < 2, α > 1 2 , αp > 1, and 0 ≤ s < 1 12 . Suppose that {X n ; n ≥ 1} is a sequence of ANA random variables with the mixing coefficients ρ -(n) < s. Assume that there exist a random variable X and some positive constant C 1 such that C 1 P(|X| > x) ≤ inf n≥1 P(|X n | > x) for all x ≥ 0. Assume further that EX n = 0 if 1 ≤ p < 2. Let {a n ; n ≥ 1} be a sequence of real numbers such that n i=1 |a i | 2 = O(n). Then (2.7) implies E|X| p < ∞ for all ε > 0.

Proof of Theorem 2.2 Noting that
For αp > 1, it follows that The proof of Theorem 2.2 is completed.
The following two theorems treat the case αp = 1. Theorem 2.3 Let 1 2 < α ≤ 1 and 0 ≤ s < 1 12 . Suppose that {X n ; n ≥ 1} is a sequence of mean zero ANA random variables with the mixing coefficients ρ -(n) ≤ s, which is stochastically dominated by a random variable X. Let {a n ; n ≥ 1} be a sequence of real numbers such that (2.24) Proof of Theorem 2.3 By applying the same notations as those in the proof of Theorem 2.1, we will first show (2.9). For 1 2 < α ≤ 1, note that 1 ≤ p = 1 α < 2 if αp = 1. Therefore, by (2.6) of Lemma 2.4 and EX n = 0, we have that The rest of the proof is similar to those of Case 1 and Case 2 in Theorem 2.1, we also have that I 1 ≤ CE|X| p < ∞ and I 2 ≤ CE|X| p < ∞. The proof of Theorem 2.3 is completed. (2.28) In view of the Borel-Cantelli lemma, we also have that Remark 2.1 Taking a n = 1 for all n ≥ 1 in Theorems 2.1-2.4 above, we can also obtain the Baum and Katz type complete convergence theorem for ANA random variables under the cases of 0 < p < 2, α > 1 2 , αp > 1 and 1 2 < α ≤ 1, αp = 1, respectively. Since ANA random variables includeρ-mixing random variables and NA random variables, the main results of this paper also hold forρ-mixing and NA cases. Hence, Theorems 2.1-2.4 extend the corresponding ones of Peligrad and Gut [4] and Cai [5] to the weighted sums.
Remark 2.2 Wu and Jiang [23] also investigated the almost sure convergence for identically distributed ANA random variables and obtained the Marcinkiewicz-Zygmund type strong law of large numbers under E|X| p < ∞ for 0 < p < 2. Compared with their result, it is worth pointing out that we establish some much stronger convergence results for weighted sums of ANA random variables without the assumption of identical distribution, which can imply the corresponding one of Wu and Jiang [23].