Strong and Weak Convergence for Asymptotically Almost Negatively Associated Random Variables

The strong law of large numbers for sequences of asymptotically almost negatively associated (AANA, in short) random variables is obtained, which generalizes and improves the corresponding one of Bai and Cheng (2000) for independent and identically distributed random variables to the case of AANA random variables. In addition, the Feller-type weak law of large number for sequences of AANA random variables is obtained, which generalizes the corresponding one of Feller (1946) for independent and identically distributed random variables.


Introduction
Many useful linear statistics based on a random sample are weighted sums of independent and identically distributed random variables.Examples include least-squares estimators, nonparametric regression function estimators, and jackknife estimates,.In this respect, studies of strong laws for these weighted sums have demonstrated significant progress in probability theory with applications in mathematical statistics.
Theorem A. Suppose that 1 < ,  < ∞, 1 ≤  < 2, and 1/ = 1/ + 1/.Let {,   ,  ≥ 1} be a sequence of independent and identically distributed random variables satisfying  = 0, and let {  , 1 ≤  ≤ ,  ≥ 1} be an array of real constants such that lim sup = 0 a.s. ( We point out that the independence assumption is not plausible in many statistical applications.So it is of interest to extend the concept of independence to the case of dependence.One of these dependence structures is asymptotically almost negatively associated, which was introduced by Chandra and Ghosal [9] as follows. Definition 1.A sequence {  ,  ≥ 1} of random variables is called asymptotically almost negatively associated (AANA, in short) if there exists a nonnegative sequence () → 0 as for all ,  ≥ 1 and for all coordinatewise nondecreasing continuous functions  and  whenever the variances exist.
It is easily seen that the family of AANA sequence contains negatively associated (NA, in short) sequences (with () = 0,  ≥ 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].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 [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; Wang et al. [11] established the law of the iterated logarithm for product sums; Ko et al. [12] studied the Hájek-Rényi type inequality; Yuan and An [13] established some Rosenthal type inequalities for maximum partial sums of AANA sequence; Wang et al. [14] obtained some strong growth rate and the integrability of supremum for the partial sums of AANA random variables; Wang et al. [15,16] studied complete convergence for arrays of rowwise AANA random variables and weighted sums of arrays of rowwise AANA random variables, respectively; Hu et al. [17] studied the strong convergence properties for AANA sequence; Yang et al. [18] investigated the complete convergence, complete moment convergence, and the existence of the moment of supermum 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 strong convergence for AANA random variables, which generalizes and improves the result of Theorem A. In addition, we will give the Feller-type weak law of large number for sequences of AANA random variables, which generalizes the corresponding one of Feller [19] for independent and identically distributed random variables.
Throughout this paper, let {  ,  ≥ 1} be a sequence of AANA random variables with the mixing coefficients {(),  ≥ 1}.  = ∑  =1   .For  > 1, let  ≐ /( − 1) be the dual number of .The symbol  denotes a positive constant which may be different in various places.Let () be the indicator function of the set .   = (  ) stands for   ≤   .
The definition of stochastic domination will be used in the paper as follows.
Definition 2. A sequence {  ,  ≥ 1} of random variables is said to be stochastically dominated by a random variable  if there exists a positive constant  such that for all  ≥ 0 and  ≥ 1.
Our main results are as follows.
Theorem 3. Suppose that 0 < , < ∞, 0 <  < 2, and 1/ = 1/ + 1/.Let {  ,  ≥ 1} be a sequence of AANA random variables, which is stochastically dominated by a random variable  and   = 0, if  > 1. Suppose that there exists a positive integer  such that Remark 4. Theorem 3 generalizes and improves Theorem A of Bai and Cheng [3] for independent and identically distributed random variables to the case of AANA random variables, since Theorem 3 removes the identically distributed condition and expands the ranges , , and , respectively.
At last, we will present the Feller-type weak law of large number for sequences of AANA random variables, which generalizes the corresponding one of Feller [19] for independent and identically distributed random variables.Theorem 5. Let  > 1/2 and {,   ,  ≥ 1} be a sequence of identically distributed AANA random variables with the mixing coefficients then

Preparations
To prove the main results of the paper, we need the following lemmas.The first two lemmas were provided by Yuan and An [13].
, where integer number k ≥ 1, then there exists a positive constant   depending only on p such that for all  ≥ 1,  (max The last one is a fundamental property for stochastic domination.The proof is standard, so the details are omitted.
where  1 and  2 are positive constants.
To prove (6), it suffices to show that It follows by Markov's inequality and the fact ||  < ∞ that It is easily seen that Hence, we have by ( 28) and (29) that which together with  1 < ∞ yields (23).This completes the proof of the theorem.(38) This completes the proof of the theorem.

Lemma 8 .
Let {  ,  ≥ 1} be a sequence of random variables, which is stochastically dominated by a random variable .