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.

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.

Let

Suppose that

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 [

A sequence

It is easily seen that the family of AANA sequence contains negatively associated (NA, in short) sequences (with

Since the concept of AANA sequence was introduced by Chandra and Ghosal [

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 [

Throughout this paper, let

The definition of stochastic domination will be used in the paper as follows.

A sequence

Our main results are as follows.

Suppose that

Theorem

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 [

Let

To prove the main results of the paper, we need the following lemmas. The first two lemmas were provided by Yuan and An [

Let

Let

If

If

The last one is a fundamental property for stochastic domination. The proof is standard, so the details are omitted.

Let

Without loss of generality, we assume that

Firstly, we will show that

For any

Secondly, we will prove that

To prove (

Denote for

The authors are most grateful to the Editor Binggen Zhang and anonymous referee for the 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 11126176), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20093401120001), the Natural Science Foundation of Anhui Province (11040606M12, 1208085QA03), the Natural Science Foundation of Anhui Education Bureau (KJ2010A035), the 211 project of Anhui University, the Academic Innovation Team of Anhui University (KJTD001B), and the Students Science Research Training Program of Anhui University (KYXL2012007).