Abstract

In this work, the complete moment convergence and L_p convergence for asymptotically almost negatively associated (AANA, in short) random variables are investigated. As an application, the complete convergence theorem for weighted sums of AANA random variables is obtained. These theorems obtained extend and improve some earlier results.

1. Introduction

Definition 1.1 A finite collection of random variables is said to be negatively associated (NA, in short) if for every pair of disjoint subsets and of

(1.1)

whenever and are any real coordinatewise nondecreasing functions such that this covariance exists. An infinite sequence of random variables is said to be NA if for every finite sub-collection is NA.

The concept of NA was introduced by Joag-Dev and Proschan ⁴, and its probability limit properties have aroused wide interest because of their numerous applications in reliability theory, percolation theory and multivariate statistical analysis. By inspecting the proof of maximal inequality for NA random variables in Matula ⁷, one can also allow negative correlations provided they are small. Primarily motivated by this, Chandra and Ghosal ^{1, 2} introduced the following dependence.

Definition 1.2 A sequence of random variables is called AANA if there exists a nonnegative sequence as such that

(1.2)

for all and for all coordinatewise nondecreasing continuous functions and whenever the variances exist.

The family of AANA sequence contains NA (in particular, independent) sequence (with ) and some more sequences of random variables which are not much deviated from being NA. Chandra and Ghosal ¹ once pointed out that NA implies AANA, but AANA does not imply NA. Namely, AANA is much weaker than NA. Since, NA has been applied to the reliability theory, multivariate statistical analysis and percolation theory, and attracted extensive attentions. Hence, extending the limit properties of NA random variables to the wider case of AANA random variables is highly desirable in the theory and applications.

For recent various results and applications of AANA random variables, we can refer to that Chandra and Ghosal ¹ obtained the Kolmogorov type inequality and the strong law of large numbers of Marcinkiewicz-Zygmund; Chandra and Ghosal ² established the almost sure convergence of weighted averages; Wang et al. ¹⁰ obtained the law of the iterated logarithm for product sums; Ko et al. ⁵ studied the Hájek-Rényi type inequality; Yuan and An ¹⁴ established some Rosenthal type inequalities; Yuan and Wu ¹⁵ studied the limiting behavior of the maximum of the partial sum under residual Cesàro alpha-integrability assumption; Wang et al. ^{11, 12}, Huang et al. ³ studied the complete convergence of weighted sums for arrays of rowwise AANA random variables and arrays of rowwise AANA random variables, respectively; Yang et al. ¹⁶ investigated the complete convergence of moving average process for AANA sequence; and Tang ⁹ studied the strong law of large numbers for general weighted sums, Shen and Wu ⁸ obtained some new complete convergence results and Feller-type weak law of large numbers, and so forth.

Recently, Liang and Su ⁶ obtained the following complete convergence result for weighted sums of NA random variables.

Theorem A Let be a sequence of NA random variables with be an array of real numbers such that and for If then for ,

(1.3)

Wang et al. ¹³ extended the result of Liang and Su ⁶ to AANA random variables as follows.

Theorem B Let be a sequence of AANA random variables with and for Let be an array of real numbers satisfying and for some and where integer number Then for and

(1.4)

This paper is motivated by Liang and Su ⁶, Wang et al. ¹³. We further study the convergence properties for AANA random variables and establish the complete moment convergence theorem and convergence theorem. As an application, the complete convergence and strong law of large numbers for weighted sums of AANA random variables are obtained. The obtained results extend and improve the above Theorem A and Theorem B.

The structure of this paper is as follows. In Section 2, some important lemmas are firstly provided and the complete moment convergence theorem for AANA random variables is presented. In Section 3, the convergence theorem is provided.

Throughout this paper, let be a sequence of AANA random variables with the mixing coefficients Let be the indicator function of the set The symbol will denote a positive constant which is not necessarily the same one in each appearance, will stand for

2. Complete Moment Convergence

In order to prove our main results, the following lemmas are needed.

Lemma 2.1 (Yuan and An ¹⁴) Let be a sequence of AANA random variables with the mixing coefficients fn be all nondecreasing (or all nonincreasing) continuous functions, then is still a sequence of AANA random variables with the mixing coefficients

(2.1)

Theorem 2.1 Let be a sequence of AANA random variables with and for all Let be a sequence of positive real numbers. For some constant and if

(2.2)

(2.3)

(2.4)

Then for

(2.5)

Theorem 2.2 Let be a sequence of AANA random variables with and for all Let be an array of real numbers such that for as Then for and

(2.6)

Corollary 2.1 Under the conditions of Theorem 2.2, then

(2.7)

Proof of Theorem 2.1 For define

From Lemma 2.1, the sequences of and are still AANA random variables. For then

(2.8)

To prove (2.5), it needs only to show that and When for

(2.9)

It follows from (2.2) that

Note that it follows from and (2.4) that

By Markov inequality, (2.1) and inequality, we can obtain that

(2.10)

Hence, the desired result of follows from (2.10) immediately.

Secondly, we will show that For define that

By the similar argument as the proof of (2.9), we can obtain that

By (2.2), it follows that

(2.11)

Note that it follows from and (2.4) that

which implies

(2.12)

For convenience, let denote

Hence,

(2.13)

Let it follows from and (2.2) that

(2.14)

Hence, the desired result of follows from (2.2) and the above statements immediately. The proof of Theorem 2.1 is completed.

Proof of Theorem 2.2 Let and in Theorem 2.1, then

(2.15)

(2.16)

It follows from and that

(2.17)

The proof of Theorem 2.2 is completed.

3. L_p Convergence

In this section, we will state the L_p convergence under some conditions.

Theorem 3.1 Let be a sequence of AANA random variables with and For suppose that for some

(3.1)

(3.2)

Then,

(3.3)

Proof of Theorem 3.1 For we use the same notations of Theorem 2.1,

It follows from inequality and that

Note that it follows from Lemma 2.2 and (3.1) that

(3.4)

By inequality, Lemma 2.2, (3.1) and (3.2), we can obtain that

(3.5)

From (3.4) and (3.5), we can obtain and as The proof of Theorem 3.1 is completed.

Take in Theorem 3.1, we can immediately obtain the following result.

Corollary 3.1 Let be a sequence of AANA random variables with and For suppose that for some

(3.6)

(3.7)

Then,

(3.8)

Corollary 3.2 Let be a sequence of AANA random variables with and For suppose that for some

(3.9)

Then (3.2) and (3.9) imply

(3.10)

Proof of Corollary 3.2 For we use the same notation of Theorem 2.1, It follows from inequality and that

(3.11)

By Markov inequality, Lemma 2.2, (3.9) and (3.2), we can obtain that

(3.12)

Take into account the definition of and (3.9), we can obtain that

(3.13)

The proof of Corollary 3.2 is completed.

Support

This work is supported by the National Nature Science Foundation of China (11526085), the Humanities and Social Sciences Foundation for the Youth Scholars of Ministry of Education of China (15YJCZH066), the Science and Technology Plan Project of Hunan Province (2016TP1020), the Construct Program of the Key Discipline in Hunan Province.

References