Lp-convergence, complete convergence, and weak laws of large numbers for asymptotically negatively associated random vectors with values in Rd

In this paper, based on the Rosenthal-type inequality for asymptotically negatively associated random vectors with values in Rd, we establish results on Lp-convergence and complete convergence of the maximums of partial sums are established. We also obtain weak laws of large numbers for coordinatewise asymptotically negatively associated random vectors with values in Rd.

Introduction

Ko et al. [1] introduced the concept of negative association (NA) for -valued random vectors.

Definition 1.1

(Ko et al. [1])

A finite sequence of -valued random vectors is said to be negatively associated (NA) if for any disjoint nonempty subsets and any nondecreasing functions f on and g on , whenever the covariance exists. Let denote the cardinality of a set A. An infinite sequence of -valued random vectors is negatively associated if every finite subsequence is negatively associated.

In the case of , the concept of negative association had already been introduced by Joag-Dev and Proschan [2]. A number of well-known multivariate distributions possess the NA property, such as the multinomial distribution, multivariate hypergeometric distribution, negatively correlated normal distribution, and joint distribution of ranks.

In addition to Definition 1.1, for random vectors in , we can define asymptotically negative association (ANA).

Definition 1.2

A sequence of -valued random vectors is said to be asymptotically negatively associated (ANA) if where , and f on and g on are any real coordinatewise nondecreasing functions.

In the case of , the concept of asymptotically negative association was proposed by Zhang [3, 4] and studied by Yuan and Wu [5].

It is obvious that a sequence of asymptotically negatively associated random variables is negatively associated if and only if . Compared to negative association, asymptotically negative association defines a strictly larger class of random variables (for detailed examples, see Zhang [3, 4]). Consequently, the study of the limit theorems for asymptotically negatively associated random variables is of much interest.

We refer to Zhang [4] for the central limit theorem, Wang and Lu [6] for some inequalities of maximums of partial sums and weak convergence, Wang and Zhang [7] for the Berry–Esseen theorem and the law of the iterated logarithm, Yuan and Wu [5] for the -convergence and complete convergence of the maximums of the partial sums, among others.

The concept of coordinatewise negative association (CNA) for random vectors with values in was introduced as follows. Let denote the inner product, and let be an orthonormal basis. A sequence of -valued random vectors is said to be coordinatewise negatively associated (CNA) if for each j , the sequence of random variables is NA, where .

As in the definition of CNA, we can define coordinatewise asymptotically negative association for random vectors with values in .

Definition 1.3

A sequence of -valued random vectors is said to be coordinatewise asymptotically negatively associated (CANA) if for each j , the sequence of random variables is asymptotically negatively associated, where for and .

It is clear that if a sequence of -valued random vectors is ANA, then it is CANA. However, in general, the converse is not true.

Let be a sequence of -valued random vectors. We consider the following inequalities for : If there exists a positive constant , such that the left-hand (right-hand) side of (1.3) is satisfied for all , , and , then the sequence is said to be coordinatewise weakly lower (upper) bounded by X. The sequence is said to be coordinatewise weakly bounded by X if it is both coordinatewise lower and upper bounded by X (see Huan et al. [8]).

In Sect. 2, we give some lemmas, which will be used to prove the main results, and in Sect. 3, we prove the -convergence and complete convergence results for the maximums of the partial sums of the sequence of ANA random vectors with values in . In addition, in Sect. 4, we establish a weak law of large numbers for CANA random vectors with values in .

Throughout the paper, the symbol C denotes a generic constant , which is not necessarily the same in each occurrence, for a sequence of random vectors, and denotes the -norm. Moreover, ≪ represents the Vinogradov symbol O, and is the indicator function.

Some lemmas

From the definition of a sequence of ANA random vectors, we have the following:

Lemma 2.1

(Yuan and Wu [5])

Nondecreasing (or nonincreasing) functions defined on disjoint subsets of a sequence of ANA random vectors with mixing coefficients is also ANA with mixing coefficients not greater than .

Wang and Lu [6] proved the following Rosenthal-type inequality for a sequence of ANA random variables in .

Lemma 2.2

For a positive integer , positive real numbers , and , if is a sequence of ANA random variables with , , and for every , then there is a positive constant such that, for all ,

Inspired by the proof of Lemma 2.3 in Li-Xin Zhang [9], we extend Lemma 2.2 to -valued ANA random vectors as follows.

Lemma 2.3

For a positive integer , positive real numbers , and , if is a sequence of -valued ANA random vectors with , , and for every , then there is a positive constant such that, for all ,

Proof

Note that and by Lemma 2.2 Hence (2.2) follows. □

From Lemma 1.2 of Kuczmaszewska [10] we obtain the following lemma.

Lemma 2.4

Let be a sequence of -valued random vectors weakly upper bounded by a random vector X, and let . Then, for some constant , implies .

The following lemma supplies us with the analytical part in the proofs of the theorems in the subsequent sections.

Lemma 2.5

(Yuan and Wu [5])

Let and be sequences of nonnegative numbers. If then for every .

Next, we will extend some -convergence and complete convergence results for the maximums of the partial sum of -valued ANA random variables in Yuan and Wu [5] to -valued random vectors.

-convergence and complete convergence for ANA random vectors with values in

The following theorem is an extension of Theorem 3.2 in Yuan and Wu [5] to random vectors with values in .

Theorem 3.1

Let be positive real numbers, and let be a positive integer. Suppose that is a sequence of -valued ANA random vectors with mixing coefficients such that . If are -valued random vectors satisfying then, for any ,

By Lemma 2.3, the Hölder inequality, and (3.1) we obtain which by Lemma 2.5 yields (3.2) for any . □

As applications of Theorem 3.1, we introduce two results that are not present in Yuan and Wu [5].

Theorem 3.2

Let be positive real numbers, and let be a positive integer. Suppose that is a sequence of -valued ANA random vectors with mixing coefficients such that . If is weakly upper bounded by a random vector X with , then, for any , (3.2) holds.

By Lemma 2.3, Lemma 2.4(i), Hölder’s inequality, and the proof of Theorem 3.1 we obtain  □

Corollary 3.3

Let be positive real numbers, and let be a positive integer. Suppose that is a sequence of -valued ANA random vectors with mixing coefficients such that . If are identically distributed random vectors with , then, for any , (3.2) holds.

A sequence of random vectors is said to converge completely to a constant a if for any , In this case, we write completely. This notion was given by Hsu and Robbins [11]. Note that the complete convergence implies the almost sure convergence in view of the Borel–Cantelli lemma.

The following theorem provides an extension of Theorem 4.2 of Yuan and Wu [5] for ANA random variables in to random vectors in .

Theorem 3.4

Let be positive real numbers, and let be a positive integer. Suppose that is a sequence of -valued ANA random vectors with mixing coefficients such that . If satisfies (3.1), then, for any ,

By Lemma 2.3, Lemma 2.5, Hölder’s inequality, and the proof of Theorem 3.1 we obtain Hence (3.4) holds. □

Remark

Note that the proof of Theorem 3.4 is a little different from that of Theorem 4.2 in Yuan and Wu [5].

As applications of Theorem 3.4, we introduce two results that are not present in Yuan and Wu [5].

Theorem 3.5

Let be positive real numbers, and let be a positive integer. Suppose that is a sequence of -valued ANA random vectors with mixing coefficients such that . If is weakly upper bounded by a random vector X with , then, for any , (3.4) holds.

As in the proof of Theorem 3.4, we obtain  □

Corollary 3.6

Let be positive real numbers, and let be a positive integer. Suppose that is a sequence of -valued ANA random vectors with mixing coefficients such that . If are identically distributed random vectors with , then, for any , (3.4) holds.

Weak law of large numbers for ANA random vectors with values in

In this section, we establish the weak laws of large numbers for -valued ANA random vectors when .

We assume that is a sequence of ANA random vectors with values in . For and , we set

Theorem 4.1

Let be positive real numbers, and let be a positive integer. Suppose that is a sequence of -valued ANA random vectors with mixing coefficients such that and . If and then we obtain the weak law of large numbers

By the standard method we obtain Thus Next, we will show that It is well known that, for all , is a sequence of -valued ANA random vectors by Lemma 2.1. Then, by the Markov inequality, Hölder’s inequality, and Lemma 2.3 we have which yields (4.6). Combining (4.5) and (4.6), the WLLN (4.3) follows. The proof is complete. □

Theorem 4.2

Let be positive real numbers, and let be a positive integer. Suppose that is a sequence of -valued CANA random vectors with mixing coefficients such that and . If is coordinatewise weakly upper bounded by a random vector X with then the WLLN (4.3) holds.

We first show that (4.5) holds. By (1.3) and (4.4) we obtain which yields (4.5). It remains to show that (4.6) holds.

Since for all , is a sequence of -valued ANA random vectors, by Lemma 2.1, Lemma 2.3, and (1.3) we have which yields (4.6). Combining (4.5) and (4.6), we obtain the WLLN (4.3). Hence the proof is complete. □

Suppose that is a sequence of -valued CNA random vectors. If is coordinatewise weakly upper bounded by a random vector X with , then the WLLN (4.3) holds.

Corollary 4.3

Let be a positive integer, and let . Suppose that is a sequence of -valued ANA random vectors with mixing coefficient such that and . If is a sequence of identically distributed random vectors with then (4.3) holds.

Theorem 4.4

Let , and let be an integer. Suppose that is a sequence of mean zero -valued ANA random vectors with mixing coefficients such that and . If is coordinatewise weakly upper bounded by a random vector X with then

It follows from (4.10) that which yields It remains to prove (4.3).

Since (4.12) implies (4.8), (4.3) follows from Theorem 4.2. Thus the proof is complete. □

Suppose that is a zero-mean sequence of -valued NA random vectors. If is coordinatewise weakly upper bounded by a random vector X with , then (4.11) holds.

Corollary 4.5

Let be an integer, and let . Suppose that is a sequence of -valued ANA random vectors with mixing coefficient such that and . If is a sequence of identically distributed random vectors with and then we obtain the WLLN (4.11).

The proof follows by substituting by in the proof of Theorem 4.4. □

Conclusions

We generalized the -convergence and complete convergence results of Yuan and Wu [5] from -valued ANA random variables to -valued random vectors by using a Rosenthal-type inequality. We also established weak laws of large numbers for CANA random vectors under . As applications, we obtained some -convergence and complete convergence results that are not present in Yuan and Wu [5] even when .