\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{p}$\end{document}Lp-convergence, complete convergence, and weak laws of large numbers for asymptotically negatively associated random vectors with values in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{d}$\end{document}Rd

In this paper, based on the Rosenthal-type inequality for asymptotically negatively associated random vectors with values in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{d}$\end{document}Rd, we establish results on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{p}$\end{document}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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{d}$\end{document}Rd.

In addition to Definition 1.1, for random vectors in R d , we can define asymptotically negative association (ANA). where dist(S, T) = min{|x -y|; x ∈ S, y ∈ T}, ρ -(S, T) = 0 ∨ Cov(f (X i , i ∈ S), g(X j , j ∈ T)) (Var f (X i , i ∈ S)) 1 2 (Var g(X j , j ∈ T)) 1 2 , and f on R |S|d and g on R |T|d are any real coordinatewise nondecreasing functions.
In the case of d = 1, 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 ρ -(1) = 0. 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 L p -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 R d was introduced as follows. Let ·, · denote the inner product, and let {e j , j ≥ 1} be an orthonormal basis. A sequence {X n , n ≥ 1} of R d -valued random vectors is said to be coordinatewise negatively associated (CNA) if for each As in the definition of CNA, we can define coordinatewise asymptotically negative association for random vectors with values in R d .
n , n ≥ 1, 1 ≤ j ≤ d} of random variables is asymptotically negatively associated, where X (j) n = X n , e j for n ≥ 1 and 1 ≤ j ≤ d.
It is clear that if a sequence of R d -valued random vectors is ANA, then it is CANA. However, in general, the converse is not true.
Let {X, X n , n ≥ 1} be a sequence of R d -valued random vectors. We consider the following inequalities for 1 ≤ j ≤ d: If there exists a positive constant C 1 , (C 2 ) such that the left-hand (right-hand) side of (1.3) is satisfied for all 1 ≤ j ≤ d, n ≥ 1, and t ≥ 0, then the sequence {X n , n ≥ 1} is said to be coordinatewise weakly lower (upper) bounded by X. The sequence {X n , n ≥ 1} 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 L p -convergence and complete convergence results for the maximums of the partial sums of the sequence of ANA random vectors with values in R d . In addition, in Sect. 4, we establish a weak law of large numbers for CANA random vectors with values in R d .
Throughout the paper, the symbol C denotes a generic constant (0 < C < ∞), which is not necessarily the same in each occurrence, S n = n i=1 X i for a sequence {X n , n ≥ 1} of random vectors, and · p denotes the L p -norm. Moreover, represents the Vinogradov symbol O, and I(·) 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 {X n , n ≥ 1} of ANA random vectors with mixing coefficients ρ -(s) is also ANA with mixing coefficients not greater than ρ -(s).
Wang and Lu [6] proved the following Rosenthal-type inequality for a sequence of ANA random variables in R 1 .
Inspired by the proof of Lemma 2.3 in Li-Xin Zhang [9], we extend Lemma 2.2 to R dvalued ANA random vectors as follows.

Lemma 2.3 For a positive integer N
Hence (2.2) follows.
From Lemma 1.2 of Kuczmaszewska [10] we obtain the following lemma.

Lemma 2.4 Let {X n , n ≥ 1} be a sequence of R d -valued random vectors weakly upper bounded by a random vector X, and let r
The following lemma supplies us with the analytical part in the proofs of the theorems in the subsequent sections.
for every n ≥ 1.
Next, we will extend some L p -convergence and complete convergence results for the maximums of the partial sum of R 1 -valued ANA random variables in Yuan and Wu [5] to R d -valued random vectors.

L p -convergence and complete convergence for ANA random vectors with values in R d
The following theorem is an extension of Theorem 3.2 in Yuan and Wu [5] to random vectors with values in R d .
As applications of Theorem 3.1, we introduce two results that are not present in Yuan and Wu [5]. A sequence of random vectors {X n , n ≥ 1} is said to converge completely to a constant a if for any > 0, ∞ n=1 P X na > < ∞.
In this case, we write X n → a 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 R 1 to random vectors in R d . 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].

Weak law of large numbers for ANA random vectors with values in R d
In this section, we establish the weak laws of large numbers for R d -valued ANA random vectors when p ≥ 2. We assume that {X n , n ≥ 1} is a sequence of ANA random vectors with values in R d . For n, i ≥ 1 and 1 ≤ j ≤ d, we set then we obtain the weak law of large numbers Proof By the standard method we obtain Next, we will show that It is well known that, for all n ≥ 1, {Y ni -EY ni , i ≥ 1} is a sequence of R d -valued ANA random vectors by Lemma 2.1. Then, by the Markov inequality, Hölder's inequality, and Lemma 2.3 we have → 0 as n → ∞ by (4.1) and (4.2) , (4.7) which yields (4.6). Combining (4.5) and (4.6), the WLLN (4.3) follows. The proof is complete.
n 0 x p-1 P X (j) > x dx (by integration by parts)  then we obtain the WLLN (4.11).
Proof The proof follows by substituting X (j) by X (j)

Conclusions
We generalized the L p -convergence and complete convergence results of Yuan and Wu [5] from R 1 -valued ANA random variables to R d -valued random vectors by using a Rosenthal-type inequality. We also established weak laws of large numbers for CANA random vectors under p ≥ 2. As applications, we obtained some L p -convergence and complete convergence results that are not present in Yuan and Wu [5] even when d = 1.