Ko Journal of Inequalities and Applications (2018) 2018:107 https://doi.org/10.1186/s13660-018-1699-6

RESEARCH

Open Access

Lp-convergence, complete convergence, and weak laws of large numbers for asymptotically negatively associated random vectors with values in Rd Mi-Hwa Ko1* *

Correspondence: [email protected] Division of Mathematics and Informational Statistics, Wonkwang University, Jeonbuk, Korea 1

Abstract 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 . MSC: 60F15 Keywords: Asymptotically negative association; Random vectors; Rosenthal-type inequality; Lp -convergence; Complete convergence; Weak laws of large numbers

1 Introduction Ko et al. [1] introduced the concept of negative association (NA) for Rd -valued random vectors. Definition 1.1 (Ko et al. [1]) A finite sequence {X1 , . . . , Xm } of Rd -valued random vectors is said to be negatively associated (NA) if for any disjoint nonempty subsets A, B ⊂ {1, . . . , m} and any nondecreasing functions f on R|A|d and g on R|B|d ,   Cov f (Xi , i ∈ A), g(Xj , j ∈ B) ≤ 0

(1.1)

whenever the covariance exists. Let |A| denote the cardinality of a set A. An infinite sequence {Xi , i ≥ 1} of Rd -valued random vectors is negatively associated if every finite subsequence is negatively associated. In the case of d = 1, 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. © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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In addition to Definition 1.1, for random vectors in Rd , we can define asymptotically negative association (ANA). Definition 1.2 A sequence {X1 , . . . , Xm } of Rd -valued random vectors is said to be asymptotically negatively associated (ANA) if   ρ – (r) = sup ρ – (S, T) : S, T ⊂ N, dist(S, T) ≥ r → 0

as r → ∞,

(1.2)

S,T

where dist(S, T) = min{|x – y|; x ∈ S, y ∈ T},  ρ – (S, T) = 0 ∨

Cov(f (Xi , i ∈ S), g(Xj , j ∈ T)) 1

 , 1

(Var f (Xi , i ∈ S)) 2 (Var g(Xj , j ∈ T)) 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 Lp -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 Rd was introduced as follows. Let ·, · denote the inner product, and let {ej , j ≥ 1} be an orthonormal basis. A sequence {Xn , n ≥ 1} of Rd -valued random vectors is said to be coordinatewise negatively associated (CNA) if for each j (1 ≤ j ≤ d), the sequence (j) (j) {Xn , n ≥ 1, 1 ≤ j ≤ d} of random variables is NA, where Xn = Xn , ej . As in the definition of CNA, we can define coordinatewise asymptotically negative association for random vectors with values in Rd . Definition 1.3 A sequence {Xn , n ≥ 1} of Rd -valued random vectors is said to be coordinatewise asymptotically negatively associated (CANA) if for each j (1 ≤ j ≤ d), the se(j) quence {Xn , n ≥ 1, 1 ≤ j ≤ d} of random variables is asymptotically negatively associated, (j) where Xn = Xn , ej for n ≥ 1 and 1 ≤ j ≤ d. It is clear that if a sequence of Rd -valued random vectors is ANA, then it is CANA. However, in general, the converse is not true. Let {X, Xn , n ≥ 1} be a sequence of Rd -valued random vectors. We consider the following inequalities for 1 ≤ j ≤ d: n    1  (j)      P Xk  > t ≤ C2 P X (j)  > t . C1 P X (j)  > t ≤ n k=1

(1.3)

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If there exists a positive constant C1 , (C2 ) 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 {Xn , n ≥ 1} is said to be coordinatewise weakly lower (upper) bounded by X. The sequence {Xn , 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 Lp -convergence and complete convergence results for the maximums of the partial sums of the sequence of ANA random vectors with values in Rd . In addition, in Sect. 4, we establish a weak law of large numbers for CANA random vectors with values in Rd . Throughout the paper, the symbol C denotes a generic constant (0 < C < ∞), which is

not necessarily the same in each occurrence, Sn = ni=1 Xi for a sequence {Xn , n ≥ 1} of random vectors, and  · p denotes the Lp -norm. Moreover, represents the Vinogradov symbol O, and I(·) is the indicator function.

2 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 {Xn , 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 R1 . 1 p/2 Lemma 2.2 For a positive integer N ≥ 1, positive real numbers p ≥ 2, and 0 ≤ r < ( 6p ) , – if {Xi , i ≥ 1} is a sequence of ANA random variables with ρ (N) ≤ r, EXi = 0, and E|Xi |p < ∞ for every i ≥ 1, then there is a positive constant D = D(p, N, r) such that, for all n ≥ 1,

 k  p/2  n  n  p   p 2 . Xi  ≤ D E|Xi | + E|Xi | E max   1≤k≤n i=1

i=1

(2.1)

i=1

Inspired by the proof of Lemma 2.3 in Li-Xin Zhang [9], we extend Lemma 2.2 to Rd valued ANA random vectors as follows. 1 p/2 Lemma 2.3 For a positive integer N ≥ 1, positive real numbers p ≥ 2, and 0 ≤ r < ( 6p ) , d – if {Xi , i ≥ 1} is a sequence of R -valued ANA random vectors with ρ (N) ≤ r, EXi = 0, and EXi p < ∞ for every i ≥ 1, then there is a positive constant D = D (p, N, r) such that, for all n ≥ 1,

k

p

Xi E max

1≤k≤n  ≤ D

i=1

n i=1

 EXi p +

n i=1

p/2 EXi 2

.

(2.2)

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Proof Note that

k

 k   k  d  d

  

  (j)  (j)  Xi = max Xi  ≤ max  Xi  max 

1≤k≤n    1≤k≤n 1≤k≤n i=1

j=1

i=1

j=1

i=1

and by Lemma 2.2  k   n p n  p  (j) 2 2  (j) p   Xi  ≤ D EXi  + EXi  E max   1≤k≤n 

i=1

i=1

 ≤D

n

i=1

EXi 

2

i=1

p 2

+

n

EXi 

p

.

i=1



Hence (2.2) follows. From Lemma 1.2 of Kuczmaszewska [10] we obtain the following lemma.

Lemma 2.4 Let {Xn , n ≥ 1} be a sequence of Rd -valued random vectors weakly upper bounded by a random vector X, and let r > 0. Then, for some constant C > 0, EXr < ∞

implies n–1 nk=1 EXk r ≤ CEXr . 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 {an , n ≥ 1} and {bn , n ≥ 1} be sequences of nonnegative numbers. If

sup n–1 n≥1

n

ai < ∞ and

∞

bn < ∞,

n=1

i=1

then n

 ai bi ≤ sup m

i=1

m≥1

–1

m i=1

ai

n

bi

(2.3)

i=1

for every n ≥ 1. Next, we will extend some Lp -convergence and complete convergence results for the maximums of the partial sum of R1 -valued ANA random variables in Yuan and Wu [5] to Rd -valued random vectors.

3 Lp -convergence and complete convergence for ANA random vectors with values in Rd The following theorem is an extension of Theorem 3.2 in Yuan and Wu [5] to random vectors with values in Rd .

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Theorem 3.1 Let p ≥ 2 be positive real numbers, and let N ≥ 1 be a positive integer. Suppose that {Xn , n ≥ 1} is a sequence of Rd -valued ANA random vectors with mixing coeffi1 p/2 cients ρ – (s) such that ρ – (N) < ( 6p ) . If {Xn , n ≥ 1} are Rd -valued random vectors satisfying 1 EXk p < ∞, n n

sup n≥1

(3.1)

k=1

then, for any δ > 12 , n–δ max Si – ESi  → 0

in Lp .

1≤i≤n

(3.2)

Proof By Lemma 2.3, the Hölder inequality, and (3.1) we obtain p  E n–δ max Si – ESi  1≤i≤n

p

i



= n–pδ E max (Xj – EXj )

1≤i≤n j=1

n

–pδ

n

 EXi – EXi  + n p

i=1

–pδ

n

p/2 EXi – EXi 

2



by (2.2)



i=1

p

n–pδ–1+ 2

n

EXi p

i=1

1 EXi p , n i=1 n

p

n–pδ+ 2 sup n≥1

which by Lemma 2.5 yields (3.2) for any δ > 12 .

(3.3)



As applications of Theorem 3.1, we introduce two results that are not present in Yuan and Wu [5]. Theorem 3.2 Let p ≥ 2 be positive real numbers, and let N ≥ 1 be a positive integer. Suppose that {Xn , n ≥ 1} is a sequence of Rd -valued ANA random vectors with mixing coeffi1 p/2 cients ρ – (s) such that ρ – (N) < ( 6p ) . If {Xn , n ≥ 1} is weakly upper bounded by a random vector X with EXp < ∞, then, for any δ > 12 , (3.2) holds. Proof By Lemma 2.3, Lemma 2.4(i), Hölder’s inequality, and the proof of Theorem 3.1 we obtain p  E n–pδ max Si – ESi  1≤i≤n

p

≤ n–pδ–1+ 2

n

EXi p

  see (3.3)

i=1 p –pδ+ 2

≤n

EXp → 0

as n → ∞ by Lemma 2.4.



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Corollary 3.3 Let p ≥ 2 be positive real numbers, and let N ≥ 1 be a positive integer. Suppose that {Xn , n ≥ 1} is a sequence of Rd -valued ANA random vectors with mixing coef1 p/2 ficients ρ – (s) such that ρ – (N) < ( 6p ) . If {Xn , n ≥ 1} are identically distributed random p vectors with EX1  < ∞, then, for any δ > 12 , (3.2) holds. A sequence of random vectors {Xn , n ≥ 1} is said to converge completely to a constant a if for any  > 0, ∞   P Xn – a >  < ∞. n=1

In this case, we write Xn → 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 R1 to random vectors in Rd . Theorem 3.4 Let p ≥ 2 be positive real numbers, and let N ≥ 1 be a positive integer. Suppose that {Xn , n ≥ 1} is a sequence of Rd -valued ANA random vectors with mixing coeffi1 p/2 cients ρ – (s) such that ρ – (N) < ( 6p ) . If {Xn , n ≥ 1} satisfies (3.1), then, for any δ > 12 , n–δ max Si – ESi  → 0 1≤i≤n

(3.4)

completely.

Proof By Lemma 2.3, Lemma 2.5, Hölder’s inequality, and the proof of Theorem 3.1 we obtain ∞  p E n–δ max Si – ESi  1≤i≤n

n=1



∞

n–pδ–1+p/2

n=1



EXk p

k=1

≤

EXk p

  see (3.3)

k=1

∞

∞

n

∞

n–pδ–1+p/2

n=k

k –pδ–1+p/2 EXk p

k=1 ∞

1 EXk p n–pδ–1+p/2 n≥1 n n=1 k=1   p <∞ since –pδ – 1 + < –1 . 2 n

≤ sup

Hence (3.4) holds.

  by Lemma 2.5 (3.5) 

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].

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Theorem 3.5 Let p ≥ 2 be positive real numbers, and let N ≥ 1 be a positive integer. Suppose that {Xn , n ≥ 1} is a sequence of Rd -valued ANA random vectors with mixing coeffi1 p/2 ) . If {Xn , n ≥ 1} is weakly upper bounded by a random cients ρ – (s) such that ρ – (N) < ( 6p p vector X with EX < ∞, then, for any δ > 12 , (3.4) holds. Proof As in the proof of Theorem 3.4, we obtain ∞  p E n–δ max Si – ESi  1≤i≤n

n=1

∞

1 EXk p n–pδ–1+p/2 n n=1 n

≤ sup n≥1

k=1

≤ EXp

∞

n–pδ–1+p/2

n=1



<∞

  see (3.5)

since –pδ – 1 +

(by Lemma 2.4)  p < –1 and EXp < ∞ . 2



Corollary 3.6 Let p ≥ 2 be positive real numbers, and let N ≥ 1 be a positive integer. Suppose that {Xn , n ≥ 1} is a sequence of Rd -valued ANA random vectors with mixing coef1 p/2 ficients ρ – (s) such that ρ – (N) < ( 6p ) . If {Xn , n ≥ 1} are identically distributed random p vectors with EX1  < ∞, then, for any δ > 12 , (3.4) holds.

4 Weak law of large numbers for ANA random vectors with values in Rd In this section, we establish the weak laws of large numbers for Rd -valued ANA random vectors when p ≥ 2. We assume that {Xn , n ≥ 1} is a sequence of ANA random vectors with values in Rd . For n, i ≥ 1 and 1 ≤ j ≤ d, we set (j)

Xi = Xi , ej ,  (j)     (j) (j) (j)  (j)  Yni = –nI Xi < –n + Xi I Xi  ≤ n + nI Xi > n , Yni =

d

and

(j)

Yni ej .

j=1

Theorem 4.1 Let p ≥ 2 be positive real numbers, and let N ≥ 1 be a positive integer. Suppose that {Xn , n ≥ 1} is a sequence of Rd -valued ANA random vectors with mixing coeffi1 p/2 ) . If cients ρ – (s) such that ρ – (N) ≤ r and 0 ≤ r < ( 6p

lim

n→∞

n d   (j)  P Xi  > n = 0

(4.1)

i=1 j=1

and

n d lim

n→∞

i=1

(j) p (j) j=1 E(|Xi | I(|Xi | ≤ n)) np

= 0,

(4.2)

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then we obtain the weak law of large numbers 1 (Xi – EYni ) →p 0 as n → ∞. n i=1 n

(4.3)

Proof By the standard method we obtain

 n  n

 1



P (Xi = Yni )

(Xi – Yni ) >  ≤ P

n i=1

i=1

≤

=

n

n d  (j) (j)  P(Xi = Yni ) = P Xi = Yni

i=1

i=1 j=1

n d  (j)     P Xi  > n → 0 as n → ∞ by (4.1) .

(4.4)

i=1 j=1

Thus 1 (Xi – Yni ) →p 0 as n → ∞. n i=1 n

(4.5)

Next, we will show that 1 (Yni – EYni ) →p 0 n i=1 n

as n → ∞.

(4.6)

It is well known that, for all n ≥ 1, {Yni – EYni , i ≥ 1} is a sequence of Rd -valued ANA random vectors by Lemma 2.1. Then, by the Markov inequality, Hölder’s inequality, and Lemma 2.3 we have

 n

1



P

(Yni – EYni ) > 

n i=1

n

p

1



≤ p p E (Yni – EYni )

 n i=1 ≤

n d C  (j) p E Yni np i=1 j=1

≤

d d n n  C   (j) p  (j)  C p  (j)  X > n + n P E Xi  I Xi  ≤ n i p p n i=1 j=1 n i=1 j=1

≤C

n n d d  (j)   (j) p  (j)   C  P  Xi  > n + p E Xi  I Xi  ≤ n n i=1 j=1 i=1 j=1

  → 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. 

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Theorem 4.2 Let p ≥ 2 be positive real numbers, and let N ≥ 1 be a positive integer. Suppose that {Xn , n ≥ 1} is a sequence of Rd -valued CANA random vectors with mixing coeffi1 p2 cients ρ – (s) such that ρ – (N) ≤ r and 0 ≤ r < ( 6p ) . If {Xn , n ≥ 1} is coordinatewise weakly upper bounded by a random vector X with d

lim

n→∞

   np–1 P X (j)  > n = 0,

(4.8)

j=1

then the WLLN (4.3) holds. Proof We first show that (4.5) holds. By (1.3) and (4.4) we obtain

 n

1

P

(Xi – Yni ) > 

n i=1  n  (Xi = Yni ) ≤P i=1

≤

n d   (j)  P Xi  > n i=1 j=1

≤C

d

   nP X (j)  > n

  by (1.3)

j=1

≤C

d

   np–1 P X (j)  > n

j=1

= o(1)

by (4.8),

(4.9)

which yields (4.5). It remains to show that (4.6) holds. Since for all n ≥ 1, {Yni – EYni , i ≥ 1} is a sequence of Rd -valued ANA random vectors, by Lemma 2.1, Lemma 2.3, and (1.3) we have

k



1



P max (Yni – EYni ) > 

1≤k≤n n i=1

p

k 



–p ≤ n E max (Yni – EYni )

1≤k≤n 

(by the Markov inequality)

i=1

≤

n d C  (j) p E Yni np i=1 j=1

≤

n n d d  (j) p  (j)   C  C p  (j)  n P X E Xi  I Xi  ≤ n > n + i p p n i=1 j=1 n i=1 j=1

=

d d   C p  (j)  C  (j) p  (j)  > n + n P X E X I X ≤n p–1 p–1 n n j=1 j=1



see (4.7)



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d  C n p–1  (j)   = p–1 x P X > x dx n j=1 0

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(by integration by parts)

d n–1  C k+1 p–1  (j)   ≤ p–1 x P X > x dx n k j=1 k=0

≤

≤

C np–1

d n

     (k + 1)p – k p P X (j)  > k

j=1 k=1

n d C  p–1  (j)   p k P X >k → 0 np–1 j=1

as n → ∞ by (4.8),

k=1

which yields (4.6). Combining (4.5) and (4.6), we obtain the WLLN (4.3). Hence the proof is complete.  Remark Suppose that {Xn , n ≥ 1} is a sequence of Rd -valued CNA random vectors. If {Xn , n ≥ 1} is coordinatewise weakly upper bounded by a random vector X with

limn→∞ dj=1 nP(|X (j) | > n) = 0, then the WLLN (4.3) holds. Corollary 4.3 Let N ≥ 1 be a positive integer, and let p ≥ 2. Suppose that {Xn , n ≥ 1} is a sequence of Rd -valued ANA random vectors with mixing coefficient ρ – (s) such that ρ – (N) ≤ r 1 p2 and 0 ≤ r < ( 6p ) . If {Xn , n ≥ 1} is a sequence of identically distributed random vectors with

lim

n→∞

d

 (j)   np–1 P X1  > n = 0,

(4.8 )

j=1

then (4.3) holds. Theorem 4.4 Let p ≥ 2, and let N ≥ 1 be an integer. Suppose that {Xn , n ≥ 1} is a sequence of mean zero Rd -valued ANA random vectors with mixing coefficients ρ – (s) such 1 p2 that ρ – (N) < r and 0 ≤ r < ( 6p ) . If {Xn , n ≥ 1} is coordinatewise weakly upper bounded by a random vector X with d  p–1  < ∞, E X (j) 

(4.10)

j=1

then 1 Xi →p 0 n i=1 n

as n → ∞.

(4.11)

Proof It follows from (4.10) that

lim

n→∞

d   p–1   E X (j)  I X (j)  > n = 0, j=1

(4.12)

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which yields

n

1

EYni

n i=1 ≤

1 EYni  n i=1

≤

1  (j)  EYni n j=1 i=1

≤

d d n n  1   (j)  1   (j)  (j)  E X i I Xi ≤ n  + nP Xi  > n n j=1 i=1 n j=1 i=1

=

n d  1   (j)  (j)  E X i I Xi > n  n j=1 i=1

n

d

+

≤

n

d n  1  (j)  nP Xi > n n j=1 i=1

(by EXi = 0)

d d         E X (j) I X (j)  > n + nP X (j)  > n j=1

≤C



by (1.3)



j=1

d d    p–1     E X (j)  I X (j)  > n + C np–1 P X (j)  > n j=1

≤ 2C

j=1

d  p–1    E X (j)  I X (j)  > n →p 0

  as n → ∞ by (4.12) .

(4.13)

j=1

It remains to prove (4.3). Since (4.12) implies (4.8), (4.3) follows from Theorem 4.2. Thus the proof is complete.  Remark Suppose that {Xn , n ≥ 1} is a zero-mean sequence of Rd -valued NA random vectors. If {Xn , n ≥ 1} is coordinatewise weakly upper bounded by a random vector X with

d (j) j=1 E|X | < ∞, then (4.11) holds. Corollary 4.5 Let N ≥ 1 be an integer, and let p ≥ 2. Suppose that {Xn , n ≥ 1} is a sequence of Rd -valued ANA random vectors with mixing coefficient ρ – (s) such that ρ – (N) ≤ r and 1 p2 ) . If {Xn , n ≥ 1} is a sequence of identically distributed random vectors with 0 ≤ r < ( 6p EX1 = 0 and d  (j) p–1  < ∞, E X1 

(4.10 )

j=1

then we obtain the WLLN (4.11). (j)

Proof The proof follows by substituting X (j) by X1 in the proof of Theorem 4.4.



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5 Conclusions We generalized the Lp -convergence and complete convergence results of Yuan and Wu [5] from R1 -valued ANA random variables to Rd -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 Lp -convergence and complete convergence results that are not present in Yuan and Wu [5] even when d = 1. Acknowledgements This paper was supported by Wonkwang University in 2018. Competing interests The author declares that there is no conflict of interests regarding the publication of this article. Authors’ contributions All authors read and approved the final manuscript.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 1 December 2017 Accepted: 25 April 2018 References 1. Ko, M.H., Kim, T.S., Han, K.H.: A note on the almost sure convergence for dependent random variables in a Hilbert space. J. Theor. Probab. 22, 506–513 (2009) 2. Joag-Dev, K., Proschan, F.: Negative association of random variables with applications. Ann. Stat. 11, 286–295 (1983) 3. Zhang, L.X.: A functional central limit theorem for asymptotically negatively dependent random fields. Acta Math. Hung. 86, 237–259 (2000) 4. Zhang, L.X.: Central limit theorems for asymptotically negatively dependent random fields. Acta Math. Sin. Engl. Ser. 16, 691–710 (2000) 5. Yuan, D.M., Wu, X.S.: Limiting behavior of the maximum of the partial sum for asymptotically negatively associated random variables under residual Cesàro alpha-integrability assumption. J. Stat. Plan. Inference 140, 2395–2402 (2010) 6. Wang, J.F., Lu, F.B.: Inequalities of maximum of partial sums and weak convergence for a class of weak dependent random variables. Acta Math. Sin. Engl. Ser. 22, 693–700 (2006) 7. Wang, J.F., Zhang, L.X.: A Berry–Esseen theorem and a law of iterated logarithm for asymptotically negatively associated sequences. Acta Math. Sin. Engl. Ser. 23, 127–136 (2007) 8. Huan, N.V., Quang, N.V., Thuan, N.T.: Baum–Katz type theorems for coordinatewise negatively associated random vectors in Hilbert spaces. Acta Math. Hung. 144, 132–149 (2014) 9. Zhang, L.X.: Strassen’s law of the iterated logarithm for negatively associated random vectors. Stoch. Process. Appl. 95, 311–328 (2001) 10. Kuczmaszewska, A.: On complete convergence in Marcinkiewicz–Zygmund type SLLN for negatively associated random variables. Acta Math. Hung. 128(1), 116–130 (2010) 11. Hsu, P.L., Robbins, H.: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33, 25–31 (1947)