# Convergence Theorems form-Coordinatewise Negatively Associated Random Vectors in Hilbert Spaces

Authors: Lyurong Shi
Journal: Complexity
 (2021)

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## Abstract

In this study, some new results on convergence properties form-coordinatewise negatively associated random vectors in Hilbert space are investigated. The weak law of large numbers, strong law of large numbers, complete convergence, and complete moment convergence for linear process of H-valued m-coordinatewise negatively associated random vectors with random coefficients are established. These results improve and generalise some corresponding ones in the literature.


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## Body

## 1. Introduction

The random variablesX1,X2,…,Xn are said to be negatively associated (NA, in short) if, for every pair of disjoint subsets A and B of 1,2,…,n and any real coordinatewise nondecreasing (or nonincreasing) functions f1 on ℝA and f2 on ℝB,(1)Covf1Xi,i∈A,f2Xj,j∈B≤0,whenever the covariance above exists, where A and B denote the cardinalities of A and B, respectively. A sequence Xi,i≥1 of random variables is NA if every finite subcollection is NA.The concept of NA random variables can be seen in Joag-Dev and Proschan [1], which also illustrated that many well-known multivariate distributions, such as multinomial, convolution of unlike multinomial, multivariate hypergeometric, permutation distribution, negatively correlated normal distribution, and joint distribution of ranks, all satisfy the NA property.Because of its wide applications, the concept of NA random variables was extended to many different directions. For example, Chandra and Ghosal [2] extended NA to asymptotically almost negative association (AANA); Hu et al. [3] extended the concept of NA random variables to m-NA random variables; Zhang and Wang [4] generalised it to a more broad case, i.e., asymptotically negative association (ANA); Zhang [5] extended it to ℝd-valued random vectors; Ko et al. [6] introduced the concept of NA random vectors taking values in real separable Hilbert spaces.Since the concept of NA random vectors was introduced, some related works were established. For the details, we refer to Miao [7], for the Hajeck-Renyi inequality, Thanh [8], for the almost sure convergence, and Hien and Thanh [9], for weak laws of large numbers, among others. For more recent studies, we refer to Hu et al. [10], Li et al. [11], Yang et al. [12], Yang et al. [13], Fang et al. [14, 15], Mei et al. [16], and Yuan et al. [17].LetH be a real separable Hilbert space with the norm ⋅ generated by an inner product ⋅,⋅. Denote Xj=X,ej, where ej,j∈ℬ is an orthonormal basis in H, X is also a random vector taking values in H, and ℬ is a subset of 1,2,….Huan et al. [18] first introduced the concept of coordinatewise negatively associated (CNA) random vectors in Hilbert spaces. Huan et al. [18] exemplified that a sequence of NA random vectors is also CNA in Hilbert spaces while the reverse is not true in general. There are also some interesting results concerning the CNA random vectors in Hilbert spaces, most of which mainly extended the following Baum–Katz type convergence theorem from independent and identically distributed (i.i.d.) random variables in classical probability space to CNA random vectors in Hilbert space.Baum–Katz theorem (see [19]): let p>1,α>1/2 and αp>1. Let Xn,n≥1 be a sequence of i.i.d. random variables with zero mean. Then, the following statements are equivalent:(i)
EX1p<∞(ii)
∑n=1∞nαp−2P∑k=1nXk>εnα<∞(iii)
∑n=1∞nαp−2Psupk≥nk−α∑i=1kXi>εnα<∞Huan et al. [18] proved the sufficient conditions of the Baum–Katz type complete convergence with 1≤p<2 and αp>1, which partially extends Theorem A to CNA random vectors in Hilbert space. Huan [20] considered the case αp=1 with 1<p<2. Ko [21] extended the result of Huan et al. [18] to the complete moment convergence, where, however, the case p=1 is wrongly proved as pointed out by Huang and Wu [22]. Ko [23] also established the complete moment convergence with αp=1 and 1≤p<2. However, there are still some mistakes for α=p=1. To be specific, it shows in equation (2.9) of [23] that ∫1uy1/α−2logydy≤Cu1/α−1logu, where we know it is wrong when α=1; furthermore, the formula ∑n=1mlogn/nα≤Clogm/nα−1 in equation (2.11) of [23] is yet invalid for α=1. One goal of this work is to further investigate the Baum–Katz type convergence theorem such as complete convergence and complete moment convergence under a much broad dependence assumption and obtain some new results including the interesting case α=p=1. Moreover, to the best of my knowledge, there was no paper investigating the limit properties of random vectors with random coefficients in Hilbert spaces. Therefore, the work will mainly focus on this topic to obtain some results which were not established before.In this paper, some new results on the weak law of large numbers, strong law of large numbers, complete convergence, and complete moment convergence for linear process ofH-valued m-coordinatewise negatively associated (m-CNA, in short) random vectors with random coefficients are established successfully. The results improve and generalise the corresponding ones of Hien and Thanh [9], Huan et al. [18], Huan [20], Ko [21], and Ko [23].In what follows, letC denote a generic positive constant whose value may vary in different lines. x+=maxx,0 and IA implies the indicator function of the set A. ℤ=…,−2,−1,0,1,2,… represents the set of integers.The paper is organized as follows. Section2 gives some preliminary definitions and lemmas. Section 3 presents the main results and their proofs. Section 4 contains the conclusion of the paper.


## 2. Preliminaries

In this section, we will present some concepts and important lemmas as below.Definition 1.
(see [18]). A sequence Xi,i≥1 of random vectors taking values in H is said to be CNA if, for each j∈ℬ, the sequence Xij,i≥1 of random variables is NA, where Xij=Xi,ej. Inspired by Hu et al. [3] and Huan et al. [18], we introduce the concept of m-CNA random vectors in Hilbert space as follows.Definition 2.
Letm≥1 be a given integer. A sequence Xn,n≥1 of random variables is said to be m-CNA if, for any n≥2 and any i1,i2,…,in such that ij−ik≥m for all 1≤k≠j≤n, we have that Xi1,Xi2,…,Xin are CNA. Obviously, if m=1, then m-CNA random vectors is CNA. Hence, the concept of m-CNA random vectors is a natural extension of that of CNA random vectors. We also present an example of m-CNA random vectors which are not CNA, as follows.Example 1.
Letξi,i≥1 be a sequence of independent random vectors, where ξi follows the standard multivariate normal distribution for each i≥1. Take Xi=ξi+θ1ξi+1−θ2ξi+2 for each i≥1, where 0<θ1,θ2<1. Then, it is easy to check that, for any j∈ℬ and i≥1, CovXij,Xi+1j=θ11−θ2>0 but CovXij,Xi+2j=−θ2<0 and CovXij,Xi+1j=0 for l≥3. That is to say, Xi,i≥1 is a sequence of m-CNA random vectors with m=2. We also introduce the following concept of the linear process of random vectors in Hilbert spaces with random coefficients.Definition 3.
Assume thatXi,−∞<i<∞ is a sequence of H-valued random vectors and Ai,−∞<i<∞ is a sequence of random variables. The sequence Yt,t≥1 of random vectors is said to be linear process with random coefficients if(2)Yt=∑i=−∞∞AiXt−i.
The following concept is often used in the literature while dealing with the convergence theorems of random vectors in Hilbert spaces.Definition 4.
A sequenceXn,n≥1 is said to be coordinatewise weakly upper bounded by X if there exists a positive constant C such that supn≥1n−1∑i=1nPXij>x≤CPXij>x, for all j∈ℬ and x≥0.Lemma 1.
LetXi,−∞<i<∞ be a sequence of m-CNA random vectors. If fi⋅,−∞<i<∞ is a sequence of coordinatewise nondecreasing (or nonincreasing) continuous real functions, then ∑j∈ℬfiXijej,−∞<i<∞ is still a sequence of m-CNA random vectors.Proof.
It follows from Definitions1 and 2 that, for any n≥2 and any i1,i2,…,in taking values in ℤ such that ij−ik≥m for all 1≤k≠j≤n, Xi1j,Xi2j,…,Xinj is a sequence of NA random variables for each j∈ℬ. By Lemma 2.1 of [24], one can see that fi1Xi1j,fi2Xi2j,…,finXinj is still a sequence of NA random variables for each j∈ℬ. Hence, by Definitions 1 and 2 again, it follows that fi1Xi1,fi2Xi2,…,finXin are CNA, i.e., ∑j∈ℬfiXij,−∞<i<∞ is m-CNA.Lemma 2.
(see [18]). Let Xn,n≥1 be a sequence of H-valued CNA random vectors with zero mean and EXn2<∞, for all n≥1. Then,(3)Emax1≤k≤n∑i=1kXi2≤2∑i=1nEXi2.Lemma 3.
LetXn,n≥1 be a sequence of H-valued m-CNA random vectors with zero mean and EXn2<∞ for all n≥1. Then,(4)Emax1≤k≤n∑i=1kXi2≤2m∑i=1nEXi2.Proof.
Notice that(5)max1≤k≤n∑i=1kXi≤∑l=0m−1maxk≥1,mk≤n∑i=1kXmi+l,where we can define without loss of generality that X0=0. From Definition 1, we see that Xmi+l,i≥0 is CNA for each l=1,2,…,m. Hence, by cr inequality and Lemma 2, we have(6)Emax1≤k≤n∑i=1kXi2≤m∑l=0m−1Emaxk≥1,mk≤n∑i=1kXmi+l2≤2m∑l=0m−1∑i≥0,mi≤nEXmi+l2≤2m∑i=0nEXi2.Lemma 4.
LetXi,−∞<i<∞ be a sequence of H-valued m-CNA random vectors with zero mean and EXi2<∞ for all −∞<i<∞. If Ai,−∞<i<∞ is a sequence of random variables independent of Xi,−∞<i<∞ and E∑i=−∞∞Ai2<∞, then(7)Emax1≤k≤n∑l=−∞∞∑i=1−lk−lAiXl2≤Csup−∞<i<∞∑l=1−in−iEXl2.Proof.
It follows by Hölder inequality and Lemma3 that(8)Emax1≤k≤n∑l=−∞∞∑i=1−lk−lAiXl2=Emax1≤k≤n∑i=−∞∞Ai∑l=1−ik−iXl2≤E∑i=−∞∞Aimax1≤k≤n∑l=1−ik−iXl2=E∑i=−∞∞Ai1/2⋅Ai1/2max1≤k≤n∑l=1−ik−iXl2≤E∑i=−∞∞Ai1/2∑i=−∞∞Aimax1≤k≤n∑l=1−ik−iXl21/22=∑s=1∞E∑i=−∞∞Ai∑i=−∞∞Aimax1≤k≤n∑l=1−ik−iXl2Is−1≤∑i=−∞∞Ai<s≤∑s=1∞sE∑i=−∞∞AiIs−1≤∑i=−∞∞Ai<sEmax1≤k≤n∑l=1−ik−iXl2≤2msup−∞<i<∞∑l=1−in−iEXl21+2∑s=2∞E∑i=−∞∞Ai2Is−1≤∑i=−∞∞Ai<s≤Csup−∞<i<∞∑l=1−in−iEXl2.
The proof is therefore complete.Lemma 5.
(see [25]). Let Xn,n≥1 be a sequence of random variables satisfying n−1∑i=1nPZi>x≤CPZ>x for a random variable Z and any x≥0. Then, for any a>0 and b>0, there exist some positive constants c1 and c2 such that(9)n−1∑i=1nEZiaIZi>b≤c1EZaIZ>b,n−1∑i=1nEXiaIZi≤b≤c2EZaIZ≤b+baPZ>b.
Following the method of Lemma 2.3 in [26], we can obtain the following inequality in Hilbert spaces.Lemma 6.
LetYi,1≤i≤n and Zi,1≤i≤n be two sequences of random vectors. Then, for any q>1, ε>0, and a>0, the following inequality holds:(10)Emax1≤k≤n∑i=1kYi+Zi−εa+≤1εq+1q−1a1−qEmax1≤k≤n∑i=1kYiq+Emax1≤k≤n∑i=1kZi.


## 3. Main Results and Their Proofs

In this section, we will present our main results. The first one is the weak law of large numbers for linear processYt,t≥1 of m-CNA random vectors in Hilbert spaces with random coefficients.Theorem 1.
LetXi,−∞<i<∞ be a sequence of zero mean H-valued m-CNA random vectors coordinatewise weakly upper bounded by a random vector X. Suppose that Ai,−∞<i<∞ is a sequence of random variables independent of Xi,−∞<i<∞ and E∑i=−∞∞Ai2<∞. Assume further ∑j∈ℬEXjIXj>n⟶0 as n⟶∞ if the cardinality ℬ<∞ or ∑j∈ℬEXj<∞ if ℬ=∞. Then,(11)1n∑t=1nYt⟶P0,asn⟶∞.Proof.
For each−∞<i<∞ and j∈ℬ, denote(12)Xnij=−nIXij<−n+XijIXij≤n+nIXij>n;Znij=Xij−Xnij=Xij+nIXij<−n+Xij−nIXij>n,Xni=∑j∈ℬXnijej,Zni=∑j∈ℬZnijej.
Note that(13)∑t=1nYt=∑t=1n∑i=−∞∞AiXt−i=∑l=−∞∞∑i=1−ln−lAiXl=∑l=−∞∞∑i=1−ln−lAiXnl−EXnl+∑l=−∞∞∑i=1−ln−lAiZnl−EZnl=∑l=−∞∞∑i=1−ln−lAiXnl−EXnl+∑t=1n∑i=−∞∞AiZn,t−i−EZn,t−i.
It is sufficient to prove that, for anyε>0,(14)P∑l=−∞∞∑i=1−ln−lAiXnl−EXnl>nε⟶0,(15)P∑t=1n∑i=−∞∞AiZn,t−i−EZn,t−i>nε⟶0.
It follows from Lemma1 that Xni,−∞<i≤∞ is still a sequence of m-CNA random vectors. Furthermore, it is easy to check that, as n⟶∞,(16)n−1∑j∈ℬ∫01xPXj>xdx≤n−1∑j∈ℬEXj⟶0,ifℬ=∞,n−1∑j∈ℬ∫01xPXj>xdx≤n−1ℬ⟶0,ifℬ<∞.
Hence, we obtain by Chebyshev inequality,E∑i=−∞∞Ai2<∞, Lemmas 4 and 5, and integration by parts that(17)P∑l=−∞∞∑i=1−ln−lAiXnl−EXnl>nε≤Cn−2E∑l=−∞∞∑i=1−ln−lAiXnl−EXnl2≤Cn−2sup−∞<i<∞∑l=1−in−iEXnl−EXnl2≤Cn−2sup−∞<i<∞∑l=1−in−i∑j∈ℬEXlj2IXlj≤n+n2PXlj>n≤Cn−1∑j∈ℬEXj2IXj≤n+n2PXj>n=Cn−1∑j∈ℬ∫0nxPXj>xdx=Cn−1∑j∈ℬ∑k=0n−1∫kk+1xPXj>xdx≤Cn−1∑j∈ℬ∫01xPXj>xdx+Cn−1∑j∈ℬ∑k=1nk+12−k2PXj>k≤Cn−1∑j∈ℬ∫01xPXj>xdx+Cn−1∑k=1n∑j∈ℬkPXj>k≤Cn−1∑j∈ℬ∫01xPXj>xdx+Cn−1∑k=1n∑j∈ℬEXjIXj>k,which converges to 0 as n⟶∞, and thus, equation (14) holds true. On the contrary, we have, by Markov inequality, Lemma 5, and Jensen inequality, that(18)P∑t=1n∑i=−∞∞AiZn,t−i−EZn,t−i>nε≤Cn−1E∑t=1n∑i=−∞∞AiZn,t−i−EZn,t−i,≤Cn−1∑t=1n∑i=−∞∞EAiEZn,t−i−EZn,t−i≤CE∑i=−∞∞Ai∑j∈ℬn−1∑t=1nEXt−ijIXt−ij>n≤CE∑i=−∞∞Ai21/2∑j∈ℬEXjIXj>n,which obtains equation (15) as n⟶∞ by the assumption of Theorem 1, and the proof is thus complete.Remark 1.
Hien and Thanh [9] obtained the weak law of large numbers for NA random vectors under the moment condition ∑j∈ℬEXj<∞. Contrasting to Corollary 2.5 of Hien and Thanh [9], Theorem 1 not only extends the assumption of NA random vectors to linear process of m-CNA random vectors with random coefficients but also improves the moment condition when |ℬ|<∞.Theorem 2.
Let1<p<2 and αp≥1. Let Xi,−∞<i<∞ be a sequence of zero mean H-valued m-CNA random vectors coordinatewise weakly upper bounded by a random vector X with ∑j∈ℬEXjp<∞. Suppose that Ai,−∞<i<∞ is a sequence of random variables independent of Xi,−∞<i<∞ and E∑i=−∞∞Ai2<∞. Then, for any ε>0,(19)∑n=1∞nαp−α−2Emax1≤k≤n∑t=1kYt−εnα+<∞,and thus,(20)∑n=1∞nαp−2Pmax1≤k≤n∑t=1kYt>εnα<∞.Proof.
Define for each−∞<i<∞ and j∈ℬ that(21)Wnij=−nIXij<−nα+XijIXij≤nα+nIXij>nα,Tnij=Xij−Wnij=Xij+nαIXij<−nα+Xij−nαIXij>nα,Wni=∑j∈ℬWnijej,Tni=∑j∈ℬTnijej.
Similar to the argument of equation (13), we have that, for each 1≤k≤n,(22)∑t=1kYt=∑l=−∞∞∑i=1−lk−lAiXl=∑l=−∞∞∑i=1−lk−lAiWnl−EWnl+∑t=1k∑i=−∞∞AiTn,t−i−ETn,t−i.
On the one hand, noting thatWni,−∞<i≤∞ is still a sequence of m-CNA random vectors by Lemma 1, we obtain by Lemmas 4 and 5 that(23)∑n=1∞nαp−2α−2Emax1≤k≤n∑l=−∞∞∑i=1−lk−lAiWnl−EWnl2≤C∑n=1∞nαp−2α−2sup−∞<i<∞∑l=1−in−i∑j∈ℬEXlj2IXlj≤nα+n2αPXlj>nα≤C∑j∈ℬ∑n=1∞nαp−2α−1EXj2IXj≤nα+C∑j∈ℬ∑n=1∞nαp−1PXj>nα≤C∑j∈ℬ∑n=1∞nαp−2α−1∑l=1nEXj2Il−1α<Xj≤lα+C∑j∈ℬEXjp≤C∑j∈ℬ∑l=1∞lαp−2αEXj2Il−1α<Xj≤lα+C∑j∈ℬEXjp≤C∑j∈ℬEXjp<∞.
On the other hand, it follows from Lemma5 and Jensen inequality that(24)∑n=1∞nαp−α−2Emax1≤k≤n∑t=1k∑i=−∞∞AiTn,t−i−ETn,t−i≤C∑n=1∞nαp−α−2∑t=1n∑i=−∞∞EAiETn,t−i−ETn,t−i≤C∑n=1∞nαp−α−1E∑i=−∞∞Ai∑j∈ℬn−1∑t=1nEXt−ijIXt−ij>nα≤C∑j∈ℬ∑n=1∞nαp−α−1E∑i=−∞∞Ai21/2EXjIXj>nα≤C∑j∈ℬ∑l=1∞EXjIlα<Xj≤l+1α∑n=1lnαp−α−1≤C∑j∈ℬ∑l=1∞lαp−αEXjIlα<Xj≤l+1α≤C∑j∈ℬEXjp<∞.
Hence, it follows from Lemma6 (with q=2) and equations (22)–(24) that(25)∑n=1∞nαp−α−2Emax1≤k≤n∑t=1kYt−εnα+≤C∑n=1∞nαp−2α−2Emax1≤k≤n∑l=−∞∞∑i=1−lk−lAiWnl−EWnl2+∑n=1∞nαp−α−2Emax1≤k≤n∑t=1k∑i=−∞∞AiTn,t−i−ETn,t−i≤C∑j∈ℬEXjp<∞,which obtains equation (19). Now, we prove equation (20). It follows from equation (19) that(26)∞>∑n=1∞nαp−α−2Emax1≤k≤n∑t=1kYt−εnα+,≥∑n=1∞nαp−α−2∫0εnαPmax1≤k≤n∑t=1kYt−εnα>sds≥ε∑n=1∞nαp−2Pmax1≤k≤n∑t=1kYt>2εnα,which combining with the arbitrariness of ε gets equation (20). The proof is complete.Remark 2.
Huan et al. [18] and Huan [20] established equation (20) for CNA random vectors, respectively, for 1≤p<2, αp>1, and 1<p<2, αp=1. Although Ko [21] extended the result of Huan et al. [18] to complete moment convergence, it only holds for 1<p<2, αp>1, as illustrated in Section 1. Hence, Theorem 2 improves and extends the results of Huan et al. [18], Huan [20], and Ko [21] from CNA random vectors to linear process of m-CNA random vectors with random coefficients. By Theorem 2, we can easily get the following conclusion.Corollary 1.
Under the conditions of Theorem2, if αp>1, we have that, for any ε>0,(27)∑n=1∞nαp−2Esupk≥nk−α∑t=1kYt−ε+<∞,and thus,(28)∑n=1∞nαp−2Psupk≥nk−α∑t=1kYt>ε<∞.Proof.
It follows from Theorem2 that(29)∑n=1∞nαp−2Esupk≥nk−α∑t=1kYt−ε+=∑l=1∞∑2l−1≤n<2lnαp−2Esupk≥nk−α∑t=1kYt−ε+,≤C∑l=1∞2lαp−1Esupk≥2l−1k−α∑t=1kYt−ε+≤C∑s=1∞Emax2s−1≤k<2sk−α∑t=1kYt−ε+∑l=1s2lαp−1≤C∑s=1∞2sαp−1Emax2s−1≤k<2s2−αs−1∑t=1kYt−ε+≤C∑s=1∞2sαp−α−1Emax1≤k<2s∑t=1kYt−ε2−α2αs+≤C∑s=0∞∑2s≤n<2s+1nαp−α−2Emax1≤k<2s∑t=1kYt−ε2−α2αs+≤C∑n=1∞nαp−α−2Emax1≤k≤n∑t=1kYt−ε2−2αnα+<∞.
Furthermore, similar to the proof of equation (20), we have that(30)∞>∑n=1∞nαp−2Esupk≥nk−α∑t=1kYt−ε+≥∑n=1∞∑n=1∞nαp−2∫0εPsupk≥nk−α∑t=1kYt−ε>sds≥ε∑n=1∞nαp−2Psupk≥nk−α∑t=1kYt>2ε.
The proof is complete.
Forp=1, we can obtain the following result.Theorem 3.
Letα≥1. Let Xi,−∞<i<∞ be a sequence of zero mean H-valued m-CNA random vectors coordinatewise weakly upper bounded by a random vector X with ∑j∈ℬEXjlnδ+11+Xj<∞ for some δ≥0. Suppose that Ai,−∞<i<∞ is a sequence of random variables independent of Xi,−∞<i<∞ and E∑i=−∞∞Ai2<∞. Then, for any ε>0,(31)∑n=1∞n−2lnδnEmax1≤k≤n∑t=1kYt−εnα+<∞,and thus,(32)∑n=1∞nα−2lnδnPmax1≤k≤n∑t=1kYt>εnα<∞.Proof.
We still use the notations and method in the proof of Theorem2. Similar to the argument of equation (23), we have(33)∑n=1∞n−α−2lnδnEmax1≤k≤n∑l=−∞∞∑i=1−lk−lAiWnl−EWnl2≤C∑j∈ℬ∑n=1∞n−α−1lnδnEXj2IXj≤nα+C∑j∈ℬ∑n=1∞nα−1lnδnPXj>nα≤C∑j∈ℬ∑l=1∞l−αlnδlEXj2Il−1α<Xj≤lα+C∑j∈ℬ∑l=1∞lαlnδlPlα<Xj≤l+1α≤C∑j∈ℬ∑l=1∞EXj2⋅Xj−1ln1+Xj1/αδIl−1α<Xj≤lα+C∑j∈ℬ∑l=1∞EXjln1+Xj1/αδIl−1α<Xj≤lα≤C∑j∈ℬ∑l=1∞EXjlnδ1+XjIl−1α<Xj≤lα≤C∑j∈ℬEXjlnδ1+Xj<∞.
Furthermore, similar to the argument of equation (24), we have(34)∑n=1∞n−2lnδnEmax1≤k≤n∑t=1k∑i=−∞∞AiTn,t−i−ETn,t−i≤C∑j∈ℬ∑l=1∞EXjIlα<Xj≤l+1α∑n=1ln−1lnδn≤C∑j∈ℬ∑l=1∞lnδ+1lEXjIlα<Xj≤l+1α≤C∑j∈ℬ∑l=1∞EXjlnδ+11+XjIlα<Xj≤l+1α≤C∑j∈ℬEXjlnδ+11+XjIXj>1≤C∑j∈ℬEXjlnδ+11+Xj<∞.
Hence, by Lemma1 and equations (22), (33), and (34), we can obtain equation (31). Following the proof of equation (20), we can also get equation (32) by equation (31). The proof is complete.Remark 3.
Ko [23] proved the complete moment convergence for coordinatewise asymptotically almost negatively associated (CAANA) random vectors with αp=1 and 1/2<α≤1. However, as stated in Section 1, the meaningful case α=1 is wrongly proved. Note that Theorem 3 also works if α=p=1. Thus, Theorem 3 fills the vacancy and extends it to some more general settings. By Theorems 2 and 3, one can obtain the following strong law of large numbers for linear process of m-CNA random vectors with random coefficients.Corollary 2.
Let1≤p<2. Let Xi,−∞<i<∞ be a sequence of zero mean H-valued m-CNA random vectors coordinatewise weakly upper bounded by a random vector X with ∑j∈ℬEXjp<∞ if p>1 or ∑j∈ℬEXjln1+Xj<∞ if p=1. Suppose that Ai,−∞<i<∞ is a sequence of random variables independent of Xi,−∞<i<∞ and E∑i=−∞∞Ai2<∞. Then, with probability 1,(35)n−1/p∑t=1kYt⟶0,asn⟶∞.Proof.
Letα=1/p. It follows from equations (20) and (32) (with δ=0) that(36)∞>∑n=1∞n−1Pmax1≤k≤n∑t=1kYt>εn1/p=∑l=0∞∑2l≤n<2l+1n−1Pmax1≤k≤n∑t=1kYt>εn1/p≥12∑l=1∞Pmax1≤k≤2l∑t=1kYt>22/pε2l−11/p.
By Borel–Cantelli lemma, the formula above implies that, asl⟶∞, with probability 1,(37)12l−11/pmax1≤k≤2l∑t=1kYt⟶0.
Meanwhile, for any fixedn, there exists positive integer l such that 2l−1≤n<2l. Hence, with probability 1, one has(38)n−1/p∑t=1kYt≤12l−11/pmax1≤k≤2l∑t=1kYt⟶0.


## 4. Conclusion

In this study, the concept ofm-CNA random vectors is introduced as a natural extension of CNA random vectors. The weak law of large numbers, complete convergence, and complete moment convergence for linear process of H-valued m-CNA random vectors with random coefficients are established. As a corollary of the complete convergence, the strong law of large numbers is also obtained. These results improve and generalise the corresponding ones of recent works such as Hien and Thanh [9], Huan et al. [18], Huan [20], Ko [21], and Ko [23].However, there are still two open problems should be conquered. In specific, the Baum–Katz type theorem is only extended under the restriction1≤p<2; the first problem is that whether it is possible to release to p≥2? Another problem is whether the moment condition ∑j∈ℬEXjln1+Xj<∞ for the strong law of large numbers with p=1 can be weakened to ∑j∈ℬEXj<∞?.



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*Source: 3462317-2021-07-20.xml*