Convergence Theorems for m -Coordinatewise Negatively Associated Random Vectors in Hilbert Spaces

In this study, some new results on convergence properties for m -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.


Introduction
e random variables X 1 , X 2 , . . . , X n 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 f 1 on R |A| and f 2 on R |B| , whenever the covariance above exists, where |A| and |B| denote the cardinalities of A and B, respectively. A sequence X i , i ≥ 1 of random variables is NA if every finite subcollection is NA. e concept of NA random variables can be seen in Joag-Dev and Proschan [1], which also illustrated that many wellknown 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 R d -valued random vectors; Ko et al. [6] introduced the concept of NA random vectors taking values in real separable Hilbert spaces.
Let H be a real separable Hilbert space with the norm ‖ · ‖ generated by an inner product 〈·, ·〉. Denote X (j) � 〈X, e j 〉, where e (j) , 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. ere 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 X n , n ≥ 1 be a sequence of i.i.d. random variables with zero mean. en, the following statements are equivalent: [18] proved the sufficient conditions of the Baum-Katz type complete convergence with 1 ≤ p < 2 and αp > 1, which partially extends eorem 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 [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. erefore, 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 of H-valued m-coordinatewise negatively associated (m-CNA, in short) random vectors with random coefficients are established successfully. e results improve and generalise the corresponding ones of Hien and anh [9], Huan et al. [18], Huan [20], Ko [21], and Ko [23].
In what follows, let C denote a generic positive constant whose value may vary in different lines. x + � max x, 0 { } and I(A) implies the indicator function of the set A. Z � . . . , − 2, − 1, 0, 1, 2, . . . { } represents the set of integers. e paper is organized as follows. Section 2 gives some preliminary definitions and lemmas. Section 3 presents the main results and their proofs. Section 4 contains the conclusion of the paper.

Preliminaries
In this section, we will present some concepts and important lemmas as below.
Definition 1 (see [18] [3] and Huan et al. [18], we introduce the concept of m-CNA random vectors in Hilbert space as follows. Definition 2. Let m ≥ 1 be a given integer. A sequence X n , n ≥ 1 of random variables is said to be m-CNA if, for any n ≥ 2 and any (i 1 , i 2 , . . . , i n ) such that |i j − i k | ≥ m for all 1 ≤ k ≠ j ≤ n, we have that (X i 1 , X i 2 , . . . , X i n ) 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. Let ξ i , i ≥ 1 be a sequence of independent random vectors, where ξ i follows the standard multivariate normal distribution for each i ≥ 1. Take We also introduce the following concept of the linear process of random vectors in Hilbert spaces with random coefficients.
Definition 3. Assume that X i , − ∞ < i < ∞ is a sequence of H-valued random vectors and A i , − ∞ < i < ∞ is a sequence of random variables. e sequence Y t , t ≥ 1 of random vectors is said to be linear process with random coefficients if (2) e following concept is often used in the literature while dealing with the convergence theorems of random vectors in Hilbert spaces.
Proof. It follows from Definitions 1 and 2 that, for any n ≥ 2 and any (i 1 , i 2 , . . . , i n ) taking values in Z such that |i j − is a sequence of NA random variables for each j ∈ B. By Lemma 2.1 of [24], Complexity NA random variables for each j ∈ B. Hence, by Definitions 1 and 2 again, it follows that [18]). Let X n , n ≥ 1 be a sequence of H-valued CNA random vectors with zero mean and E‖X n ‖ 2 < ∞, for all n ≥ 1. en, Lemma 3. Let X n , n ≥ 1 be a sequence of H-valued m-CNA random vectors with zero mean and E‖X n ‖ 2 < ∞ for all n ≥ 1. en, where we can define without loss of generality that X 0 � 0. From Definition 1, we see that X mi+l , i ≥ 0 is CNA for each (l � 1, 2, . . . , m). Hence, by c r inequality and Lemma 2, we have Proof. It follows by Hölder inequality and Lemma 3 that e proof is therefore complete.

□ Complexity
Lemma 5 (see [25]). Let X n , n ≥ 1 be a sequence of random variables satisfying n − 1 n i�1 P(|Z i | > x) ≤ CP(|Z| > x) for a random variable Z and any x ≥ 0. en, for any a > 0 and b > 0, there exist some positive constants c 1 and c 2 such that Following the method of Lemma 2.3 in [26], we can obtain the following inequality in Hilbert spaces. Lemma 6. Let Y i , 1 ≤ i ≤ n and Z i , 1 ≤ i ≤ n be two sequences of random vectors. en, for any q > 1, ε > 0, and a > 0, the following inequality holds:

Main Results and Their Proofs
In this section, we will present our main results. e first one is the weak law of large numbers for linear process Y t , t ≥ 1 of m-CNA random vectors in Hilbert spaces with random coefficients.
Proof. For each − ∞ < i < ∞ and j ∈ B, denote Note that

Complexity
It is sufficient to prove that, for any ε > 0, It follows from Lemma 1 that X ni , − ∞ < i ≤ ∞ is still a sequence of m-CNA random vectors. Furthermore, it is easy to check that, as n ⟶ ∞, Hence, we obtain by Chebyshev inequality, E( ∞ i�− ∞ A i ) 2 < ∞, Lemmas 4 and 5, and integration by parts that 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 which obtains equation (15) as n ⟶ ∞ by the assumption of eorem 1, and the proof is thus complete.
□ Remark 1. Hien and anh [9] obtained the weak law of large numbers for NA random vectors under the moment condition j∈B E|X (j) | < ∞. Contrasting to Corollary 2.5 of Hien and anh [9], eorem 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| < ∞.

a sequence of zero mean H-valued m-CNA random vectors coordinatewise weakly upper bounded by a random vector
and thus, Proof. Define for each − ∞ < i < ∞ and j ∈ B that Similar to the argument of equation (13), we have that, On the one hand, noting that W ni , − ∞ < i ≤ ∞ is still a sequence of m-CNA random vectors by Lemma 1, we obtain by Lemmas 4 and 5 that 6 Complexity On the other hand, it follows from Lemma 5 and Jensen inequality that Hence, it follows from Lemma 6 (with q � 2) and equations (22)-(24) that which obtains equation (19). Now, we prove equation (20). It follows from equation (19) that which combining with the arbitrariness of ε gets equation (20). e proof is complete.

Corollary 1.
Under the conditions of eorem 2, if αp > 1, we have that, for any ε > 0, and thus, Proof. It follows from eorem 2 that Furthermore, similar to the proof of equation (20), we have that 8 Complexity e proof is complete. For p � 1, we can obtain the following result. □ Theorem 3. Let α ≥ 1. Let X i , − ∞ < i < ∞ be a sequence of zero mean H-valued m-CNA random vectors coordinatewise weakly upper bounded by a random vector X with and thus, Proof. We still use the notations and method in the proof of eorem 2. Similar to the argument of equation (23), we have Furthermore, similar to the argument of equation (24), Hence, by Lemma 1 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). e 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 eorem 3 also works if α � p � 1. us, eorem 3 fills the vacancy and extends it to some more general settings. By eorems 2 and 3, one can obtain the following strong law of large numbers for linear process of m-CNA random vectors with random coefficients.

Conclusion
In this study, the concept of m-CNA random vectors is introduced as a natural extension of CNA random vectors. e 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. ese results improve and generalise the corresponding ones of recent works such as Hien and anh [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 restriction 1 ≤ p < 2; the first problem is that whether it is possible to release to p ≥ 2? Another problem is whether the moment condition j∈B E|X (j) |ln(1 + |X (j) |) < ∞ for the strong law of large numbers with p � 1 can be weakened to j∈B E|X (j) | < ∞?.

Data Availability
No data were used to support the findings of the study.

Conflicts of Interest
e author declares no conflicts of interest.