Convergence Theorems for <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2724395pt" id="M1" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 13.7215 8.14084" width="13.7215pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-110" d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"/></g></svg>-</span>Coordinatewise Negatively Associated Random Vectors in Hilbert Spaces

Shi, Lyurong

doi:https://doi.org/10.1155/2021/3462317

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Finite-time Control of Complex Systems and Their Applications 2021

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Research Article | Open Access

Volume 2021 | Article ID 3462317 | https://doi.org/10.1155/2021/3462317

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

Lyurong Shi¹

Academic Editor: Xiaodi Li

Received07 May 2021

Revised02 Jun 2021

Accepted14 Jun 2021

Published20 Jul 2021

Abstract

In this study, some new results on convergence properties for -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 -coordinatewise negatively associated random vectors with random coefficients are established. These results improve and generalise some corresponding ones in the literature.

1. Introduction

The random variables are said to be negatively associated (NA, in short) if, for every pair of disjoint subsets and of and any real coordinatewise nondecreasing (or nonincreasing) functions on and on ,whenever the covariance above exists, where and denote the cardinalities of and , respectively. A sequence 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 -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 -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].

Let be a real separable Hilbert space with the norm generated by an inner product . Denote , where is an orthonormal basis in , is also a random vector taking values in , and is a subset of .

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 and . Let be a sequence of i.i.d. random variables with zero mean. Then, the following statements are equivalent:(i)(ii)(iii)

Huan et al. [18] proved the sufficient conditions of the Baum–Katz type complete convergence with and , which partially extends Theorem A to CNA random vectors in Hilbert space. Huan [20] considered the case with . Ko [21] extended the result of Huan et al. [18] to the complete moment convergence, where, however, the case is wrongly proved as pointed out by Huang and Wu [22]. Ko [23] also established the complete moment convergence with and . However, there are still some mistakes for . To be specific, it shows in equation (2.9) of [23] that , where we know it is wrong when ; furthermore, the formula in equation (2.11) of [23] is yet invalid for . 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 . 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 of H-valued -coordinatewise negatively associated (-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, let denote a generic positive constant whose value may vary in different lines. and implies the indicator function of the set . represents the set of integers.

The 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.

2. Preliminaries

In this section, we will present some concepts and important lemmas as below.

Definition 1. (see [18]). A sequence of random vectors taking values in is said to be CNA if, for each , the sequence of random variables is NA, where . Inspired by Hu et al. [3] and Huan et al. [18], we introduce the concept of -CNA random vectors in Hilbert space as follows.

Definition 2. Let be a given integer. A sequence of random variables is said to be -CNA if, for any and any such that for all , we have that are CNA. Obviously, if , then -CNA random vectors is CNA. Hence, the concept of -CNA random vectors is a natural extension of that of CNA random vectors. We also present an example of -CNA random vectors which are not CNA, as follows.

Example 1. Let be a sequence of independent random vectors, where follows the standard multivariate normal distribution for each . Take for each , where . Then, it is easy to check that, for any and , but and for . That is to say, is a sequence of -CNA random vectors with . We also introduce the following concept of the linear process of random vectors in Hilbert spaces with random coefficients.

Definition 3. Assume that is a sequence of -valued random vectors and is a sequence of random variables. The sequence of random vectors is said to be linear process with random coefficients ifThe following concept is often used in the literature while dealing with the convergence theorems of random vectors in Hilbert spaces.

Definition 4. A sequence is said to be coordinatewise weakly upper bounded by if there exists a positive constant such that , for all and .

Lemma 1. Let be a sequence of -CNA random vectors. If is a sequence of coordinatewise nondecreasing (or nonincreasing) continuous real functions, then is still a sequence of -CNA random vectors.

Proof. It follows from Definitions 1 and 2 that, for any and any taking values in such that for all , is a sequence of NA random variables for each . By Lemma 2.1 of [24], one can see that is still a sequence of NA random variables for each . Hence, by Definitions 1 and 2 again, it follows that are CNA, i.e., is -CNA.

Lemma 2. (see [18]). Let be a sequence of -valued CNA random vectors with zero mean and , for all . Then,

Lemma 3. Let be a sequence of -valued -CNA random vectors with zero mean and for all . Then,

Proof. Notice thatwhere we can define without loss of generality that . From Definition 1, we see that is CNA for each . Hence, by inequality and Lemma 2, we have

Lemma 4. Let be a sequence of -valued -CNA random vectors with zero mean and for all . If is a sequence of random variables independent of and , then

Proof. It follows by Hölder inequality and Lemma 3 thatThe proof is therefore complete.

Lemma 5. (see [25]). Let be a sequence of random variables satisfying for a random variable and any . Then, for any and , there exist some positive constants and such thatFollowing the method of Lemma 2.3 in [26], we can obtain the following inequality in Hilbert spaces.

Lemma 6. Let and be two sequences of random vectors. Then, for any , , and , the following inequality holds:

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 process of -CNA random vectors in Hilbert spaces with random coefficients.

Theorem 1. Let be a sequence of zero mean -valued -CNA random vectors coordinatewise weakly upper bounded by a random vector . Suppose that is a sequence of random variables independent of and . Assume further as if the cardinality or if . Then,

Proof. For each and , denoteNote thatIt is sufficient to prove that, for any ,It follows from Lemma 1 that is still a sequence of -CNA random vectors. Furthermore, it is easy to check that, as ,Hence, we obtain by Chebyshev inequality, , Lemmas 4 and 5, and integration by parts thatwhich converges to 0 as , and thus, equation (14) holds true. On the contrary, we have, by Markov inequality, Lemma 5, and Jensen inequality, thatwhich obtains equation (15) as 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 . 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 -CNA random vectors with random coefficients but also improves the moment condition when .

Theorem 2. Let and . Let be a sequence of zero mean -valued -CNA random vectors coordinatewise weakly upper bounded by a random vector with . Suppose that is a sequence of random variables independent of and . Then, for any ,and thus,

Proof. Define for each and thatSimilar to the argument of equation (13), we have that, for each ,On the one hand, noting that is still a sequence of -CNA random vectors by Lemma 1, we obtain by Lemmas 4 and 5 thatOn the other hand, it follows from Lemma 5 and Jensen inequality thatHence, it follows from Lemma 6 (with ) and equations (22)–(24) thatwhich obtains equation (19). Now, we prove equation (20). It follows from equation (19) thatwhich combining with the arbitrariness of 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 , , and , . Although Ko [21] extended the result of Huan et al. [18] to complete moment convergence, it only holds for , , 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 -CNA random vectors with random coefficients. By Theorem 2, we can easily get the following conclusion.

Corollary 1. Under the conditions of Theorem 2, if , we have that, for any ,and thus,

Proof. It follows from Theorem 2 thatFurthermore, similar to the proof of equation (20), we have thatThe proof is complete.
For , we can obtain the following result.

Theorem 3. Let . Let be a sequence of zero mean -valued -CNA random vectors coordinatewise weakly upper bounded by a random vector with for some . Suppose that is a sequence of random variables independent of and . Then, for any ,and thus,

Proof. We still use the notations and method in the proof of Theorem 2. Similar to the argument of equation (23), we haveFurthermore, similar to the argument of equation (24), we haveHence, 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). The proof is complete.

Remark 3. Ko [23] proved the complete moment convergence for coordinatewise asymptotically almost negatively associated (CAANA) random vectors with and . However, as stated in Section 1, the meaningful case is wrongly proved. Note that Theorem 3 also works if . 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 -CNA random vectors with random coefficients.

Corollary 2. Let . Let be a sequence of zero mean -valued -CNA random vectors coordinatewise weakly upper bounded by a random vector with if or if . Suppose that is a sequence of random variables independent of and . Then, with probability 1,

Proof. Let . It follows from equations (20) and (32) (with ) thatBy Borel–Cantelli lemma, the formula above implies that, as , with probability 1,Meanwhile, for any fixed , there exists positive integer such that . Hence, with probability 1, one has

4. Conclusion

In this study, the concept of -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 -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 restriction ; the first problem is that whether it is possible to release to ? Another problem is whether the moment condition for the strong law of large numbers with can be weakened to ?.

Data Availability

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

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

This work was supported by the Key Research Projects of Natural Science in Colleges and Universities of Anhui (KJ2020A0908) and the Key Research Projects of Natural Science of Wuhu Institute of Technology (Wzyzrzd202006).

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Copyright © 2021 Lyurong Shi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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