Limiting Behavior of the Partial Sum for Negatively Superadditive Dependent Random Vectors in Hilbert Space

Abstract

In this paper, We study the complete convergence and L_p- convergence for the maximum of the partial sum of negatively superadditive dependent random vectors in Hilbert space. The results extend the corresponding ones of Ko (Ko, 2020) to -valued negatively superadditive dependent random vectors.

1. Introduction

Alam and Saxena [1] introduced the concept of negative association as follows: a finite family of random variables is said to be negatively associated (NA) for every pair of disjoint subsets of and of , whenever and are coordinate-wise nondecreasing functions and the covariance exists. An infinite family is NA if every finite subfamily is NA.

A function is said to be superadditive if for all , where is for componentwise maximum and is for componentwise minimum. This notion was introduced by Kemperman [2].

Based on the above superadditive function, the concept of negatively superadditive dependent (NSD) random variables was introduced by Hu [3] as follows: a random vector is said to be negatively superadditive dependent (NSD) if , where are independent such that and have the same distribution for each and is a superadditive function such that the above expectations exist. A sequence of random variables is said to be NSD if for any , is NSD. Hu [3] gave an example for illustrating that negatively superadditive dependence (NSD) does not imply negative association (NA), and Christofides and Vaggelatou [4] indicated that NA implies NSD. So, the NSD structure is an extension of the NA structure. Sometimes, the NSD structure is more useful than the NA structure and NSD random variables have wide applications in reliability theory and multivariate statistical analysis. For this reason, studying the limit theorems for NSD random variables is much significant.

Let be a real separable Hilbert space with the norm generated by an inner product . Let be an orthonormal basis in , be an -valued random vector, and be denoted by .

Ko et al. [5] and Huan et al. [6] introduced the concept of negatively associated random vectors taking values in , Ko [7] introduced the concept of asymptotically negatively associated -valued random vectors, and Dung et al. [8] introduced the concept of pairwise NQD -valued random vectors.

As the concept of -valued NA random vectors was introduced by Ko et al. [5], Son et al. [9] presented the concept of -valued negatively superadditive dependent (NSD) random vectors as follows: a sequence of -valued random vectors is said to be NSD if for any , the sequence of -valued random vectors is negatively superadditive dependent (NSD).

Let be a sequence of -valued random vectors. We will use the following inequalities:

If there exists a positive constant such that the left-hand side (right-hand side) of (1) is satisfied for all and , then the sequence is said to be weakly lower (upper) bounded by . The sequence is said to be weakly bounded by if it is both lower and upper bounded by .

In this paper, we show the complete convergence results and -convergence of the maximum of the partial sums for NSD random vectors in Hilbert space. We also consider residual Cesàro alpha-integrability and strongly residual Cesàro alpha-integrability for NSD random vectors in Hilbert space.

Throughout the paper, the symbol denotes a generic constant which is not necessarily the same in each occurrence, and denotes the -norm. Moreover, represents the Vinogradov symbol and is the indicator function.

2. Some Lemmas

The following lemmas will be useful to prove the main results.

Lemma 1. (see [3]). If is negatively superadditive dependent (NSD) and are all nondecreasing functions, then is also NSD.

Lemma 2. (see [10]). Let be a sequence of NSD random variables with and for all . Then, for all , there is a positive constant such thatWe extend Lemma 2 to a sequence of Hilbert valued random vectors as follows.

Lemma 3. Let be a sequence of -valued NSD random vectors with and for every , where for any and . If is a sequence of NSD random variables with and for each , then there is a positive constant such that

Proof. Inspired by the proof of Lemma 1.7 of Huan et al. [6] and from Lemma 2, we have thatIt is obvious that if is an NSD sequence of -valued random vectors, where for any , then is a sequence of NSD random variables for each . However, the reverse is not true in general.

Lemma 4. (see [11]). Let and be sequences of non-negative numbers. If , then for every .

Lemma 5. (see [12]). Let be a sequence of -valued random vectors, weakly upper bounded by a random vector . Let for some .And(i)If , then (ii)(iii)

3. Main Results

A sequence of random vectors is said to converge completely to a constant if for any , .

In this case, we write completely. This notion was given by Hsu and Robbins [13]. Note that complete convergence implies the almost sure convergence in view of the Borel–Cantelli lemma.

Based on Lemma 3, we will extend the complete convergence results of the maximum of the partial sum of NSD random variables to the case of -valued random vectors.

Theorem 1. Let be a sequence of -valued NSD random vectors. If a sequence satisfiesthen for any ,where .

Proof. For each , let be the integer such that . Then, we obtain thatHence, it is enough to proveBy Lemma 3, the Hölder inequality, Lemma 4, and (6), we have thatwhich yields (9). Hence, the desired result (7) follows.
Based on Lemma 3, we will extend some -convergence of NSD random variables to the case of -valued random vectors.

Theorem 2. Let be a sequence of -valued NSD random vectors satisfying (6); then, for any ,

Proof. Let for any . Note that for any is a sequence of NSD random variables. Then, is a sequence of -valued NSD random variables by Lemma 1. By Lemma 3, Hölder’s inequality, and (6), we obtainwhich yields (11) for any . The proof of theorem is completed.
We consider -convergence of weakly upper bounded -valued NSD random vectors.

Theorem 3. Let be a sequence of -valued NSD random vectors which is weakly upper bounded by a random vector with . Then, for any , (11) holds.

Proof. By Lemma 3, Hölder’s inequality, , the proof of Theorem 2, and Lemma 5 (i), we obtainHence, (11) holds.
We will extend two special kinds of uniform integrability which were introduced by Chandra and Goswami [14] to -valued random vectors.

Definition 1 (see [15]). For , a sequence of random vectors in Hilbert space is said to be residually Cesàro alpha-integrable (RCI ()) ifClearly, is RCI for any if is identically distributed with and is RCI () for any if is stochastically dominated by a non-negative random vector with (see [15]).

Theorem 4. Let a random vector for any be non-negative where every component of is non-negative random variable for each . Let be a sequence of -valued NSD non-negative random vectors. If is RCI () for some , then for any ,

Proof. Let such that for all , where . For each , letDefine, for each and . Then, .
Note that is a sequence of NSD random variables and thatObviously, for each and , and are monotone transforms of the random variable by (16) and (17), respectively. Thus, and are NSD sequences of -valued random variables by Lemma 1. and are also NSD sequences of zero mean random variables. By Lemma 3 and the first condition of the RCI () property of (14) of the sequence , we obtainBy Lemma 3, the Hölder inequality and relation (19), and the second condition of the RCI () property (14) of the sequence , we obtainThus, by (20) and (21), we havewhich yields (15). The proof of theorem is completed.

Definition 2. (see [15]). For , a sequence of -valued random vectors is said to be strongly residually Cesàro alpha-integrable (SRCI ()) ifNote that is SRCI for any , provided that is stochastically dominated by a non-negative random vector with for .

Theorem 5. Define a -valued non-negative random vector as in Theorem 4 Let be a sequence of -valued NSD non-negative random vectors. If is SRCI for some , then for any ,

Proof. For each , let be the integer such that . Then, we have thatIt is sufficient to proveDefine , and as in the proof of Theorem 4. To prove (26), we will first show that completely. In other words, we will proveBy Lemma 3 and the Hölder inequality, we have thatIn view of the first condition of the SRCI property (23) of the sequence and Lemma 4, we conclude thatwhich yields (27). Next, we prove thatNamely, we will prove thatBy Lemma 3, the Hölder inequality, and (19) and the second condition of the SRCI property (23) of the sequence , we havewhich yields (31). Thus, by (27) and (31), the desired result (26) follows. The proof of Theorem 5 is completed.

4. Conclusions

In this article, we obtain the maximal moment inequality for a sequence of -valued NSD random vectors by extending the maximal moment inequality for a sequence of NSD random variables in Wang et al. [10] (see Lemma 3). Using this maximal moment inequality, we investigate the complete convergence results (see Theorem 1), -convergence (see Theorems 2 and 3), and residual Cesàro alpha-integrability and strongly residual Cesàro alpha-integrability (see Theorems 4 and 5) for -valued NSD random vectors.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This study was supported by Wonkwang University in 2020.