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\(L_{p}\)-convergence, complete convergence, and weak laws of large numbers for asymptotically negatively associated random vectors with values in \(\mathbb{R}^{d}\)
- Mi-Hwa Ko^{1}Email author
https://doi.org/10.1186/s13660-018-1699-6
© The Author(s) 2018
- Received: 1 December 2017
- Accepted: 25 April 2018
- Published: 8 May 2018
Abstract
In this paper, based on the Rosenthal-type inequality for asymptotically negatively associated random vectors with values in \(\mathbb{R}^{d}\), we establish results on \(L_{p}\)-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 \(\mathbb{R}^{d}\).
Keywords
- Asymptotically negative association
- Random vectors
- Rosenthal-type inequality
- \(L_{p}\)-convergence
- Complete convergence
- Weak laws of large numbers
MSC
- 60F15
1 Introduction
Ko et al. [1] introduced the concept of negative association (NA) for \(\mathbb{R}^{d}\)-valued random vectors.
Definition 1.1
(Ko et al. [1])
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.
In addition to Definition 1.1, for random vectors in \(\mathbb{R}^{d}\), we can define asymptotically negative association (ANA).
Definition 1.2
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 \(\rho^{-}(1)=0\). 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 \(L_{p}\)-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 \(\mathbb{R}^{d}\) was introduced as follows. Let \(\langle \cdot ,\cdot \rangle \) denote the inner product, and let \(\{e_{j}, j\geq 1\}\) be an orthonormal basis. A sequence \(\{X_{n}, n \geq 1\}\) of \(\mathbb{R}^{d}\)-valued random vectors is said to be coordinatewise negatively associated (CNA) if for each j \((1\leq j \leq d)\), the sequence \(\{X_{n}^{(j)}, n\geq 1, 1\leq j\leq d\}\) of random variables is NA, where \(X_{n}^{(j)}=\langle X_{n}, e_{j}\rangle \).
As in the definition of CNA, we can define coordinatewise asymptotically negative association for random vectors with values in \(\mathbb{R} ^{d}\).
Definition 1.3
A sequence \(\{X_{n}, n\geq 1\}\) of \(\mathbb{R}^{d}\)-valued random vectors is said to be coordinatewise asymptotically negatively associated (CANA) if for each j \((1\leq j\leq d)\), the sequence \(\{X_{n}^{(j)}, n\geq 1, 1\leq j \leq d\}\) of random variables is asymptotically negatively associated, where \(X_{n}^{(j)} =\langle X_{n}, e_{j}\rangle \) for \(n\geq 1\) and \(1\leq j\leq d\).
It is clear that if a sequence of \(\mathbb{R}^{d}\)-valued random vectors is ANA, then it is CANA. However, in general, the converse is not true.
In Sect. 2, we give some lemmas, which will be used to prove the main results, and in Sect. 3, we prove the \(L_{p}\)-convergence and complete convergence results for the maximums of the partial sums of the sequence of ANA random vectors with values in \(\mathbb{R}^{d}\). In addition, in Sect. 4, we establish a weak law of large numbers for CANA random vectors with values in \(\mathbb{R}^{d}\).
Throughout the paper, the symbol C denotes a generic constant \((0< C<\infty )\), which is not necessarily the same in each occurrence, \(S_{n}=\sum_{i=1}^{n} X_{i}\) for a sequence \(\{X_{n}, n\geq 1\}\) of random vectors, and \(\Vert \cdot \Vert _{p}\) denotes the \(L_{p}\)-norm. Moreover, ≪ represents the Vinogradov symbol O, and \(I(\cdot )\) 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 \(\{X_{n}, n\geq 1\}\) of ANA random vectors with mixing coefficients \(\rho^{-}(s)\) is also ANA with mixing coefficients not greater than \(\rho^{-}(s)\).
Wang and Lu [6] proved the following Rosenthal-type inequality for a sequence of ANA random variables in \(\mathbb{R}^{1}\).
Lemma 2.2
Inspired by the proof of Lemma 2.3 in Li-Xin Zhang [9], we extend Lemma 2.2 to \(\mathbb{R}^{d}\)-valued ANA random vectors as follows.
Lemma 2.3
Proof
From Lemma 1.2 of Kuczmaszewska [10] we obtain the following lemma.
Lemma 2.4
Let \(\{X_{n}, n\geq 1\}\) be a sequence of \(\mathbb{R}^{d}\)-valued random vectors weakly upper bounded by a random vector X, and let \(r>0\). Then, for some constant \(C>0\), \(E\Vert X \Vert ^{r}< \infty \) implies \(n^{-1}\sum_{k=1}^{n} E\Vert X_{k} \Vert ^{r}\leq CE\Vert X \Vert ^{r}\).
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])
Next, we will extend some \(L_{p}\)-convergence and complete convergence results for the maximums of the partial sum of \(\mathbb{R}^{1}\)-valued ANA random variables in Yuan and Wu [5] to \(\mathbb{R}^{d}\)-valued random vectors.
3 \(L_{p}\)-convergence and complete convergence for ANA random vectors with values in \(\mathbb{R}^{d}\)
The following theorem is an extension of Theorem 3.2 in Yuan and Wu [5] to random vectors with values in \(\mathbb{R}^{d}\).
Theorem 3.1
Proof
As applications of Theorem 3.1, we introduce two results that are not present in Yuan and Wu [5].
Theorem 3.2
Let \(p\geq 2\) be positive real numbers, and let \(N\geq 1\) be a positive integer. Suppose that \(\{X_{n}, n\geq 1\}\) is a sequence of \(\mathbb{R}^{d}\)-valued ANA random vectors with mixing coefficients \(\rho^{-}(s)\) such that \(\rho^{-}(N)<(\frac{1}{6p})^{p/2}\). If \(\{X_{n}, n\geq 1\}\) is weakly upper bounded by a random vector X with \(E\Vert X \Vert ^{p}<\infty \), then, for any \(\delta >\frac{1}{2}\), (3.2) holds.
Proof
Corollary 3.3
Let \(p\geq 2\) be positive real numbers, and let \(N\geq 1\) be a positive integer. Suppose that \(\{X_{n}, n\geq 1\}\) is a sequence of \(\mathbb{R}^{d}\)-valued ANA random vectors with mixing coefficients \(\rho^{-}(s)\) such that \(\rho^{-}(N)<(\frac{1}{6p})^{p/2}\). If \(\{X_{n}, n\geq 1\}\) are identically distributed random vectors with \(E\Vert X_{1} \Vert ^{p}<\infty \), then, for any \(\delta >\frac{1}{2}\), (3.2) holds.
The following theorem provides an extension of Theorem 4.2 of Yuan and Wu [5] for ANA random variables in \(\mathbb{R}^{1}\) to random vectors in \(\mathbb{R}^{d}\).
Theorem 3.4
Proof
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].
Theorem 3.5
Let \(p\geq 2\) be positive real numbers, and let \(N\geq 1\) be a positive integer. Suppose that \(\{X_{n}, n\geq 1\}\) is a sequence of \(\mathbb{R}^{d}\)-valued ANA random vectors with mixing coefficients \(\rho^{-}(s)\) such that \(\rho^{-}(N)<(\frac{1}{6p})^{p/2}\). If \(\{X_{n}, n\geq 1\}\) is weakly upper bounded by a random vector X with \(E\Vert X \Vert ^{p}<\infty \), then, for any \(\delta >\frac{1}{2}\), (3.4) holds.
Proof
Corollary 3.6
Let \(p\geq 2\) be positive real numbers, and let \(N\geq 1\) be a positive integer. Suppose that \(\{X_{n}, n\geq 1\}\) is a sequence of \(\mathbb{R}^{d}\)-valued ANA random vectors with mixing coefficients \(\rho^{-}(s)\) such that \(\rho^{-}(N)<(\frac{1}{6p})^{p/2}\). If \(\{X_{n}, n\geq 1\}\) are identically distributed random vectors with \(E\Vert X_{1} \Vert ^{p}<\infty \), then, for any \(\delta >\frac{1}{2}\), (3.4) holds.
4 Weak law of large numbers for ANA random vectors with values in \(\mathbb{R}^{d}\)
In this section, we establish the weak laws of large numbers for \(\mathbb{R}^{d}\)-valued ANA random vectors when \(p\geq 2\).
Theorem 4.1
Proof
Theorem 4.2
Proof
Remark
Suppose that \(\{X_{n}, n\geq 1\}\) is a sequence of \(\mathbb{R}^{d}\)-valued CNA random vectors. If \(\{X_{n}, n\geq 1\}\) is coordinatewise weakly upper bounded by a random vector X with \(\lim_{n\rightarrow \infty }\sum_{j=1}^{d} nP(\vert X^{(j)} \vert >n)=0\), then the WLLN (4.3) holds.
Corollary 4.3
Theorem 4.4
Proof
Since (4.12) implies (4.8), (4.3) follows from Theorem 4.2. Thus the proof is complete. □
Remark
Suppose that \(\{X_{n}, n\geq 1\}\) is a zero-mean sequence of \(\mathbb{R}^{d}\)-valued NA random vectors. If \(\{X_{n}, n \geq 1\}\) is coordinatewise weakly upper bounded by a random vector X with \(\sum_{j=1}^{d} E\vert X^{(j)} \vert <\infty \), then (4.11) holds.
Corollary 4.5
Proof
The proof follows by substituting \(X^{(j)}\) by \(X_{1}^{(j)}\) in the proof of Theorem 4.4. □
5 Conclusions
We generalized the \(L_{p}\)-convergence and complete convergence results of Yuan and Wu [5] from \(\mathbb{R}^{1}\)-valued ANA random variables to \(\mathbb{R}^{d}\)-valued random vectors by using a Rosenthal-type inequality. We also established weak laws of large numbers for CANA random vectors under \(p\geq 2\). As applications, we obtained some \(L_{p}\)-convergence and complete convergence results that are not present in Yuan and Wu [5] even when \(d=1\).
Declarations
Acknowledgements
This paper was supported by Wonkwang University in 2018.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The author declares that there is no conflict of interests regarding the publication of this article.
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Authors’ Affiliations
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