Hájek–Rényi inequality for m-asymptotically almost negatively associated random vectors in Hilbert space and applications
Abstract
In this paper, we obtain the Hájek–Rényi inequality and, as an application, we study the strong law of large numbers for H-valued m-asymptotically almost negatively associated random vectors with mixing coefficients \(\{q(n), n\geq1\}\) such that \(\sum_{n=1}^{\infty}q(n)^{2}<\infty\).
Keywords
Asymptotically almost negative association Hilbert space Hájek–Rényi inequality Strong law of large numbers Mixing coefficientsMSC
60F151 Introduction
Let H be a real separable Hilbert space with the norm \(\Vert \cdot \Vert \) generated by an inner product \(\langle\cdot,\cdot\rangle\). Let \(\{ e_{j}, j\geq1\}\) be an orthonormal basis in H. For an H-valued random vector X, we denote \(X^{(j)}=\langle X, e_{j}\rangle\).
Ko et al. [1] also extended the concept of NA for \(\mathbb {R}^{d}\)-valued random vectors to random vectors with values in a real separable Hilbert space as follows: A sequence \(\{X_{n}, n\geq1\}\) of random vectors taking values in a real separable Hilbert space \((H, \langle\cdot,\cdot\rangle)\) is called NA if for some orthonormal basis \(\{e_{k}, k\geq1\}\) in H and for any \(d\geq1\), the d-dimensional sequence \(\{(\langle X_{n}, e_{1}\rangle, \dots, \langle X_{n}, e_{d}\rangle), n\geq1\}\) is NA.
The definitions of NA random vectors in \(\mathbb{R}^{d}\) and in a Hilbert space can be applied to asymptotically almost negative association (AANA).
Definition 1.1
In the case \(d=1\), the concept of AANA was introduced by Chandra and Ghosal [3, 4]. Obviously, AANA random variables contain independent random variables (with \(q(n)=0\) for \(n\geq1\)) and NA random variables. Chandra and Ghosal [3] pointed out that NA implies AANA, but AANA does not imply NA. Because NA has been applied to the reliability theory, multivariate statistical analysis, and percolation theory, the extension of the limit properties of NA random variables to AANA random variables is of interest in theory and applications.
Since the concept of AANA was introduced, various investigations have been established by many authors. For more detail, we can refer to Chandra and Ghosal [3, 4], Ko et al. [5], Yuan and An [6, 7], Wang et al. [8], Tang [9], Shen and Wu [10], and so forth.
Definition 1.2
A sequence \(\{X_{i}, i\geq1\}\) of random vectors taking values in a real separable Hilbert space \((H, \langle\cdot,\cdot\rangle)\) is said to be AANA if for some orthonormal basis \(\{e_{k}, k\geq1\}\) in H and for any \(d\geq1\), the d-dimensional sequence \(\{(\langle X_{i}, e_{1}\rangle,\ldots,\langle X_{i}, e_{d}\rangle), i\geq1\}\) is AANA.
We now introduce the notion of m-asymptotically almost negative association (m-AANA) in \(\mathbb{R}^{d}\) and in a Hilbert space.
Definition 1.3
In the case \(d=1\), Nam, Hu, and Volodin [11] introduced the concept of m-AANA and investigated maximal inequalities and strong law of large numbers.
The family of m-AANA sequence contains AANA (with \(m=1\)), NA, and independent sequences as particular cases.
Definition 1.4
Let \(m\geq1\) be an integer. A sequence \(\{X_{i}, i\geq1\}\) of random vectors taking values in a real separable Hilbert space \((H, \langle\cdot,\cdot\rangle)\) is said to be m-AANA if for some orthonormal basis \(\{e_{k}, k\geq1\}\) in H and for any \(d\geq1\), the d-dimensional sequence \(\{(\langle X_{i}, e_{1}\rangle,\ldots,\langle X_{i}, e_{d}\rangle), i\geq1\}\) is m-AANA.
In this paper, we obtain the Hájek–Rényi inequality, and, as an application, we give the strong law of large numbers for m-AANA random vectors in a Hilbert space. In particular, we extend the results in Nam, Hu, and Volodin [11] to a Hilbert space.
2 Some lemmas
We start with the property of m-asymptotically almost negatively associated (m-AANA) random variables, which can be easily obtained from the definition of m-AANA random variables.
Lemma 2.1
Let\(\{X_{n}, n\geq1\}\)be a sequence ofm-AANA random variables with mixing coefficients\(\{q(n), n\geq1\}\), and let\(\{f_{n}, n\geq1\}\)be a sequence of nondecreasing continuous functions. Then\(\{f_{n}(X_{n}), n\geq1\}\)is still a sequence ofm-AANA random variables with mixing coefficients\(\{q(n), n\geq1\}\).
Lemma 2.2
(Yuan and An [6])
Nam et al. [11] extended Lemma 2.2 to the case of m-AANA sequence.
Lemma 2.3
(Nam et al. [11])
From Lemma 2.3 we obtain the Rosenthal-type inequality for m-AANA random vectors with coefficients \(\{q(n), n\geq1\}\) such that \(\sum_{n=1}^{\infty}q^{2}(n)<\infty\) in a Hilbert space.
Lemma 2.4
Proof
Remark
3 Main results
Theorem 3.1
Proof
Remark
Proof
See the proof of Theorem 1 in Fazekas [16]. □
Note that by using (3.4) we can also prove Theorem 3.1.
From Theorem 3.1 we can get the following more generalized Hájek–Rényi inequality.
Theorem 3.2
Proof
Remark
Proof
See the proof of Theorem 3 in Fazekas [16]. □
4 Applications
Theorem 4.1
Proof
Theorem 4.2
Proof
Remark
Note that Theorem 4.2 can be also proved by the Hájek–Rényi-type maximal inequality for moments (3.9) (see Theorem 2 of Fazekas [16]).
From Theorem 3.2 we have the following:
Corollary 4.3
From Theorems 4.1 and 4.2 we have the following:
Corollary 4.4
5 Conclusions
Notes
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 interest regarding the publication of this article.
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