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The Hájek-Rènyi inequality and strong law of large numbers for ANA random variables
Journal of Inequalities and Applications volume 2014, Article number: 521 (2014)
Abstract
In this paper, the Hájek-Rènyi inequality and the strong law of large numbers for asymptotically negatively associated random variables are obtained. In particular, the classical Marcinkiewicz strong law of large numbers for negatively associated random variables is generalized to the case of asymptotically negative association.
MSC:60F05, 60F15.
1 Introduction
Let be a probability space and be a sequence of random variables defined on it.
A finite family of random variables is said to be negatively associated (NA) if for every pair of disjoint subsets and any real coordinatewise nondecreasing functions f on and g on
An infinite family of random variables is negatively associated if every finite subfamily is negatively associated. This concept was introduced by Joag-Dev and Proschan [1].
A new kind of dependence structure called asymptotically negative association was proposed by Zhang [2, 3] which is a useful weakening of the definition of negative association (see also Yuan and Wu [4]).
Definition (Yuan and Wu [4])
A sequence of random variables is said to be asymptotically negatively associated (ANA) if
where
and is the set of nondecreasing functions.
It is obvious that a sequence of asymptotically negatively associated random variables is negatively associated if and only if . Compared to negative association, asymptotically negative association defines a strictly larger class of random variables (for detailed examples, see Zhang [2]).
Consequently, the study of the limit theorems for asymptotically negatively associated random variables is of much interest.
For example, Zhang [3] proved the central limit theorem, Wang and Lu [5] obtained some inequalities of the maximum of partial sums and weak convergence, Wang and Zhang [6] established the law of the iterated logarithm, and Yuan and Wu [4] showed the limiting behavior of the maximum of partial sums.
Hájek and Rènyi [7] proved that if is a sequence of independent random variables with and , , and is a sequence of positive nondecreasing real numbers, then for any and for any positive integer ,
Since then, this inequality has been of concern for more and more authors (e.g., Chow [8] and Gan [9] for martingales, Liu et al. [10] for negatively associated random variables and Kim et al. [11] for asymptotically almost negatively associated random variables).
Inspired by Kim et al. [11], we will obtain the Hájek-Rènyi inequality type for asymptotically negatively associated random variables and prove the strong law of large numbers by using this inequality.
2 Hájek-Rènyi inequality for ANA random variables
Lemma 2.1 (Yuan and Wu [4])
Let be a sequence of asymptotically negatively associated (ANA) random variables and a sequence of positive numbers. Then is still a sequence of ANA random variables.
From Wang and Lu’s [5] Rosenthal type inequality for asymptotically negatively associated random variables we obtain the following.
Lemma 2.2 Let and N be a positive integer. Let be a sequence of asymptotically negatively associated random variables with , , and . Then, for all there is a positive constant such that
Theorem 2.3 Let be a sequence of positive nondecreasing real numbers and a sequence of mean zero, square integrable ANA random variables. Let . Then, for
where D is a positive constant defined in Lemma 2.2.
Proof First note that is a sequence of mean zero, square integrable ANA random variables by Lemma 2.1. Thus satisfies (1.2) for all coordinatewise increasing continuous functions f and g. Without loss of generality, set . Since
we get
and
From (2.3) we have
Hence by Lemma 2.2 the desired result (2.2) follows. □
From Theorem 2.3, we can get the following more generalized Hájek-Rènyi type inequality.
Theorem 2.4 Let be a sequence of positive nondecreasing real numbers. Let and N be a positive integer. Let be a sequence of mean zero and square integrable ANA random variables with and . Let , . Then, for and for any positive integer we have
where D is a positive constant defined in Lemma 2.2.
Proof By Theorem 2.3 we have
Hence the proof is complete. □
3 Strong law of large numbers for ANA random variables
Using the Hájek-Rènyi inequality for ANA random variables we will prove the strong law of large number for ANA random variables.
Theorem 3.1 Let and N be a positive integer. Let be a sequence of positive nondecreasing real numbers and a sequence of mean zero, square integrable random variables with and . Let , , and . Assume
Then, for any
-
(A)
,
-
(B)
implies a.s. as .
Proof (A) Note that
By Theorem 2.3, it follows from (3.1) that
where D is a positive constant defined in Lemma 2.2.
Hence the proof of (A) is complete.
(B) By Theorem 2.4 we get
But by assumption (3.1) we have
By the Kronecker lemma and (3.1) we get
Hence, by combining (3.1), (3.2), and (3.3) we have
i.e., a.s. as . □
Corollary 3.2 Let and N be a positive integer. Let be a sequence of mean zero, square integrable ANA random variables with and . Then, for
for all and , where D is a constant defined in Lemma 2.2,
Corollary 3.3 Let and N be a positive integer. Let be a sequence of mean zero and square integrable ANA random variables with and . Assume that
where , . Then, for
-
(A)
a.s. as ,
-
(B)
for any , where .
Finally, we consider almost convergence for weighted sums of ANA random variables as applications of Theorem 3.1.
Theorem 3.4 Let and N be a positive integer. Let be an array of real numbers with , , , and be a sequence of positive nondecreasing real numbers such that and let be a sequence of mean zero, square integrable ANA random variables satisfying , , and (3.1). Then
Proof Define
Then we obtain
and
Note that if an array of real numbers satisfies and for every fixed i then, for every sequence of real numbers with as
(See Kim et al. [11] for more details.) □
Hence, from the above fact and (3.5)-(3.9), the desired result (3.4) follows.
Theorem 3.5 Let and N be a positive integer. Let be an array of real numbers with , , , and be a sequence of positive nondecreasing real numbers such that and let be a sequence of mean zero, square integrable ANA random variables with and , where , . Then, for some
Proof By putting from Corollary 3.3 and Theorem 3.4, the result follows and the proof is omitted. □
Now we prove the Marcinkiewicz strong law of large numbers for ANA random variables by using Theorem 3.1. The method of proof is the same as that used in the classical Marcinkiewicz strong law of large numbers for i.i.d. random variables (see Stout [[12], Theorem 3.2.3]).
Theorem 3.6 Let and N be a positive integer. Let be a sequence of identically distributed ANA random variables with , for some and . Then
Sketch of proof To prove (3.10) it suffices to show that
and
where and .
Note that and are sequences of identically distributed ANA random variables. We only show (3.11). Equation (3.12) can be proved similarly.
Set , . Then is a sequence of identically distributed ANA random variables.
Note that .
So
We will prove
Notice that
By Kronecker’s lemma and (3.15) we see that (3.14) is true.
We also have
By Theorem 3.1 and (3.13)-(3.16) the proof of (3.11) is complete. □
Theorem 3.7 Let and N be a positive integer. Let be an array of real numbers with and let be a sequence of identically distributed ANA random variables with random variables with , , and for . Then, for some
Proof Basically, using the ideas in the proof of Theorem 3.4 and Theorem 3.6, we can obtain (3.17) and the proof is omitted. □
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Acknowledgements
The author wishes to thank the editor and the referees for their valuable comments. This paper was supported by Wonkwang University in 2014.
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Ko, MH. The Hájek-Rènyi inequality and strong law of large numbers for ANA random variables. J Inequal Appl 2014, 521 (2014). https://doi.org/10.1186/1029-242X-2014-521
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DOI: https://doi.org/10.1186/1029-242X-2014-521