Open Access

An Exponential Inequality for Negatively Associated Random Variables

Journal of Inequalities and Applications20092009:649427

https://doi.org/10.1155/2009/649427

Received: 15 October 2008

Accepted: 7 May 2009

Published: 19 May 2009

Abstract

An exponential inequality is established for identically distributed negatively associated random variables which have the finite Laplace transforms. The inequality improves the results of Kim and Kim (2007), Nooghabi and Azarnoosh (2009), and Xing et al. (2009). We also obtain the convergence rate for the strong law of large numbers, which improves the corresponding ones of Kim and Kim, Nooghabi and Azarnoosh, and Xing et al.

1. Introduction

Let be a sequence of random variables defined on a fixed probability space The concept of negatively associated random variables was introduced by Alam and Saxena [1] and carefully studied by Joag-Dev and Proschan [2]. A finite family of random variables is said to be negatively associated if for every pair of disjoint subsets and of
(1.1)

whenever and are coordinatewise increasing and the covariance exists. An infinite family of random variables is negatively associated if every finite subfamily is negatively associated. As pointed out and proved by Joag-Dev and Proschan [2], a number of well-known multivariate distributions possess the negative association property, such as multinomial, convolution of unlike multinomial, multivariate hypergeometric, Dirichlet, permutation distribution, negatively correlated normal distribution, random sampling without replacement, and joint distribution of ranks.

The exponential inequality plays an important role in various proofs of limit theorems. In particular, it provides a measure of convergence rate for the strong law of large numbers. The counterpart of the negative association is positive association. The concept of positively associated random variables was introduced by Esary et al. [3]. The exponential inequalities for positively associated random variables were obtained by Devroye [4], Ioannides and Roussas [5], Oliveira [6], Sung [7], Xing and Yang [8], and Xing et al. [9]. On the other hand, Kim and Kim [10], Nooghabi and Azarnoosh [11], and Xing et al. [12] obtained exponential inequalities for negatively associated random variables.

In this paper, we establish an exponential inequality for identically distributed negatively associated random variables by using truncation method (not using a block decomposition of the sums). Our result improves those of Kim and Kim [10], Nooghabi and Azarnoosh [11], and Xing et al. [12]. We also obtain the convergence rate for the strong law of large numbers.

2. Preliminary lemmas

To prove our main results, the following lemmas are needed. We start with a well known lemma. The constant can be taken as that of Marcinkiewicz-Zygmund (see Shao [13]).

Lemma 2.1.

Let be a sequence of negatively associated random variables with mean zero and finite th moments, where Then there exists a positive constant depending only on such that
(2.1)

If then it is possible to take

The following lemma is due to Joag-Dev and Proschan [2]. It is still valid for any

Lemma 2.2.

Let be a sequence of negatively associated random variables. Then for any
(2.2)

The following lemma plays an essential role in our main results.

Lemma 2.3.

Let be negatively associated mean zero random variables such that
(2.3)
for a sequence of positive constants Then for any
(2.4)

Proof.

From the inequality for all we have
(2.5)
since for all It follows by Lemma 2.2 that
(2.6)

3. Main results

Let be a sequence of random variables and be a sequence of positive real numbers. Define for
(3.1)

Note that for For each fixed are bounded by If are negatively associated random variables, then are also negatively associated random variables, since are monotone transformations of

Lemma 3.1.

Let be a sequence of identically distributed negatively associated random variables. Let be as in (3.1). Then for any
(3.2)

Proof.

Noting that we have by Lemma 2.3 that
(3.3)

The following lemma gives an exponential inequality for the sum of bounded terms.

Lemma 3.2.

Let be a sequence of identically distributed negatively associated random variables. Let be as in (3.1). Then for any such that
(3.4)

Proof.

By Markov's inequality and Lemma 3.1, we have that for any
(3.5)
Putting note that we get
(3.6)
Since are also negatively associated random variables, we can replace by in the above statement. That is,
(3.7)
Observing that
(3.8)

the result follows by (3.6) and (3.7).

Remark 3.3.

From [14, Lemma  3.5] in Yang, it can be obtained an upper bound which is greater than our upper bound.

The following lemma gives an exponential inequality for the sum of unbounded terms.

Lemma 3.4.

Let be a sequence of identically distributed negatively associated random variables with for some Let be as in (3.1). Then, for any the following statements hold:

(i)

(ii)

Proof.
  1. (i)
    By Markov's inequality and Lemma 2.1, we get
    (3.9)
     
The rest of the proof is similar to that of [12, Lemma  4.1] in Xing et al. and is omitted.
  1. (ii)

    The proof is similar to that of (i) and is omitted.

     

Now we state and prove one of our main results.

Theorem 3.5.

Let be a sequence of identically distributed negatively associated random variables with for some Let where is a sequence of positive numbers such that
(3.10)
Then
(3.11)

Proof.

Note that and It follows by Lemmas 3.2 and 3.4 that
(3.12)

In Theorem 3.5, the condition on is (3.10). But, Kim and Kim [10], Nooghabi and Azarnoosh [11], and Xing et al. [12] used as only We give some examples satisfying the condition (3.10) of Theorem 3.5.

Example 3.6.

Let where Then as and so the upper bound of (3.11) is The corresponding upper bound was obtained by Kim and Kim [10] and Nooghabi and Azarnoosh [11]. Since our upper bound is much lower than it, our result improves the theorem in Kim and Kim [10] and Nooghabi and Azarnoosh [11, Theorem  5.1].

Example 3.7.

Let By Example 3.6 with the upper bound of (3.11) is The corresponding upper bound was obtained by Xing et al. [12]. Hence our result improves Xing et al. [12, Theorem  5.1].

By choosing and in Theorem 3.5, we obtain the following result.

Theorem 3.8.

Let be a sequence of identically distributed negatively associated random variables with for some Let Then
(3.13)

Remark 3.9.

By the Borel-Cantelli lemma, converges almost surely with rate The convergence rate is faster than the rate obtained by Xing et al. [12].

The following example shows that the convergence rate is unattainable in Theorem 3.8.

Example 3.10.

Let be a sequence of i.i.d. random variables. Then are negatively associated random variables with for any Set Then is also It is well known that (see Feller [15, page 175]). Thus we have that
(3.14)

which implies that the series diverges.

Declarations

Acknowledgments

The author would like to thank the referees for the helpful comments and suggestions that considerably improved the presentation of this paper. This work was supported by the Korea Science and Engineering Foundation (KOSEF) Grant funded by the Korea government (MOST) (no. R01-2007-000-20053-0).

Authors’ Affiliations

(1)
Department of Applied Mathematics, Pai Chai University

References

  1. Alam K, Saxena KML: Positive dependence in multivariate distributions. Communications in Statistics: Theory and Methods 1981,10(12):1183–1196. 10.1080/03610928108828102MathSciNetView ArticleGoogle Scholar
  2. Joag-Dev K, Proschan F: Negative association of random variables, with applications. The Annals of Statistics 1983,11(1):286–295. 10.1214/aos/1176346079MathSciNetView ArticleMATHGoogle Scholar
  3. Esary JD, Proschan F, Walkup DW: Association of random variables, with applications. Annals of Mathematical Statistics 1967,38(5):1466–1474. 10.1214/aoms/1177698701MathSciNetView ArticleMATHGoogle Scholar
  4. Devroye L: Exponential inequalities in nonparametric estimation. In Nonparametric Functional Estimation and Related Topics (Spetses, 1990), NATO Advanced Science Institutes Series C. Volume 335. Edited by: Roussas G. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:31–44.View ArticleGoogle Scholar
  5. Ioannides DA, Roussas GG: Exponential inequality for associated random variables. Statistics & Probability Letters 1999,42(4):423–431. 10.1016/S0167-7152(98)00240-5MathSciNetView ArticleMATHGoogle Scholar
  6. Oliveira PE: An exponential inequality for associated variables. Statistics & Probability Letters 2005,73(2):189–197. 10.1016/j.spl.2004.11.023MathSciNetView ArticleMATHGoogle Scholar
  7. Sung SH: A note on the exponential inequality for associated random variables. Statistics & Probability Letters 2007,77(18):1730–1736. 10.1016/j.spl.2007.04.012MathSciNetView ArticleMATHGoogle Scholar
  8. Xing G, Yang S: Notes on the exponential inequalities for strictly stationary and positively associated random variables. Journal of Statistical Planning and Inference 2008,138(12):4132–4140. 10.1016/j.jspi.2008.03.024MathSciNetView ArticleMATHGoogle Scholar
  9. Xing G, Yang S, Liu A: Exponential inequalities for positively associated random variables and applications. Journal of Inequalities and Applications 2008, 2008:-11.Google Scholar
  10. Kim T-S, Kim H-C: On the exponential inequality for negative dependent sequence. Communications of the Korean Mathematical Society 2007,22(2):315–321. 10.4134/CKMS.2007.22.2.315View ArticleMATHGoogle Scholar
  11. Nooghabi HJ, Azarnoosh HA: Exponential inequality for negatively associated random variables. Statistical Papers 2009,50(2):419–428. 10.1007/s00362-007-0081-4MathSciNetView ArticleMATHGoogle Scholar
  12. Xing G, Yang S, Liu A, Wang X: A remark on the exponential inequality for negatively associated random variables. Journal of the Korean Statistical Society 2009,38(1):53–57. 10.1016/j.jkss.2008.06.005MathSciNetView ArticleMATHGoogle Scholar
  13. Shao Q-M: A comparison theorem on moment inequalities between negatively associated and independent random variables. Journal of Theoretical Probability 2000,13(2):343–356. 10.1023/A:1007849609234MathSciNetView ArticleMATHGoogle Scholar
  14. Yang S: Uniformly asymptotic normality of the regression weighted estimator for negatively associated samples. Statistics & Probability Letters 2003,62(2):101–110.MathSciNetView ArticleMATHGoogle Scholar
  15. Feller W: An Introduction to Probability Theory and Its Applications. Vol. I. 3rd edition. John Wiley & Sons, New York, NY, USA; 1968:xviii+509.Google Scholar

Copyright

© Soo Hak Sung. 2009

This article is published under license to BioMed Central Ltd. 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.