# An Exponential Inequality for Negatively Associated Random Variables

- Soo Hak Sung
^{1}Email author

**2009**:649427

**DOI: **10.1155/2009/649427

© Soo Hak Sung. 2009

**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

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.

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.

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

Lemma 2.3.

Proof.

## 3. Main results

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.

Proof.

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

Lemma 3.2.

Proof.

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)

- (i)By Markov's inequality and Lemma 2.1, we get(3.9)

- (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.

Proof.

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.

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.

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

## References

- 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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

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.