# An Exponential Inequality for Negatively Associated Random Variables

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## 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 Open image in new window for the strong law of large numbers, which improves the corresponding ones of Kim and Kim, Nooghabi and Azarnoosh, and Xing et al.

### Keywords

Convergence Rate Multivariate Distribution Fixed Probability Infinite Family Finite Family## 1. Introduction

whenever Open image in new window and Open image in new window 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 Open image in new window 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 Open image in new window can be taken as that of Marcinkiewicz-Zygmund (see Shao [13]).

Lemma 2.1.

If Open image in new window then it is possible to take Open image in new window

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

Lemma 2.2.

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

Lemma 2.3.

Proof.

## 3. Main results

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

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 Open image in new window which is greater than our upper bound.

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

Lemma 3.4.

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

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

Example 3.6.

Let Open image in new window where Open image in new window Then Open image in new window as Open image in new window and so the upper bound of (3.11) is Open image in new window The corresponding upper bound Open image in new window 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 Open image in new window By Example 3.6 with Open image in new window the upper bound of (3.11) is Open image in new window The corresponding upper bound Open image in new window was obtained by Xing et al. [12]. Hence our result improves Xing et al. [12, Theorem 5.1].

By choosing Open image in new window and Open image in new window in Theorem 3.5, we obtain the following result.

Theorem 3.8.

Remark 3.9.

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

The following example shows that the convergence rate Open image in new window is unattainable in Theorem 3.8.

Example 3.10.

which implies that the series Open image in new window diverges.

## Notes

### 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).

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