An Exponential Inequality for Negatively Associated Random Variables

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.

• (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.

• (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.