On the Exponential Inequality for Weighted Sums of a Class of Linearly Negative Quadrant Dependent Random Variables

Abstract

The exponential inequality for weighted sums of a class of linearly negative quadrant dependent random variables is established, which extends and improves the corresponding ones obtained by Ko et al. (2007) and Jabbari et al. (2009). In addition, we also give the relevant precise asymptotics.

1. Introduction

Lehmann [1] introduced a natural definition of negative dependence: two random variables and are said to be negative quadrant dependent (NQD, say) if for all . Based on the concept of NQD, another notion of negative dependence was formulated by Newman [2] as follows: a sequence of random variables is said to be linearly negative quadrant dependent (LNQD, say) if, for any disjoint subsets and of and positive ’s, , and are NQD. Recall that a finite family of random variables is said to be negatively associated (NA, say) if, for every pair of disjoint subsets and of , whenever and are coordinatewise increasing and the covariance exists. An infinite family is NA if every finite subfamily is NA. The concept of negative association was introduced by Joag-Dev and Proschan [3]. It is obvious to observe that NA sequences are LNQD and LNQD sequences are not necessarily NA, as it can be seen from the examples in Newman [2] or Joag-Dev and Proschan [3]. Hence, it is of interest to investigate the exponential inequality and its relevant result for LNQD sequences.

It is well-known that the exponential inequalities for partial sum play a very important role in various proofs of limit theorems. One can refer to Yang and Wang [4], T.-S. Kim and H.-C. Kim [5], Sung [6], Jabbari et al. [7], Xing et al. [8], Sung [9], and so on for further comprehension. As for the limit results about LNQD sequence, one can refer to Newman [2], Zhang [10], H. Kim and T. Kim [11], Wang and Zhang [12], and references therein.

Recently, Ko et al. [13] gave a Bernstein-Hoeffding type inequality for uniformly bounded LNQD random variables, by which they obtained the almost sure convergence rate of , where as . Motivated by the paper above, we establish the exponential inequality for weighted sums of uniformly bounded LNQD random variables. The result obtained extends and improves the corresponding ones given by Ko et al. [13] and Jabbari et al. [7]. Furthermore, we give the precise asymptotics with respect to the rate .

Throughout this paper, we always suppose that and denote positive constants independent of but whose value may vary over cases, denotes the integral part of , , , and and denote . This paper is organized as follows. Section 2 contains our main results. Section 3 contains the corresponding proofs.

2. Main Results

In this section, our main results will be given. For formulation of the theorems obtained, some assumptions are needed, which are listed below.(A1)Let be a sequence of stationary LNQD random variables with and let be a triangular array of numbers satisfying , where and are generic positive constants.(A2)Let be a positive integer sequence satisfying and as .

Theorem 1. Suppose that the assumption (A1) holds. Then for any , one has where and are positive constants.

Taking in Theorem 1, then by Borel-Cantelli lemma, we have

Corollary 2. Assume that the assumptions (A1) and (A2) hold. Let , for some . Suppose that is as in (3). Then one has In particular, one has when for .

Remark 3. (1) Theorem 1 generalizes Theorem 2.1 in Ko et al. [13] to weighted case. On the other hand, by (5), we can obtain that the strong convergence rate of is , which is obviously faster than the corresponding one that Ko et al. [13] obtained, where and as .
(2) Since LNQD sequences are strictly weaker than NA sequences, as mentioned in Section 1, Theorem 1 extends Theorem 2.1 in Jabbari et al. [7] from strictly stationary negatively associated setting to weighted LNQD case. In addition, by the analysis mentioned above, we know that the strong convergence rate of is much faster than the relevant one Jabbari et al. [7] obtained only for the special case of geometrically decreasing covariances.
(3) For the sequence of extended negatively dependent (END, say) or wide orthant dependent (WOD, say) random variables, similar results can also be obtained.

Regarding the convergence rate obtained in (5), we can give the following precise asymptotics.

Theorem 4. Let be a sequence of identically distributed LNQD random variables with and for some , , for some , and Then for , one has where denotes the standard normal random variable.

3. Proofs

Firstly, the following lemma is needed, which will be used in what follows.

Lemma 5 (see [10]). Let be an LNQD sequence with zero mean and finite th moment. Then, for , there exists a positive constant which only depends on such that for any .

Next, we give some notations used later. Define for and for . Set and then define for and Clearly, and .

Now, we can obtain the following lemma.

Lemma 6. Let be a sequence of stationary LNQD random variables with and let be a triangular array of numbers satisfying , where and are generic positive constants. If for some , then on account of Definitions (9) and (10), one has

Proof. Without loss of generality, assume that . Applying and , we have by that Therefore, in terms of the result above and the concept of LNQD, Also, since and , is LNQD. Therefore, from Lemma 3.1 in Ko et al. [13], we have Combining (13) and (14), we can get the desired result. Similarly, we can get the same result for . The proof is completed.

By Lemma 6, we can easily get the following result.

Lemma 7. Let be a sequence of stationary LNQD random variables with and let be a triangular array of numbers satisfying , where and are generic positive constants. If for some , then, for any and some ,

Proof. Applying Markov inequality and Lemma 6, we obtain Optimizing the exponent in the term of this upper-bound, we find that , so that this exponent becomes equal to as desired. The proof is completed.

Lemma 8 (see [10]). Under the conditions of Theorem 4, one has where is a standard normal random variable.

Lemma 9 (see [12]). Under the conditions of Theorem 4, we have for where denotes the standard normal distribution function.

Based on the above lemmas, the proofs of Theorems 1 and 4 can be given as follows.

Proof of Theorem 1. Let which satisfies ; then it follows from Lemma 7 that which completes the proof.

Proof of Theorem 4. Without loss of generality, set in what follows. Since it is sufficient to prove that for any .
To prove (21), we need only to show that by noting for any . It is easy to observe that for any , which implies (21).
Next, we will prove (22). Let , where and . Obviously, thus it suffices to prove that respectively. We consider firstly . Set . Noticing Lemma 8, we have as . It follows that which implies that . Turn to . By Lemma 9, it follows that for sufficiently small . Thus we obtain Hence, when . On the other hand, noting that and imply , we can obtain uniformly for . Thus we have , when . Combining the earlier results together yields (26). The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the Scientific Research Foundation of Guangxi Colleges and Universities (no. 2013LX019), the National Science Foundation of China (11061007), and the Natural Science Foundation of Guangxi (2011GXNSFA018133).