An Exponential Inequality for Negatively Associated Random Variables

Open Access
Research Article

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

Let Open image in new window be a sequence of random variables defined on a fixed probability space Open image in new window 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 Open image in new window is said to be negatively associated if for every pair of disjoint subsets Open image in new window and Open image in new window of Open image in new window

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.

Let Open image in new window be a sequence of negatively associated random variables with mean zero and finite Open image in new window th moments, where Open image in new window Then there exists a positive constant Open image in new window depending only on Open image in new window such that

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.

Let Open image in new window be a sequence of negatively associated random variables. Then for any Open image in new window

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

Lemma 2.3.

Let Open image in new window be negatively associated mean zero random variables such that
for a sequence of positive constants Open image in new window Then for any Open image in new window

Proof.

since Open image in new window for all Open image in new window It follows by Lemma 2.2 that

3. Main results

Let Open image in new window be a sequence of random variables and Open image in new window be a sequence of positive real numbers. Define for Open image in new window

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.

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

Proof.

Noting that Open image in new window we have by Lemma 2.3 that

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

Lemma 3.2.

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

Proof.

By Markov's inequality and Lemma 3.1, we have that for any Open image in new window
Since Open image in new window are also negatively associated random variables, we can replace Open image in new window by Open image in new window in the above statement. That is,
Observing that

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) Open image in new window

(ii) Open image in new window

Proof.
  1. (i)
    By Markov's inequality and Lemma 2.1, we get
     
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 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 where Open image in new window is a sequence of positive numbers such that

Proof.

Note that Open image in new window and Open image in new window It follows by Lemmas 3.2 and 3.4 that

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.

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 Then

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.

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

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).

References

  1. 1.
    Alam K, Saxena KML: Positive dependence in multivariate distributions. Communications in Statistics: Theory and Methods 1981,10(12):1183–1196. 10.1080/03610928108828102MathSciNetCrossRefGoogle Scholar
  2. 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/1176346079MathSciNetCrossRefMATHGoogle Scholar
  3. 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/1177698701MathSciNetCrossRefMATHGoogle Scholar
  4. 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.CrossRefGoogle Scholar
  5. 5.
    Ioannides DA, Roussas GG: Exponential inequality for associated random variables. Statistics & Probability Letters 1999,42(4):423–431. 10.1016/S0167-7152(98)00240-5MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Oliveira PE: An exponential inequality for associated variables. Statistics & Probability Letters 2005,73(2):189–197. 10.1016/j.spl.2004.11.023MathSciNetCrossRefMATHGoogle Scholar
  7. 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.2007.04.012MathSciNetCrossRefMATHGoogle Scholar
  8. 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.2008.03.024MathSciNetCrossRefMATHGoogle Scholar
  9. 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. 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.2007.22.2.315CrossRefMATHGoogle Scholar
  11. 11.
    Nooghabi HJ, Azarnoosh HA: Exponential inequality for negatively associated random variables. Statistical Papers 2009,50(2):419–428. 10.1007/s00362-007-0081-4MathSciNetCrossRefMATHGoogle Scholar
  12. 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.2008.06.005MathSciNetCrossRefMATHGoogle Scholar
  13. 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:1007849609234MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Yang S: Uniformly asymptotic normality of the regression weighted estimator for negatively associated samples. Statistics & Probability Letters 2003,62(2):101–110.MathSciNetCrossRefMATHGoogle Scholar
  15. 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 information

© 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.

Authors and Affiliations

  1. 1.Department of Applied MathematicsPai Chai UniversityTaejonSouth Korea

Personalised recommendations