An Exponential Inequality for Negatively Associated Random Variables
- Soo Hak Sung1Email author
https://doi.org/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.
Keywords
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:
- (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.









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
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