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Some probability inequalities for a class of random variables and their applications
Journal of Inequalities and Applications volume 2013, Article number: 57 (2013)
Abstract
Some probability inequalities for a class of random variables are presented. As applications, we study the complete convergence for it. Our main results generalize the corresponding ones for negatively associated random variables and negatively orthant dependent random variables.
MSC:60E15, 60F15.
1 Introduction
Let be a sequence of random variables defined on a fixed probability space . The exponential inequality for the partial sums 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 main purpose of the paper is to present some probability inequalities for a class of random variables. As applications, we will give some complete convergence for a class of random variables.
Firstly, we will recall the definitions of negatively orthant dependent random variables and acceptable random variables.
Definition 1.1 A finite collection of random variables is said to be negatively orthant dependent (NOD, in short) if the following two inequalities:
and
hold for all real numbers . An infinite sequence is said to be NOD if every finite subcollection is NOD.
The notion of NOD random variables was introduced by Lehmann [1] and developed in Joag-Dev and Proschan [2]. Obviously, independent random variables are NOD. Joag-Dev and Proschan [2] pointed out that negatively associated (NA, in short) random variables are NOD, but neither NUOD nor NLOD implies NA. They also presented an example in which possesses NOD, but does not possess NA. So, we can see that NOD is weaker than NA.
Recently, Giuliano Antonini et al. [3] introduced the following notion of acceptability.
Definition 1.2 We say that a finite collection of random variables is acceptable if for any real λ,
An infinite sequence of random variables is acceptable if every finite subcollection is acceptable.
As is mentioned in Giuliano Antonini et al. [3], a sequence of NOD random variables with a finite Laplace transform or finite moment generating function near zero (and hence a sequence of negatively associated random variables with finite Laplace transform, too) provides us an example of acceptable random variables. For example, Xing et al. [4] consider a strictly stationary negatively associated sequence of random variables. According to the sentence above, the sequence of strictly stationary and negatively associated random variables is acceptable. Hence, the model of acceptable random variables is more general than models considered in the previous literature. Studying the limiting behavior of acceptable random variables is of interest.
The main purpose of the paper is to present some exponential probability inequalities for a sequence of acceptable random variables and give some applications by using these exponential probability inequalities. For more details about the exponential probability inequality, one can refer to Wang et al. [5–7], Sung [8], Sung et al. [9] and Xing et al. [4, 10], and so forth.
The paper is organized as follows. The exponential probability inequalities for a sequence of acceptable random variables are presented in Section 2, and the complete convergence for it is obtained in Section 3. Our results are based on some moment conditions, while the main results of Sung et al. [9] are under the condition of moment and identical distribution.
Throughout the paper, let be a sequence of acceptable random variables and denote for each .
2 Probability inequalities for acceptable random variables
Theorem 2.1 Let be a sequence of acceptable random variables and be a sequence of positive numbers with for each . For fixed , if there exists a positive number T such that
then
Proof For each x, by Markov’s inequality, Definition 1.2 and (2.1), we can see that
which implies that
For fixed , if , then
if , then
The desired result (2.2) follows from (2.4)-(2.6) immediately. □
Corollary 2.1 Let be a sequence of acceptable random variables and be a sequence of positive numbers with for each . For fixed , if there exists a positive number T such that
then
and
Proof It is easily seen that is still a sequence of acceptable random variables. By Theorem 2.1, we can see that
which implies that (2.8) is valid. Finally, (2.9) follows (2.2) and (2.8) immediately. □
Corollary 2.2 Let be a sequence of acceptable random variables with and for each . Denote for each . For fixed , if there exists a positive number H such that
for any positive integer , then
Proof By (2.11), we can see that
for . When , it follows that
where . Take and . Hence, the conditions of Corollary 2.1 are satisfied. Therefore, (2.12) follows from Corollary 2.1 immediately. □
3 Complete convergence for acceptable random variables
In this section, we will present some complete convergence for a sequence of acceptable random variables by using the probability inequality.
Theorem 3.1 Let be a sequence of acceptable random variables with and for each . Denote , . For fixed , suppose that there exists a positive number H such that (2.11) holds true. If for any ,
and
where is a sequence of positive numbers, then completely as .
Proof By Corollary 2.2, we have for any ,
which implies that
This completes the proof of the theorem. □
It is easily seen that (3.2) holds if . So, we have the following corollary.
Corollary 3.1 Let be a sequence of acceptable random variables with and for each . Denote , . Suppose that conditions (2.11) and (3.1) hold with . Then completely as .
Theorem 3.2 Let be a sequence of acceptable random variables with and for each . Denote and
If for any ,
then completely as .
Proof By Markov’s inequality and Definition 1.2, for any and ,
Taking in the inequality above, we can get that
It follows from the inequality above and (2.5) that
which completes the proof of the theorem. □
Hanson and Wright (1971) obtained a bound on tail probabilities for quadratic forms in independent random variables using the following condition: for all and all , there exist positive constants M and γ such that
Wright [11] proved that the bound established by Hanson and Wright [12] for independent symmetric random variables also holds when the random variables are not symmetric but condition (3.4) is valid. We will study the complete convergence for a sequence of acceptable random variables under condition (3.4). The main result is as follows.
Theorem 3.3 Let be a sequence of acceptable random variables satisfying condition (3.4) for all and all , where M and γ are positive constants. Suppose that there exists a positive constant C not depending on n such that
Then for all , completely as .
Proof By Markov’s inequality and assumption (3.5), we have that for any ,
In the following, we will estimate . By (3.4), we can see that
Hence,
This completes the proof of the theorem. □
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Acknowledgements
The authors are most grateful to the editor and the anonymous referee for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11201001, 11171001, 11126176 and 11226207), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20093401120001), the Natural Science Foundation of Anhui Province (1308085QA03, 11040606M12, 1208085QA03), the 211 project of Anhui University and the Students Science Research Training Program of Anhui University (KYXL2012007).
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Shen, A., Wu, R. Some probability inequalities for a class of random variables and their applications. J Inequal Appl 2013, 57 (2013). https://doi.org/10.1186/1029-242X-2013-57
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DOI: https://doi.org/10.1186/1029-242X-2013-57
Keywords
- acceptable random variables
- probability inequality
- complete convergence