- Open Access
Some probability inequalities for a class of random variables and their applications
© Shen and Wu; licensee Springer 2013
- Received: 2 November 2012
- Accepted: 1 February 2013
- Published: 19 February 2013
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.
- acceptable random variables
- probability inequality
- complete convergence
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.
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  and developed in Joag-Dev and Proschan . Obviously, independent random variables are NOD. Joag-Dev and Proschan  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.  introduced the following notion of acceptability.
An infinite sequence of random variables is acceptable if every finite subcollection is acceptable.
As is mentioned in Giuliano Antonini et al. , 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.  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 , Sung et al.  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.  are under the condition of moment and identical distribution.
Throughout the paper, let be a sequence of acceptable random variables and denote for each .
The desired result (2.2) follows from (2.4)-(2.6) immediately. □
which implies that (2.8) is valid. Finally, (2.9) follows (2.2) and (2.8) immediately. □
where . Take and . Hence, the conditions of Corollary 2.1 are satisfied. Therefore, (2.12) follows from Corollary 2.1 immediately. □
In this section, we will present some complete convergence for a sequence of acceptable random variables by using the probability inequality.
where is a sequence of positive numbers, then completely as .
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 .
then completely as .
which completes the proof of the theorem. □
Wright  proved that the bound established by Hanson and Wright  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.
Then for all , completely as .
This completes the proof of the theorem. □
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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