Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 9037258, 7 pages
https://doi.org/10.1155/2019/9037258

Research Article

Some Types of Convergence for Negatively Dependent Random Variables under Sublinear Expectations

Ruixue Wang and Qunying Wu

College of Science, Guilin University of Technology, Guilin 541004, China

Correspondence should be addressed to Qunying Wu; wqy666@glut.edu.cn

Received 13 October 2018; Revised 20 November 2018; Accepted 16 December 2018; Published 2 January 2019

Academic Editor: Zuonong Zhu

Copyright © 2019 Ruixue Wang and Qunying Wu. 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.

Abstract.

In this paper, we research complete convergence and almost sure convergence under the sublinear expectations. As applications, we extend some complete and almost sure convergence theorems for weighted sums of negatively dependent random variables from the traditional probability space to the sublinear expectation space.

1. Introduction

As is well known that limit theorems are important research topics in probability and statistics, classical limit theorems only hold in the case of model certainty. However, in financial and other field research, statistical model uncertainty, risk random fluctuation, and other problems are emerging, and the classical probabilistic space cannot meet the need of solving such problems. Academician Peng [1–3] of Shandong University creatively proposed the concept and system of the sublinear expectation. Since it was introduced, it has been widely concerned and studied by a large number of scholars, and many excellent results have been achieved. For instance, Chen [4] studied Kolmogorov’s strong law of larger numbers. Zhang and Chen [5] established a weighted central limit theorem for independent random variables under sublinear expectation. Zhang [6–8] got the expectation inequalities, Rosenthals inequalities, and strong law of larger numbers. Yu and Wu [9] obtained complete convergence for weighted sums of END random variables under sublinear expectations, and so on.

Complete convergence and almost sure convergence are important problems in limit theorems. The initial definition of complete convergence was presented by Hsu and Robbins [10] in 1947. From then on, lots of results on complete convergence for different sequences have been found under classical probability space. For instance, Wu and Jiang [11] obtained complete convergence for negatively associated sequences of random variables. Wang et al. [12, 13] studied complete convergence for martingale difference sequence and complete convergence for a type of random variables satisfying Rosenthal-type inequality. Some results on almost sure convergence can be found in Sung et al. [14, 15], Wu [16], Chen and Gan [17], Xu and Yu [18], and so on. In this article, we establish the complete convergence and almost sure convergence for weighted sums of ND random variables under sublinear expectations. The results obtained by Sung [15] have been generalized to the sublinear expectation space.

2. Preliminaries

We also use the framework and notations of Peng [1–3]. Here, we do not discuss notations, concepts, and properties of the sublinear expectation space in detail. By the properties of sublinear expectations and capacity, we can get Markov inequality:

Definition 1 ([6]). (i) (Negative dependence) [6] In a sublinear expectation space , a random vector is said to be negatively dependent (ND) to another random vector under if for each pair of test functions and we have , whenever , are coordinate-wise nondecreasing or , are coordinate-wise nonincreasing with , ,  ,  ,  .
(ii) (ND random variables) [6] Let be a sequence of random variables in the sublinear expectation space . are said to be negatively dependent if is negatively dependent to for each .

Obviously, if is a sequence of ND random variables and functions are all nondecreasing (resp., all nonincreasing), then is also a sequence of ND random variables.

The symbol stands for a positive constant which may have different values in different places. Let denote that there exists a constant such that for sufficiently large .

Lemma 2. Let be an array of row-wise ND random variables with . Suppose that
(a) ,
(b) .
Then, for any ,

Proof. Taking in Lemma 2.2 of [9], we can get conditions (a) and (b). The condition (iii) of Lemma 2.2 of [9] is clearly satisfied for and . So, we can get Lemma 2.

Lemma 3. Assume , , . Then

Proof. is equivalent to . Note that

Therefore, (3) holds. Then, let

Thus, (4) holds.

Lemma 4 (Borel-Cantelli Lemma, Zhang [6], Lemma 3.9). Let be a sequence of events in . Suppose that is a countably subadditive capacity. If , then , where .

In the sublinear expectation space, because of the uncertainty of expectation and capacity, the study of complete convergence and almost sure convergence is much more complex and difficult. As we all know, in the probability space, there is an equality: . However, the expression does not exist in the sublinear expectation space. This needs to modify the indicator function by functions in . To this end, for , we define the even function , for all , and if , if . Then

3. Main Results

Theorem 5. Let , and be an array of row-wise negatively dependent random variables under sublinear expectations. There exist a r.v. and a constant satisfying

Let be an array of positive constants such that

Then, for any positive whole number satisfying , and ,

and

In particular, if , then

Remark 6. Condition (8) is similar to stochastic domination condition in Theorem 2.1 of [15]. Condition (9) corresponds to the moment condition of probability space. Condition (11) is similar to condition (ii) of Theorem 2.1 of [15]. Our results extend Theorem 2.1 of [15] from the traditional probability space to sublinear expectation space.

Theorem 7. Suppose that , and is countably subadditive. Let be a sequence of negatively dependent random variables. There exist a r.v. and a constant satisfying

Let be an array of positive constants such that

Then

and

In particular, if , then

Theorem 8. Suppose that , and is countably subadditive. Let be a sequence of negatively dependent random variables. There exist a r.v. and a constant satisfying (15), (16). Let be an array of positive constants such that (17), (18) hold. Then we also can get (20), (21), and (22).

Theorem 9. Suppose that , and is countably subadditive. Let be an array of row-wise negatively dependent random variables. There exist a r.v. and a constant satisfying (8) and (9). Let be an array of positive constants such that

Then

and

In particular, if , then

Remark 10. Theorems 7–9 extend Corollary 2.1, Corollary 2.2, and Corollary 2.5 of Sung [15] from the probability space to sublinear expectation space.

Proof of Theorem 5. To prove our main results, we just need to prove (12). Because of considering instead of in (12), we can get (13). Without loss of generality, we can assume that .
For fixed , denote for that

Then, we have

On account of the arbitrariness of , it follows that

Because of , by the definition of ND, is also a sequence of ND random variables. Obviously, is nondecreasing by . Therefore, is also a sequence of ND random variables. Firstly, we prove .
For the array , we can make use of Lemma 2, noting that , and condition (a) of Lemma 2 holds.
By (7) and the inequality, for any ,

Then, by (1), (8), (9), (10), (11), (31), we have

Now, conditions (a) and (b) of Lemma 2 hold. So, by Lemma 2, we can get that

So to prove , it suffices to prove that

By , (1), (7), (8), (9), (10), (31), we can get

So, we get (34).
Now, we prove . By the condition of , means that there is at least one that makes . By the definition of ND, (1), (7), (8), (9), (10), we can obtain that

When , we can get .
In the same way, we know that , so, also means that there is at least one that makes . Using the same method of proving , we can have . At last, we prove . Using the similar method to . By the value of , we can get . Hence, by (1), (7), (8), (9), (10), we have

Because of , we can get . Thus, this completes the proof of Theorem 5.

Proof of Theorem 7. Firstly, we prove (20). Without loss of generality, we assume that .
By Theorem 5, ,

Thus, we can get

Equation (39) follows from Lemma 4, and is countably subadditive. , on account of the arbitrariness of , it implies

Thus, to prove (20), we just need to check that

By (7) and (3) of Lemma 3, we can get

Thus, follows from Lemma 4 and being countably subadditive. By (17), it is easily checked that

Hence, (41) holds; because of the arbitrariness of , we can get (20). Considering instead of in (20), we can obtain

We know that , . Replacing with in (44), we can get (21). When , combine (20) with (21), and we will have (22). Thus, we obtain the result of Theorem 7.

Proof of Theorem 8. For , by (16), (18), we obtain that

We get condition (19) of Theorem 7. So the result of Theorem 8 can be proved.

Proof of Theorem 9. Without loss of generality, we still assume that . By (23), obviously

Then, by Theorem 5, to prove (25), we just need to prove that

By (4) of Lemma 3, (7), (8), (23), it is easy to check that

Therefore, (47) holds, then, we obtain (25). Considering instead of in (25), we have

Replacing with in (49), we can get (26). When , combine (25) with (26), and we will have (27). Therefore, we obtain the result of Theorem 9.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (11661029).

References

S. Peng, “G-expectation, G-Brownian motion and related stochastic calculus of Ito type,” Stochastic Analysis and Applications, pp. 541–567, 2006.

S. G. Peng, “Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations,” Science China Mathematics, vol. 52, no. 7, pp. 1391–1411, 2009.

S. Peng, “Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation,” Stochastic Processes and Their Applications, vol. 118, no. 12, pp. 2223–2253, 2008.

Z. Chen, “Strong laws of large numbers for sub-linear expectations,” Science China Mathematics, vol. 59, no. 5, pp. 945–954, 2016.

D. Zhang and Z. Chen, “A weighted central limit theorem under sublinear expectations,” Communications in Statistics—Theory and Methods, vol. 43, no. 3, pp. 566–577, 2014.

L. Zhang, “Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm,” Science China Mathematics, vol. 59, no. 12, pp. 2503–2526, 2016.

L. Zhang, “Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications,” Science China Mathematics, vol. 59, no. 4, pp. 751–768, 2016.

L. X. Zhang, “Strong limit theorems for extended independent and extended negatively dependent random variables under non-linear expectations,” 2016.

D. Yu and Q. Wu, “Complete convergence for weighted sums of extended negatively dependent random variables under sub-linear expectations,” Communications in Statistics—Theory and Methods, vol. 47, no. 19, pp. 4741–4750, 2018.

P. L. Hsu and H. Robbins, “Complete convergence and the law of large numbers,” Proceedings of the National Acadamy of Sciences of the United States of America, vol. 33, pp. 25–31, 1947.

Q. Wu and Y. Jiang, “Complete convergence and complete moment convergence for negatively associated sequences of random variables,” Journal of Inequalities and Applications, Paper No. 157, 10 pages, 2016.

X. J. Wang and S. H. Hu, “Complete convergence and complete moment convergence for martingale difference sequence,” Acta Mathematica Sinica, vol. 30, no. 1, pp. 119–132, 2014.

X. Wang and Y. Wu, “On complete convergence and complete moment convergence for a class of random variables,” Journal of the Korean Mathematical Society, vol. 54, no. 3, pp. 877–896, 2017.

B. D. Choi and S. H. Sung, “Almost sure convergence theorems of weighted sums of random variables,” Stochastic Analysis and Applications, vol. 5, no. 4, pp. 365–377, 1987.

S. H. Sung, “A strong limit theorem for weighted sums of negatively dependent random variables,” Communications in Statistics—Theory and Methods, vol. 44, no. 2, pp. 428–439, 2015.

Q. Wu, “A strong limit theorem for weighted sums of sequences of negatively dependent random variables,” Journal of Inequalities and Applications, vol. 1, pp. 1–8, 2010.

P. Chen and S. Gan, “Limiting behavior of weighted sums of i.i.d. random variables,” Statistics & Probability Letters, vol. 77, no. 16, pp. 1589–1599, 2007.

S. Xu and Y. Miao, “Almost sure convergence of weighted sums for negatively associated random variables,” Communications in Statistics—Theory and Methods, vol. 43, no. 10-12, pp. 2581–2594, 2014.