Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 7945431, 8 pages
https://doi.org/10.1155/2019/7945431

Research Article

On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

Xiaochen Ma 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 22 January 2019; Accepted 21 March 2019; Published 11 April 2019

Academic Editor: Allan C. Peterson

Copyright © 2019 Xiaochen Ma 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 article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

1. Introduction

There are many uncertainties in the fields of economy, statistics, engineering, etc., which cannot be accurately predicted or described by traditional additive probability measures. In reality, many uncertain phenomena do not satisfy the linear additivity condition, so the application of classical limit theory is limited to some extent; a growing number of people abnegate the traditional tool of additivity of probability and instead use the new tool of nonadditive probability measure to portray problems with uncertainty. Taking financial derivatives as an example, the measures of risk are generally nonlinear and do not satisfy the linear additivity; the wrong assessment and management of its risk will lead to serious consequences. Recently, Peng [1–3] proposed a generalized sublinear expectation space theory. The sublinear expectation is not dependent on the corresponding probability and can be defined directly by the real functions satisfying monotonicity, constant preserving, subadditivity, and positive homogeneity. And combining the theory of partial differential equations, Peng has given the concepts of the identical distribution, independence, and maximum distribution of the random variables under sublinear expectation framework and has established a relatively complete axiom system. About other properties of the sublinear expectations theory, we can refer to Peng [4–6].

Under sublinear expectation space, a suite of serviceable consequences have been established. For example, Zhang [7–9] acquired the exponential inequalities, Rosenthal’s inequalities, Kolmogorovs strong law of larger numbers (SLLNs) and Hartman-Wintners law of iterated logarithm, Yu and Wu [10] obtained complete convergence for weighted sums of extended negatively dependent random variables under a specific moment condition, Cheng [11] established strong law of large numbers (SLLNs) under a general moment condition and so on. And many strong laws of large numbers for weighted sums of some sequences have been obtained; one can refer to Chen [12], Wu and Jiang [13], Zhang [14], Chen et al. [15], and Chen and Liu [16]. These authors only researched Kolmogorov’s SLLNs or Marcinkiewicz’s SLLNs for some sequences of random variables. However, the strong law of large numbers for the weighted sums of extended negatively dependent (END) random variables under sublinear expectation has less related results.

In this article, we will give some conditions for strong law of large numbers for weighted sums of END random variables. By the truncated random variable and some moment inequalities, we testify that the SLLNs for the weighted sums hold for END random variables. In addition, our results extend the corresponding results of Shen et al. [17] relative to the classical probability space. In the next part of this article, we provide some definitions under the sublinear expectations containing identical distribution, extended negatively dependent (END). In part 3, we give the main results. The proof of these theorems is given in the last part.

2. Preliminaries

The research of this article uses the framework of Peng [1–3]. For brevity, we omit these definitions. By definition of and , we have

Under the sublinear expectation space, Chen [12] and Zhang [7, 8] researched the almost sure convergence of sequence. Hence, we define the almost sure as follows.

Definition 1 (Wu and Jiang 2017 [13]). A sequence of random variables is called to converge to almost surely , showed by a.s. as , if .
can be substituted by and severally. By and for any , it is quite clear that a.s. signifies a.s. , but a.s. does not signify a.s. . Further,

Definition 2 (Peng 2006 [1], 2008 [6]). Let and be two random vectors defined severally in sublinear expectation spaces . They are named identically distributed, showed by if

whenever the sublinear expectations are finite. A sequence of random variables is called to be identically distributed if for each .

Definition 3 (Zhang 2016 [7]). A sequence of random variables is called to be upper (resp., lower) extended negatively dependent (END) if there is some dominating constant such that

whenever the nonnegative functions , are all nondecreasing (resp., all nonincreasing). They are named extended negatively dependent if they are both upper extended negatively dependent and lower extended negatively dependent.

In the following passage, let be a random variable sequence in . will signify a positive constant that may have different effects in different places. Let indicate that there exists a constant such that for sufficiently large , and indicates an indicator function.

Lemma 4 (Zhang 2016, Lemma 3.9 [8]). Let be a sequence of events in . Assume that is a countably subadditive capacity. If , then

Lemma 5 (Zhang 2016, Theorem 3.1 [9]). Let be a sequence of upper extended negatively dependent (END) random variables in with . Then,

where ; there exists a constant such that for all and .

where .

3. Main Results

Theorem 6. Suppose that and is countably subadditive. Let be a sequence of upper extended negatively dependent (END) random variables and identically distributed. Assume that

Let and be sequences of positive numbers with , , such that

Then,

Further, if is extended negatively dependent random variables, then

In particular, if is extended negatively dependent random variables and , then

Theorem 7. Suppose that , and is countably subadditive. Let be a sequence of extended negatively dependent (END) random variables and identically distributed. Assume that

Let and be sequences of positive numbers with for sufficiently , . Put , for , such that

Then,

Further, if is extended negatively dependent random variables, then

In particular, if is extended negatively dependent random variables and , then

4. Proof

Proof of Theorem 6. When we replace with in (11), we can acquire (12). So, we just need to prove (11). And we suppose that . It can be easily proved that (8) is isovalent for any ,

Note that

Hence, (19) implies

For upper extended negatively dependent random variables , there are truncated functions belonging to and are nondecreasing, such that the sequences of truncated random variables are also upper extended negatively dependent. Let for any , for any ,

Then, is also a sequence of upper extended negatively dependent random variables by and being nondecreasing.
Note that

Thus, to testify (11), it suffices to verify that

It shall be pointed out that, in the classical linear expectation space, we know that the equality of is set up. Nevertheless, in the sublinear expectation space, is defined through continuous functions in and the indicator function does not satisfy continuity. Hence, this expression is no longer effective and we need to amend the indicator function by functions by substituting the original indicator function into the sublinear expectation space. Hence, we define the function as follows.
For , let be an even function and such that for all , if if and are nonincreasing for any . Then,

By (1), (19), and (24),

Thus, follows from the Borel-Cantelli lemma (Lemma 4) and being countably subadditive. It follows that from (10)

Now, we prove . For any , by the inequality and (25), ,

Because is a sequence of upper extended negatively dependent random variables with , using (6) in Lemma 5, let , for , by (10), we obtain for ,

Let such that for all and if if or . Then,

By the inequality, (28) and (29), for all , we have

By the Markov inequality, (19), and , we get

For every , there exists an such that ; thus, by (1), (21), (25), (30), and for all , we get

So, we have

Next, we prove . Since . So, by (8), (28), and Markov inequality,

Since , so , for . Hence,

Combining (29), (34), the Borel-Cantelli lemma (Lemma 4), and arbitrariness of , we obtain

Finally, we prove . By , . So, by (8), (25),

Once again, using (8) and (9), by Toeplitz lemma, we get

Hence, we get

Together with (27) and (37), (24) holds. The proof of Theorem 6 is completed.

Proof of Theorem 7.. When we replace with in (16), we can obtain (17). So, we just need to prove (16). And we assume that . Obviously, by the same methods as (19) and (21), (14) implies

Denote

By the definition of in Theorem 6, we have

Let . By (14), (15), (25), (41), for all , and being countably subadditive, we have

which implies from the Borel-Cantelli lemma (Lemma 4). Hence, in order to testify (16), we just need to testify

By the proof of (44), for any , we get

Firstly, we estimate (45). By , is also a sequence of upper extended negatively dependent random variables with . Using (7) in Lemma 5 for , for every , we obtain

Observe that, by (1), (15), the same methods as (28), (41), (47), for all , and being countably subadditive,

By the Borel-Cantelli lemma (Lemma 4) and arbitrariness of , we obtain

Secondly, we estimate (46). Since and , it follows that

Observe that, by for all , and being countably subadditive,

So, we can get

from (47), (51), and (52). Consequently,

which implies (46) from Kronecker’s lemma. Hence, the proof of Theorem 7 is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11661029) and the Support Program of the Guangxi China Science Foundation (2018GXNSFAA281011). The authors are very grateful to the referees and the editors for the helpful comments and suggestions.

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