Journal of Applied Mathematics

Volume 2014 (2014), Article ID 412758, 5 pages

http://dx.doi.org/10.1155/2014/412758

## A Strong Law of Large Numbers for Weighted Sums of i.i.d. Random Variables under Capacities

School of Mathematics, Honghe University, Mengzi 661199, China

Received 16 February 2014; Accepted 3 June 2014; Published 16 June 2014

Academic Editor: Abdel-Maksoud A. Soliman

Copyright © 2014 Defei Zhang and Ping He. 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

With the notion of independent identically distributed (i.i.d.) random variables under sublinear expectations initiated by Peng, a strong law of large numbers for weighted sums of i.i.d. random variables under capacities induced by sublinear expectations is obtained.

#### 1. Introduction

The strong law of large numbers plays important role in the development of probability theory and mathematical statistics; many studies about the extension of it have been completed by many authors. For example, Chow and Lai [1], Stout [2], Choi and Sung [3], Cuzick [4], Rosalsky and Sreehari [5], Wu [6], Bai and Cheng [7], Bai et al. [8], and so forth investigated the almost sure limiting behavior of weighted sums of i.i.d. random variables. In fact, the additivity of probability and expectations is not reasonable in many areas of applications because many uncertain phenomena cannot be well modeled using additive probabilities or linear expectations (see, e.g., Chen and Epstein [9], Huber and Strassen [10], and Wakker [11]). In the case of nonadditive probabilities, Marinacci [12] proved several limit laws for nonadditive probabilities and Maccheroni and Marinacci [13] obtained a strong law of large numbers for totally monotone capacities.

Recently, motivated by the risk measures, superhedge pricing, and modelling uncertainty in finance, Peng [14] introduced the notion of sublinear expectation space, which is a generalization of probability space. Together with the notion of sublinear expectation, Peng also introduced the notions about i.i.d., -normal distribution, and -Brownian motion. Under this framework, the weak law of large numbers and the central limit theorems under sublinear expectations were obtained in the studies by Peng in [15, 16]. Soon thereafter, Denis et al. [17] introduced the function spaces and capacity related to a sublinear expectation. Chen et al. [18] proved a strong law of large numbers for nonadditive probabilities.

A natural question is the following: can we investigate strong laws of large numbers for weighted sums of random variables under capacities? Indeed, the goal of this paper is to discuss the strong laws of large numbers for weighted sums of i.i.d. random variables under capacities. Under some assumptions, we obtain a strong law of large numbers for weighted sums of i.i.d. random variables under capacities.

The paper is organized as follows: in Section 2, we give some definitions and lemmas that are useful in this paper. In Section 3, we give our main results including the proofs.

#### 2. Preliminaries

In this section, we present some preliminaries in the theory of sublinear expectations and capacities. More details of this section can be found in the studies by Chen et al. [18] and Peng [19].

Let be a measurable space, and let be the set of random variables on .

*Definition 1. *A sublinear expectation is a functional satisfying the following: (i)monotonicity: if ;(ii)constant preserving: for ;(iii)subadditivity: ;(iv)positive homogeneity: for .

Artzner et al. [20] showed that a sublinear expectation can be expressed as a supremum of linear expectations. That is, if is a sublinear expectation on , then there exists a set (say ) of probability measures such that
For this , following Huber and Strassen [10], we define a pair of capacities denoted by
Obviously,
where is the complement set of .

It is easy to check that and are two continuous capacities in the sense of the following definition.

*Definition 2. *A set function : is called a continuous capacity if it satisfies(1), ;(2), whenever and ;(3), if ;(4), if , where .

*Definition 3 (see Peng [19]). *
*Identical Distribution*. Let and be two -dimensional random vectors in . They are called identically distributed, denoted by , if, for each measurable function on such that , one has
*Independence*. A random vector , , is said to be independent of another random vector , , under if, for each measurable function on with and for each , one has

*Remark 4. *A sequence of random variables is said to be i.i.d., if and is independent of for each .

The following lemma shows the relation between Peng's independence and pairwise independence in the study by Marinacci in [12].

Lemma 5 (see Chen et al. [18]). *Suppose that are two random variables. is a sublinear expectation and is the pair of capacities induced by . If random variable is independent of under , then is also pairwise independent of under capacities and ; that is, for all subsets and ,
**
holds for both capacities and .**Borel-Cantelli lemma is still true for capacities and under some assumptions.*

Lemma 6 (see Chen et al. [18]). *Let be a sequence of events in .*(1)*If , then .*(2)*Suppose that are pairwise independent with respect to ; that is,
**If , then .*

#### 3. Main Results

In this section, we give our main results including the proofs.

Theorem 7. *Let be a sequence of i.i.d. random variables in satisfying , and for any **
Let be an array of constants satisfying
**
Then, for , if
**
we have
**
where .**Moreover, for , if , we have
**
where .**In order to prove Theorem 7, we need the following lemma.*

Lemma 8. *Let be a sequence of i.i.d. random variables satisfying (8), and let be an array of constants. Truncate at and denote . Suppose that the following conditions hold:*(1)*, for some and some constant ;*(2)*, for some and some sequence of constants such that .**Then,
*

*Proof. *From the inequality for all , we have
for any . Let be given. We set and obtain by (1) and (2) in Lemma 8 and (8) that
for all large . For the large , it follows by the Markov inequality and (15) that
Using Lemma 6, we have
The proof is complete.

*Proof of Theorem 7. *The proof of (11) is similar to that of (12); we only prove (12). We denote , , and for and , where . Denote and for and , where . Then,

For , we will apply Lemma 8 to the random variable and weight . Note that
Hence, by Lemma 8, we have

For , we observe that
Namely,
By replacing by , we have
Letting , we have

For , note that
Then, by Lemma 8, we have

For , assumption (8) implies . Hence, by Lemma 6, is bounded . It follows that
as .

Thus,

On the other hand, ; and we note that
and the condition : from (20), (24), (26), and (28), we conclude that
which implies
Also,

Similarly, considering the sequence , from (29), we have
Note that . So
Therefore, the proof of Theorem 7 is complete.

*The following theorem shows that if the norming constant is stronger than that of Theorem 7, then condition (8) in Theorem 7 can be replaced by a weaker condition.*

*Theorem 9. Let be a sequence of i.i.d. random variables in satisfying , and for some
Let be an array of constants satisfying (9) in Theorem 7. Then, for and , if , we have
where .*

Moreover, for and , if , we have where .

*Proof. *We can prove that Lemma 8 is also true except that (8) and (1) of Lemma 8 are replaced by (35) and the following condition.

, for some and some sequence of constants such that .

For the case , we let and for and . For the case , we let , , and . The rest of the proof is similar to that of Theorem 7 and is omitted.

*Remark 10. *If in Theorem 9, then, for and , we have
where . Moreover, if , we have
where . Since if and , then and , we also have
where . The result is similar to Theorem 2.2 of Bai and Cheng [7].

*Remark 11. *If in Theorem 7, we can get the classical strong law of large numbers for weighted sums of i.i.d. random variables as follows:

*Conflict of Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgments*

*The authors thank Professor Zengjing Chen for his helpful discussion and suggestions and the reviewers for their valuable comments and suggestions to improve the presentation of this paper. This work is partially supported by the National Science Foundation of China (11301160), the Scientific Research Foundation of Yunnan Province (2013FZ116), the Scientific Research Foundation of Yunnan Province Education Committee (2011C120), the Reserve Talents Foundation of Honghe University (2014HB0204), and the Curricula Construction Foundations of Honghe University (ZYDT1308, ZDKC1111).*

*References*

- Y. S. Chow and T. L. Lai, “Limiting behavior of weighted sums of independent random variables,”
*Annals of Probability*, vol. 1, no. 5, pp. 810–824, 1973. View at Google Scholar · View at MathSciNet - W. F. Stout,
*Almost Sure Convergence*, Academic Press, New York, NY, USA, 2nd edition, 1974. View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - J. Cuzick, “A strong law for weighted sums of i.i.d. random variables,”
*Journal of Theoretical Probability*, vol. 8, no. 3, pp. 625–641, 1995. View at Publisher · View at Google Scholar · View at MathSciNet - A. Rosalsky and M. Sreehari, “On the limiting behavior of randomly weighted partial sums,”
*Statistics & Probability Letters*, vol. 40, no. 4, pp. 403–410, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - W. B. Wu, “On the strong convergence of a weighted sum,”
*Statistics & Probability Letters*, vol. 44, no. 1, pp. 19–22, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - Z. D. Bai and P. E. Cheng, “Marcinkiewicz strong laws for linear statistics,”
*Statistics & Probability Letters*, vol. 46, no. 2, pp. 105–112, 2000. View at Publisher · View at Google Scholar · View at MathSciNet - Z. D. Bai, P. E. Cheng, and C. H. Zhang, “An extension of the Hardy-Littlewood strong law,”
*Statistica Sinica*, vol. 7, no. 4, pp. 923–928, 1997. View at Google Scholar · View at MathSciNet - Z. Chen and L. Epstein, “Ambiguity, risk, and asset returns in continuous time,”
*Econometrica*, vol. 70, no. 4, pp. 1403–1443, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - P. Huber and V. Strassen, “Minimax tests and the Neyman-Pearson Lemma for capacities,”
*The Annals of Statistics*, vol. 1, no. 2, pp. 251–263, 1973. View at Google Scholar · View at MathSciNet - P. Wakker, “Testing and characterizing properties of nonadditive measures through violations of the sure-thing principle,”
*Econometrica*, vol. 69, no. 4, pp. 1039–1059, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - M. Marinacci, “Limit laws for non-additive probabilities and their frequentist interpretation,”
*Journal of Economic Theory*, vol. 84, no. 2, pp. 145–195, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - F. Maccheroni and M. Marinacci, “A strong law of large numbers for capacities,”
*The Annals of Probability*, vol. 33, no. 3, pp. 1171–1178, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - S. Peng, “
*G*-Expectation,*G*-Brownian motion and related stochastic calculus of Itô type,” in*Stochastic Analysis and Applications: The Abel Symposium 2005*, vol. 2 of*Abel Symposia*, pp. 541–567, Springer, Berlin, Germany, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - S. Peng, “Law of large numbers and central limit theorem under nonlinear expectations,” In press, http://arxiv.org/abs/math/0702358.
- S. Peng, “A new central limit theorem under sublinear expectations,” In press, http://arxiv.org/abs/0803.2656.
- L. Denis, M. Hu, and S. Peng, “Function spaces and capacity related to a sublinear expectation: application to
*G*-Brownian motion paths,”*Potential Analysis*, vol. 34, no. 2, pp. 139–161, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Chen, P. Wu, and B. Li, “A strong law of large numbers for non-additive probabilities,”
*International Journal of Approximate Reasoning*, vol. 54, no. 3, pp. 365–377, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - S. Peng, “Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and
*G*-Brownian motion,” In press, http://arxiv.org/abs/1002.4546. - P. Artzner, F. Delbaen, J. Eber, and D. Heath, “Thinking coherently,”
*Risk*, vol. 10, no. 1, pp. 68–71, 1997. View at Google Scholar

*
*