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

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).