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

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 andMaccheroni andMarinacci [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., G-normal distribution, and G-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.


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.

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].
Artzner et al. [20] showed that a sublinear expectation can be expressed as a supremum of linear expectations.That is, if Ê is a sublinear expectation on H, then there exists a set (say P) of probability measures such that For this P, 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.
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 ,  ∈ H are two random variables.Ê is a sublinear expectation and (, ) is the pair of capacities induced by Ê.If random variable  is independent of  under Ê, then  is also pairwise independent of  under capacities  and ; that is, for all subsets  and  ⊂ ,  ( ∈ ,  ∈ ) =  ( ∈ )  ( ∈ ) (6) 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 {  ,  ≥ 1} be a sequence of events in F.

Main Results
In this section, we give our main results including the proofs.
Proof.From the inequality   ≤ 1 +  + (1/2) 2  || for all  ∈ , we have for any  > 0. Let  > 0 be given.We set  = 2 log / 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.