A General Central Limit Theorem Under Sublinear Expectations

Motivated by the coherent risk measures (cf.[2,3]) and uncertain volatility models in finance (see, e.g. [13]), Peng [14,16] recently introduced the notion of sublinear expectation which is not based on a classical probability space. Under the sublinear expectations, a random variable X in a sublinear expectation space (Ω,(?),E) (for its definition, see Section 2 of this paper) is said to be of G-normal distribution with zero mean (cf. [16,19]), if for each Y which is an independent copy of X, it holds that Just as the classical normal distributions in probability theory, for a G-normal dis-tributed random variable X, we have (cf. [16]) E[ψ(X)]= u (1,0), (?)ψ∈Cb,Lip(R), where v. (t, x) is the unique viscosity solution for the following heat equation where In the theory of sublinear expectations, the heat equation above often plays a role of characteristic function in probability theory.On the basis of G-normal distribution, G-Brownian motion can be defined, and the corresponding stochastic calculus with respect to the G-Brownian motions and the related Ito's formula can also be established (cf. [14,16]). Since the importance of law of large numbers (LLN) and central limit theorem (CLT) in probability theory, Peng [15,17] has shown the corresponding LLN and CLT under sublinear expectations, which indicate that G-normal distributions play the same important role in the theory of sublinear expectations as the normal distributions in the classical probability theory.Due to the significance of sublinear expectations in finance and statistics, the the-ory of sublinear expectations has been attracting more and more attentions in both pure and applied mathematics (see, e.g. [7], [9], [20], [22] and [23]).The purpose of this paper is to investigate one of the very important fundamental results in the theory of sublinear expectations-Central Limit Theorem. Until now, all the results on central limit theorems under sublinear expectations require that the sequence of random variables is independent and identically distributed. Analogous to the CLT in the probability theory, a natural question is whether one can weaken the hypothesis of identical distributions for the CLT under sublinear expectations?In this paper, without the hypothesis of identical distributions, we prove two new central limit theorems under the sublinear expectations, which extends Peng's results.