Quasi Sure Central Limit Theorem

Peng (2006) initiated the notion of independent identically distributed random variables and a new kind of central limit theorem under sub-linear expectations. In this paper, we drive a new kind of almost sure central limit theorem for non-additive probabilities (also called quasi sure central limit theorem), which is a quai sure convergence version of Peng's central limit theorem. Moreover, this result is a natural extension of the classical almost sure central limit theorem to the case where the probability is no longer additive, and can be considered as a simulation method to some contingent claims with distribution uncertainty. Meanwhile, we prove a new kind of strong law of large numbers for non-additive probabilities without the independent identically distributed assumption.