Law of large numbers and central limit theorem for independent and non-identical distributed random variables under convex expectations dominated by sub-linear expectations

Abstract

Motivated by some interesting problems in mathematical economics, quantum mechanics and finance, non-linear expectations have been used to describe the phenomena which which have the stochastic characteristic of non-additivity. In this paper, we study two limit theorems for random variables under convex expectations, which are dominated by sub-linear expectations. Firstly, a central limit theorem (Theorem 3.1) is proved for independent and non-identical distributed random variables with only finite second order moments. Secondly, a law of large numbers (Theorem 3.2) is proved for independent and non-identical distributed random variables with only finite first order moments. These results include and extend some existing results. Furthermore, we give an example for the application of Theorem 3.2.