An invariance principle of strong law of large numbers under nonadditive probabilities

Abstract

In the framework of nonadditive probabilities or sublinear expectations, the Kolmogorov’s strong law of large numbers (SLLN) states that for a sequence of independent and identically distributed (IID) random variables, limit points of its sample mean quasi-surely fall inside an interval given by a pair of lower and upper means. In this article, we will investigate a cluster set of limit points of a sequence of stochastic processes, which are given by linear interpolating of the sample mean of IID random variables under sublinear expectations, and show an invariance principle. The invariance principle will strengthen the Kolmogorov’s SLLN under nonadditive probabilities in some extent.