On the Convergence Rates Under Sub-Linear Expectation

Abstract

In this paper, we establish the convergence rates for the law of large numbers and the law of the logarithm for identically distributed random variables with either negative dependence or independence in the framework of sub-linear expectation. We propose a novel subsequence partitioning method to address the convergence of the series of capacities. This method allows us to verify the convergence of capacity series for weighted sum sequences under suitable moment conditions in the sub-linear expectation setting. Under the assumption of countable sub-additivity of the upper capacity, we derive sufficient Choquet moment and tail conditions for the above problems, which yield convergence rates of order 1/n and (logn)/n. Our results generalize the corresponding conclusions in classical probability theory.