Strong representation of weak convergence

Abstract

Skorokhod’s representation theorem states that if on a Polish space, there is a weakly convergent sequence of probability measures , as n → ∞, then there exist a probability space and a sequence of random elements X n such that X n → X almost surely and X n has the distribution function μ n , n = 0, 1, 2, … We shall extend the Skorokhod representation theorem to the case where if there are a sequence of separable metric spaces S n , a sequence of probability measures µ n and a sequence of measurable mappings φ n such that , then there exist a probability space and S n -valued random elements X n defined on Ω, with distribution μ n and such that φ n (X n ) → X 0 almost surely. In addition, we present several applications of our result including some results in random matrix theory, while the original Skorokhod representation theorem is not applicable.