Strong Representation of Weak Convergence.

Abstract

Skorokhod's representation theorem states that if on a Polish space, there is defined a weakly convergent sequence of probability measures µ n w → µ 0 , as n → ∞, then there exist a probability space (Ω, F , P ) 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, • • • .In this paper, 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 µ n ϕ -1 n w → µ 0 , then there exist a probability space (Ω, F , P ) 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.

Locations

• arXiv (Cornell University) - View - PDF