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 $μ_n\stackrel{w}\toμ_0,$ as $n\to \infty$, then there exist a probability space $(Ω, \mathscr F, P)$ and a sequence of random elements $X_n$ such that $X_n\to X$ almost surely and $X_n$ has the distribution function $μ_n$, $n=0,1,2,\cdots$. 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 $μ_nφ_n^{-1}\stackrel {w}\toμ_0$, then there exist a probability space $(Ω,\mathscr F,P)$ and $S_n$-valued random elements $X_n$ defined on $Ω$, with distribution $μ_n$ and such that $φ_n(X_n)\to X_0$ 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.