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 $\mu_n\stackrel{w}\to\mu_0,$ as $n\to \infty$, then there exist a probability space $(\Omega, \mathscr F, P)$ and a sequence of random elements $X_n$ such that $X_n\to X$ almost surely and $X_n$ has the distribution function $\mu_n$, $n=0,1,2,\cdots$. 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 $\mu_n$ and a sequence of measurable mappings $\varphi_n$ such that $\mu_n\varphi_n^{-1}\stackrel {w}\to\mu_0$, then there exist a probability space $(\Omega,\mathscr F,P)$ and $S_n$-valued random elements $X_n$ defined on $\Omega$, with distribution $\mu_n$ and such that $\varphi_n(X_n)\to X_0$ 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.