Skorokhod's representation theorem

Remarks

On $S = \R$ there is a short proof: let $U$ be uniform on $(0, 1)$, and set $X_n = F_n^{-1}(U)$ and $X = F^{-1}(U)$, where $F_n^{-1}$ and $F^{-1}$ are the quantile functions of $\mu_n$ and $\mu$. Weak convergence implies $F_n^{-1}(u) \to F^{-1}(u)$ at each continuity point $u$ of $F^{-1}$, and $F^{-1}$ is monotone, so it has only countably many discontinuities, a null set for $U$. The general proof builds the variables from fine partitions of $S$ into small sets whose boundaries have $\mu$-measure $0$. The theorem gives quick proofs of facts about weak convergence: for example, if $X_n \to X$ in distribution and $h$ is continuous, then $h(X_n) \to h(X)$ in distribution, which is the continuous mapping theorem. It also gives $E[\abs{X}] \le \liminf_n E[\abs{X_n}]$ for real variables, by Fatou's lemma. The new variables only have the right laws; the original random variables, if any, need not converge almost surely.

Going deeper

For a student

The theorem is a standard device in weak convergence theory. It appears in proofs of the continuous mapping theorem and of results on uniform integrability and convergence of moments. It is also used for stochastic processes with values in function spaces. In the real case it is just the quantile coupling. It should not be confused with Skorokhod's embedding theorem, which embeds a random walk into a Brownian motion.