# What is Skorohod's Represntation Theorem Saying?

From Wikipedia:

Let $\mu_n, n \in N$ be a sequence of probability measures on a metric space S; suppose that $\mu_n$ converges weakly to some probability measure $\mu$ on S as $n \to \infty$. Suppose also that the support of μ is separable. Then there exist random variables $X_n, X$ defined on a common probability space $(\Omega,F, P)$ such that

$X_n \xrightarrow{d}\ \mu_n$ (i.e. $μ_n$ is the distribution/law of $X_n$);

$X \xrightarrow{d}\ \mu$ (i.e. $\mu$ is the distribution/law of X); and

$X_n \xrightarrow{\mathrm{a.s.}} x$

Can someone provide a simple, concrete example of how one would use this theorem?

One example is a version of the continuous mapping theorem which states that if $X_n \rightsquigarrow X$ then $f(X_n) \rightsquigarrow f(X)$ for a continuous function $f$. Using the a.s. representation (Skorohod's Representation theorem) there is a sequence of random variables $Y_n$ and a random variable $Y$ defined on a common probability space having the same laws as $X_n$ and $X$ s.t. $Y_n\xrightarrow{a.s.}Y$. The rest is pretty straightforward...