Skorokhod's representation theorem

In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probability space. It is named for the Ukrainian mathematician A. V. Skorokhod.

Statement

Let $(\mu _{n})_{n\in \mathbb {N} }$ be a sequence of probability measures on a metric space $S$ such that $\mu _{n}$ converges weakly to some probability measure $\mu _{\infty }$ on $S$ as $n\to \infty$ . Suppose also that the support of $\mu _{\infty }$ is separable. Then there exist $S$ -valued random variables $X_{n}$ defined on a common probability space $(\Omega ,{\mathcal {F}},\mathbf {P} )$ such that the law of $X_{n}$ is $\mu _{n}$ for all $n$ (including $n=\infty$ ) and such that $(X_{n})_{n\in \mathbb {N} }$ converges to $X_{\infty }$ , $\mathbf {P}$ -almost surely.

References

Billingsley, Patrick (1999). Convergence of Probability Measures. New York: John Wiley & Sons, Inc. ISBN 0-471-19745-9. (see p. 7 for weak convergence, p. 24 for convergence in distribution and p. 70 for Skorokhod's theorem)