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

• Convergence in distribution

• 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)