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 ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ 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.

See also

• Convergence in distribution

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)