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 $({\mu }_n{)}_{n\in \mathbb{\mathbb{N}}}$ be a sequence of probability measures on a metric space $S$ such that ${\mu }_n$ converges weakly to some probability measure ${\mu }_{\mathrm{\infty }}$ on $S$ as $n\to \mathrm{\infty }$. Suppose also that the support of ${\mu }_{\mathrm{\infty }}$ is separable. Then there exist $S$-valued random variables $X_n$ defined on a common probability space $(\mathrm{\Omega },{ℱ},\mathbf{𝐏})$ such that the law of $X_n$ is ${\mu }_n$ for all $n$ (including $n=\mathrm{\infty }$) and such that $(X_n{)}_{n\in \mathbb{\mathbb{N}}}$ converges to $X_{\mathrm{\infty }}$, $\mathbf{𝐏}$-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. https://archive.org/details/convergenceofpro0000bill.  (see p. 7 for weak convergence, p. 24 for convergence in distribution and p. 70 for Skorokhod's theorem)