Skorokhod's representation theorem

Template:Short description 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 ${\mu }_{\mathrm{\infty }}$ 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 $(\mathrm{\Omega },ℱ,𝐏)$ such that the law of $X_n$ is ${\mu }_n$ for all $n$ (including $n=\mathrm{\infty }$) and such that $(X_n{)}_{n\in \mathbb{N}}$ converges to $X_{\mathrm{\infty }}$, $𝐏$-almost surely.

See also

• Convergence in distribution

References

• Template:Cite book (see p. 7 for weak convergence, p. 24 for convergence in distribution and p. 70 for Skorokhod's theorem)