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 of the theorem

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 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 $X_n$ converges to $X_\infty$, $\mathbf{P}$-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. (see p. 7 for weak convergence, p. 24 for convergence in distribution and p. 70 for Skorokhod's theorem)