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 Soviet mathematician A. V. Skorokhod.

Statement

Let $(\mu_n)_$ 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 $S$-valued random variables $X_n$ defined on a common probability space $(\Omega,\mathcal,\mathbf)$ such that the law of $X_n$ is $\mu_n$ for all $n$ (including $n=\infty$) and such that $(X_n)_$ converges to $X_\infty$, $\mathbf$-almost surely.

See also

* Convergence in distribution

References

* {{cite book , last=Billingsley , first=Patrick , title=Convergence of Probability Measures , url=https://archive.org/details/convergenceofpro0000bill , url-access=registration , publisher=John Wiley & Sons, Inc. , location=New York , year=1999 , isbn = 0-471-19745-9 (see p. 7 for weak convergence, p. 24 for convergence in distribution and p. 70 for Skorokhod's theorem)

Probability theorems
Theorems in statistics