Skorokhod's representation theorem

Let $(\backslash mu\_n)\_\{n\; \backslash in\; \backslash mathbb\{N\}\}$ be a sequence of probability measures on a metric space $S$ such that $\backslash mu\_n$ converges weakly to some probability measure $\backslash mu\_\backslash infty$ on $S$ as $n\; \backslash to\; \backslash infty$. Suppose also that the support of $\backslash mu\_\backslash infty$ is separable. Then there exist $S$-valued random variables $X\_n$ defined on a common probability space $(\backslash Omega,\backslash mathcal\{F\},\backslash mathbf\{P\})$ such that the law of $X\_n$ is $\backslash mu\_n$ for all $n$ (including $n=\backslash infty$) and such that $(X\_n)\_\{n\; \backslash in\; \backslash mathbb\{N\}\}$ converges to $X\_\backslash infty$, $\backslash mathbf\{P\}$-almost surely.

• Convergence in distribution

• {{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)

Category:Theorems in probability theory

Category:Theorems in statistics