Skorokhod's representation theorem

From Wikipedia, the free encyclopedia

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 sequence of random variables defined on a common probability space. It is named for the Ukrainian mathematician A.V. Skorokhod.

[edit] Statement of the theorem

Let $(\mu_{n})_{n = 1}^{\infty}$ be a sequence of probability measures on a topological space $S$ ; suppose that $μ n$ converges weakly to some probability measure $μ$ on $S$ as $n \to \infty$ . Suppose also that the support of $μ$ is separable. Then there exist random varables $X n, X$ defined on a common probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that

$(X_{n})_{*} (\mathbb{P}) = \mu_{n}$ (i.e. $μ n$ is the distribution/law of $X n$ );
$X_{*} (\mathbb{P}) = \mu$ (i.e. $μ$ is the distribution/law of $X$ ); and
$X_{n} (\omega) \to X (\omega)$ as $n \to \infty$ for every $\omega \in \Omega$ .