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 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 $(\mu _{n})_{n=1}^{\infty }$ be a sequence of probability measures on a topological space $S$; suppose that $\mu _{n}$ converges weakly to some probability measure $\mu$ on $S$ as $n\to \infty$. Suppose also that the support of $\mu$ 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. $\mu _{n}$ is the distribution/law of $X_{n}$);
• $X_{*}(\mathbb {P} )=\mu$ (i.e. $\mu$ is the distribution/law of $X$); and
• $X_{n}(\omega )\to X(\omega )$ as $n\to \infty$ for every $\omega \in \Omega$.

References

• Billingsley, Patrick (1999). Convergence of Probability Measures. John Wiley & Sons, Inc., New York. ISBN 0-471-19745-9.