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.

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

• Convergence in distribution

• 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)

• Theorems in probability theory
• Theorems in statistics

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