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.

Statement

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

See also

• Convergence in distribution

References

• Billingsley, Patrick (1999). Convergence of Probability Measures. New York: John Wiley & Sons, Inc.. ISBN 0-471-19745-9. https://archive.org/details/convergenceofpro0000bill.  (see p. 7 for weak convergence, p. 24 for convergence in distribution and p. 70 for Skorokhod's theorem)