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 of the theorem

Let μn, n ∈ N be a sequence of probability measures on a metric space S; suppose that μn converges weakly to some probability measure μ on S as n → ∞. Suppose also that the support of μ is separable. Then there exist random variables Xn, X defined on a common probability space (Ω, FP) such that

• $x_n \xrightarrow{d}\ \mu_n$ (i.e. μn is the distribution/law of Xn);
• $x \xrightarrow{d}\ \mu$ (i.e. μ is the distribution/law of X); and
• $x_n \xrightarrow{\mathrm{a.s.}} x$

References

• 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)