Skorokhod's representation theorem

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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[edit]

Let , be a sequence of probability measures on a metric space such that converges weakly to some probability measure on as . Suppose also that the support of is separable. Then there exist random variables defined on a common probability space such that the law of is for all (including ) and such that converges to , -almost surely.

See also[edit]

References[edit]

  • 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)