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 Soviet mathematician A. V. Skorokhod.
Statement of the theorem
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.