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, F, P) such that
- (Xn)∗(P) = μn (i.e. μn is the distribution/law of Xn);
- X∗(P) = μ (i.e. μ is the distribution/law of X); and
- Xn(ω) → X(ω) as n → ∞ for every ω ∈ Ω.