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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 sequence of random variables defined on a common probability space. It is named for the Ukrainian mathematician A.V. Skorokhod.
## Statement of the theorem

## References

Let μ_{n}, n ∈ N be a sequence of probability measures on a topological space S; suppose that μ_{n} 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 X_{n}, X defined on a common probability space (Ω, F, P) such that

- (X
_{n})_{∗}(P) = μ_{n}(i.e. μ_{n}is the distribution/law of X_{n}); - X
_{∗}(P) = μ (i.e. μ is the distribution/law of X); and - X
_{n}(ω) → X(ω) as n → ∞ for every ω ∈ Ω.

- Billingsley, Patrick (1999).
*Convergence of Probability Measures*. New York: John Wiley & Sons, Inc.. (see theorem 29.6)

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Last updated on Saturday February 16, 2008 at 10:06:57 PST (GMT -0800)

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