Skorokhod's representation theorem - Alchetron, the free social encyclopedia

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 μ n , n ∈ N be a sequence of probability measures on a metric space S such 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 defined on a common probability space ( Ω , F , P ) such that the law of X n is μ n for all n (including n = ∞ ) and such that X n converges to X ∞ , P -almost surely.

References

Skorokhod's representation theorem Wikipedia

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