The Skorokhod Representation Theorem states that if a sequence of random variables converges in distribution to a limit, then that whole sequence, together with the limit, can be replaced by a new sequence of random variables, all defined on a common probability space, that has exactly the same distributions but converges pointwise, almost surely, rather than merely in distribution. Named for Anatoliy Skorokhod, it is a widely used technical device for upgrading distributional convergence to a stronger, pathwise form of convergence.
Facts
StatementIf probability measures on a metric space converge weakly to a limit measure with separable support, then there exist random variables on a common probability space with those laws that converge almost surely. 1 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Skorokhod's representation theorem (Wikipedia)
Sources
1. Skorokhod's representation theorem (Wikipedia)
Statement
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 S-valued 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 ) n ∈ N converges to X ∞, P-almost surely.
In Branch: Probability and Statistics, Lead sentence
In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of prob
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