Skorokhod representation theorem

A theorem stating that weakly convergent probability measures on a Polish space can be represented by random variables on one probability space that converge almost surely.

Weak convergence of probability measures: A sequence of probability measures converges weakly when its integrals against every bounded continuous function converge. This is the hypothesis the representation converts into almost-sure convergence.
Convergence in distribution: Random variables converge in distribution when their probability laws converge weakly. It is the probabilistic formulation of the theorem's weak-convergence premise.
Anatoliy Skorokhod: Anatoliy Skorokhod was a Ukrainian probabilist known for foundational work in stochastic processes and weak convergence. His work gave the representation theorem its name and original form.
Continuous mapping theorem: The continuous mapping theorem transfers convergence in distribution through functions continuous at relevant limit points. A represented almost-sure limit can yield distributional limits after applying suitable functions.
Skorokhod representation theorem for nonseparable spaces: Representation results for nonseparable spaces require additional hypotheses because the classical Polish-space theorem may fail. It marks the boundary of the classical theorem's Polish-space assumptions.
Coupling (probability theory): A joint probability distribution with specified marginal distributions is called a coupling. The theorem constructs a coupling of all the measures in the sequence.
Borel measure: A Borel measure is a measure defined on the sigma-algebra generated by open sets of a topological space. The theorem represents Borel probability measures on the underlying space.
Patrick Billingsley: Patrick Billingsley was an American probabilist whose books shaped modern treatments of probability limits. His exposition helped make the theorem standard in convergence theory.
Weak convergence of stochastic processes: Weak convergence of stochastic processes is convergence in distribution of their laws on a function space. The theorem can turn process-level weak convergence into almost-sure convergence on a shared space.
Convergence almost surely: Almost-sure convergence is pointwise convergence of random variables outside a null event. It names the conclusion and contrasts with mere convergence of laws.

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