the Skorokhod representation theorem
/ skuh-ruh-KHOD /
Weak convergence is a statement about laws, about expectations of test functions, and it lives on possibly different probability spaces for each n. Sometimes you would much rather argue with almost-sure convergence, which is far more flexible (it commutes with continuous functions, with monotone and dominated convergence, with all the pointwise machinery). The Skorokhod representation theorem says you may, almost: it lets you realize a weakly convergent sequence of laws as random variables on ONE common probability space that converge almost surely.
Suppose mu_n => mu weakly in P(S) where S is a separable metric space (Polish is the standard hypothesis). Then there exist a single probability space (Omega, F, P) and S-valued random variables Y_n, Y on it such that Y_n has law mu_n, Y has law mu, and Y_n -> Y almost surely (pointwise for P-almost every omega, in the metric of S). Crucially the original X_n need not converge in any pointwise sense, and may live on unrelated spaces; the theorem builds NEW representatives with the same marginal laws but coupled so that they converge pathwise. The construction (in the canonical real-line case) couples via the quantile transform, using a single uniform random variable to invert each distribution function simultaneously.
Its great service is to convert distributional statements into pathwise ones. The continuous mapping theorem, for instance, becomes nearly trivial: if g is continuous and X_n => X, take Skorokhod representatives Y_n -> Y a.s., then g(Y_n) -> g(Y) a.s. by continuity, hence g(X_n) => g(X). Many delicate weak-convergence arguments collapse to a one-line a.s. argument after a Skorokhod coupling. The honest caveat: the coupling is on an auxiliary space and tells you nothing about the joint law of the original variables; you have only matched the marginals.
On R, let F_n, F be CDFs with F_n -> F at continuity points (so mu_n => mu). Take Omega = (0,1) with Lebesgue measure, U(omega) = omega, and set Y_n = F_n^{-1}(U), Y = F^{-1}(U) using generalized inverses. Each Y_n has law mu_n, Y has law mu, and Y_n(omega) -> Y(omega) for almost every omega, an explicit Skorokhod coupling by the quantile transform.
The quantile-transform coupling: one uniform variable drives all the inverse CDFs, forcing almost-sure convergence.
The theorem produces NEW variables with the right marginal laws, not the original ones, and on an auxiliary space. It says nothing about the joint distribution of the X_n, so do not infer any dependence structure from the constructed coupling.