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  "body": "\n## Statement\n\nLet $S$ be a [[separable|0Z2P41SVDP]] [[metric space|0VKD1TP7F3]] with its Borel σ-algebra.\nLet $\\mu, \\mu_1, \\mu_2, \\dots$ be [[probability measures|059TJN0NZ1]] on $S$ such that $\\mu_n$ [[converges weakly|MBW9JCGZC1]] to $\\mu$.\nThen there are a probability space $(\\Omega, \\mathcal{F}, P)$ and random variables $X, X_1, X_2, \\dots$ on it with values in $S$ with the following properties.\nThe variable $X$ has law $\\mu$, each $X_n$ has law $\\mu_n$, and $X_n \\to X$ [[almost surely|0AAXWGT5TD]].\n\n## Remarks\n\nOn $S = \\R$ there is a short proof: let $U$ be uniform on $(0, 1)$, and set $X_n = F_n^{-1}(U)$ and $X = F^{-1}(U)$, where $F_n^{-1}$ and $F^{-1}$ are the [[quantile functions|0X60VE98HN]] of $\\mu_n$ and $\\mu$.\nWeak convergence implies $F_n^{-1}(u) \\to F^{-1}(u)$ at each continuity point $u$ of $F^{-1}$, and $F^{-1}$ is monotone, so it has only countably many discontinuities, a null set for $U$.\nThe general proof builds the variables from fine partitions of $S$ into small sets whose boundaries have $\\mu$-measure $0$.\nThe theorem gives quick proofs of facts about weak convergence: for example, if $X_n \\to X$ in distribution and $h$ is continuous, then $h(X_n) \\to h(X)$ in distribution, which is the [[continuous mapping theorem|CFRN0G7GDB]].\nIt also gives $E[\\abs{X}] \\le \\liminf_n E[\\abs{X_n}]$ for real variables, by Fatou's lemma.\nThe new variables only have the right laws; the original random variables, if any, need not converge almost surely.\n\n## Explanations\n\n### L1 human\n\nConvergence in distribution is a weak notion: it only says that the laws get close, not the random values.\nSkorokhod's theorem says that one can always rebuild the random variables, with the same laws, on a new space where the values themselves converge.\nThis lets one use strong tools for almost sure convergence when only convergence in law is known.\n\n### L2 human\n\nThe theorem is a standard device in weak convergence theory.\nIt appears in proofs of the [[continuous mapping theorem|CFRN0G7GDB]] and of results on uniform integrability and convergence of moments.\nIt is also used for stochastic processes with values in function spaces.\nIn the real case it is just the quantile coupling.\nIt should not be confused with Skorokhod's embedding theorem, which embeds a random walk into a Brownian motion.\n",
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