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  "body": "\n## Statement\n\nLet $(S, d)$ and $(S', d')$ be [[metric spaces|0VKD1TP7F3]] with their Borel $\\sigma$-algebras.\nLet $X, X_1, X_2, \\dots$ be random elements of $S$.\nLet $g \\colon S \\to S'$ be a [[measurable map|YYNCJSR0PJ]], and let $D_g$ be the set of points of $S$ at which $g$ is not [[continuous|0X3WGX95FE]].\nAssume that $P(X \\in D_g) = 0$.\n\n1. If $X_n \\to X$ [[in distribution|0SRYJGXKR4]], then $g(X_n) \\to g(X)$ in distribution.\n2. If the $X_n$ and $X$ are defined on one probability space and $X_n \\to X$ [[in probability|0QA1HBD9Z4]], then $g(X_n) \\to g(X)$ in probability.\n3. If the $X_n$ and $X$ are defined on one probability space and $X_n \\to X$ [[almost surely|0AAXWGT5TD]], then $g(X_n) \\to g(X)$ almost surely.\n\n## Remarks\n\nPart 3 is direct: on the event where $X_n \\to X$ and $X \\notin D_g$, the continuity of $g$ at $X$ gives $g(X_n) \\to g(X)$, and this event has probability $1$.\nPart 1 follows from the [[portmanteau theorem|4SSJ589G6T]]: for a closed set $F \\subseteq S'$, the closure of $g^{-1}(F)$ lies in $g^{-1}(F) \\cup D_g$.\nPart 2 follows from part 3 applied along subsequences, since convergence in probability holds exactly when every subsequence has a further subsequence that converges almost surely.\n\nExample: if $X_n \\to Z$ in distribution with $Z$ standard normal, then $X_n^2$ converges in distribution to a chi-squared variable with one degree of freedom.\n\nThe hypothesis $P(X \\in D_g) = 0$ cannot be dropped.\nTake $X_n = 1/n$, $X = 0$ and $g = \\mathbf{1}_{(0, \\infty)}$.\nThen $X_n \\to X$ surely, but $g(X_n) = 1$ for all $n$ while $g(X) = 0$.\nHere $D_g = \\set{0}$ and $P(X \\in D_g) = 1$.\n\n## Explanations\n\n### L1 human\n\nIf random quantities settle down to a limit, then any continuous function of them settles down to the same function of the limit.\nThis holds for each of the three usual kinds of convergence.\nThe function may even jump at some points, as long as the limit almost never lands on such a point.\n\n### L2 human\n\nThe theorem is the main tool to move limit results from one statistic to another.\nWith [[Slutsky's theorem|GCV10B5HES]] and the [[central limit theorem|0EPZC4Z1FC]], it gives the limit laws of sums, products, quotients and squares of asymptotically normal statistics.\nIt also underlies the delta method, where $g$ is a differentiable function.\nFor random paths, applied with $g$ the supremum map, it turns [[Donsker's theorem|818832PMSV]] into a limit law for the maximum of a random walk.\n",
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