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  "body": "\n## Statement\n\nLet $X$ be a [[metric space|0VKD1TP7F3]] and let $\\mathcal{M}$ be a set of Borel [[probability measures|059TJN0NZ1]] on $X$.\nCall $\\mathcal{M}$ relatively compact if every sequence in $\\mathcal{M}$ has a subsequence that [[converges weakly|MBW9JCGZC1]] to some Borel probability measure on $X$.\n\n1. If $\\mathcal{M}$ is [[tight|MNMJC8YVBS]], then $\\mathcal{M}$ is relatively compact.\n2. If $X$ is [[separable|0Z2P41SVDP]] and [[complete|0BY52JW6NK]], and $\\mathcal{M}$ is relatively compact, then $\\mathcal{M}$ is tight.\n\n## Remarks\n\nTight means that for every $\\varepsilon \u003e 0$ there is a [[compact set|0R6EWFKRCF]] $K$ with $\\mu(K) \u003e 1 - \\varepsilon$ for all $\\mu \\in \\mathcal{M}$.\nFor a compact space $X$, part 1 follows from the Riesz representation theorem and the weak-* compactness of the unit ball of the dual of $C(X)$.\nThe general case reduces to compact sets by tightness.\nFor $X = \\R$, part 1 is close to Helly's selection theorem for distribution functions.\n\nExample: the normal distributions $N(m, \\sigma^2)$ with $\\abs{m} \\le 1$ and $0 \u003c \\sigma \\le 1$ form a tight family, using $K = [-M, M]$ with $M$ large.\nSo every sequence of them has a weakly convergent subsequence.\nThe family of Dirac measures $\\delta_n$, $n \\in \\N$, on $\\R$ is not tight, and no subsequence converges weakly.\nThe mass escapes to infinity.\n\n## Explanations\n\n### L1 human\n\nA family of probability measures is tight when no mass runs away to infinity.\nThe theorem says that tightness is exactly what one needs to extract convergent subsequences.\nIt plays the role of the Bolzano–Weierstrass theorem for measures.\n\n### L2 human\n\nIn probability, one proves convergence in distribution in two steps.\nFirst one shows tightness, which gives limit points by part 1.\nThen one shows that all limit points are the same measure, for example through characteristic functions or finite-dimensional distributions.\nThis scheme proves Donsker's invariance principle for random walks.\nOn a Polish space, the set of probability measures with the weak topology is itself a Polish space, metrized by the Lévy–Prokhorov metric.\n",
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