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  "body": "\n## Definition\n\nLet $E$ be a topological space with its [[Borel σ-algebra|0WQXY32Q8D]], and let $\\mathcal{M}$ be a set of [[probability measures|059TJN0NZ1]] on $E$.\n$\\mathcal{M}$ is *tight* if for every $\\varepsilon \u003e 0$ there is a [[compact set|0R6EWFKRCF]] $K \\subseteq E$ such that\n$$\\mu(E \\setminus K) \u003c \\varepsilon \\qquad \\text{for all } \\mu \\in \\mathcal{M}.$$\nA sequence of random variables with values in $E$ is tight if the set of their laws is tight.\n\n## Remarks\n\nOn $\\R^d$, a family $\\mathcal{M}$ is tight if and only if $\\sup_{\\mu \\in \\mathcal{M}} \\mu(\\set{x : \\abs{x} \u003e R}) \\to 0$ as $R \\to \\infty$, because the compact sets are the closed bounded sets.\nExample: the [[normal|05AQE0QR0D]] laws $N(m, 1)$ with $\\abs{m} \\le 1$ form a tight family. Given $\\varepsilon$, choose $r$ with $P(\\abs{Z} \u003e r) \u003c \\varepsilon$ for $Z$ standard normal; then $K = [-1 - r, 1 + r]$ works, since $N(m, 1)(\\R \\setminus K) \\le P(\\abs{Z} \u003e r)$.\nNon-example: the point masses $\\delta_n$, $n \\in \\N$, on $\\R$. Each compact $K$ lies in some $[-R, R]$, and then $\\delta_n(\\R \\setminus K) = 1$ for $n \u003e R$.\nAlso non-tight: the normal laws $N(0, n)$, $n \\ge 1$, whose mass spreads out.\nA single Borel probability measure on a [[Polish space|584GXXY73M]] is tight, and so is every finite family.\nOn a Polish space, [[Prokhorov's theorem|D6FMEQYA76]] says that $\\mathcal{M}$ is tight if and only if it is relatively compact for [[weak convergence|MBW9JCGZC1]].\nEvery sequence of real random variables that converges in distribution is tight.\n",
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