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  "body": "\n## Statement\n\nLet $X_1, X_2, \\dots$ be [[i.i.d.|00MRN0FARV]] real random variables, and let $M_n = \\max(X_1, \\dots, X_n)$.\nAssume that there are constants $a_n \u003e 0$ and $b_n \\in \\R$ such that $(M_n - b_n) / a_n$ [[converges in distribution|0SRYJGXKR4]] to a random variable $Y$.\nAssume that the law of $Y$ is not concentrated at one point.\nLet $G$ be the [[cumulative distribution function|0H8H7ZTHW2]] of $Y$.\nThen there are $a \u003e 0$ and $b \\in \\R$ such that $x \\mapsto G(ax + b)$ is one of the following three functions:\n\n1. the Gumbel distribution function $\\Lambda(x) = \\exp(-e^{-x})$ for $x \\in \\R$;\n2. a Fréchet distribution function $\\Phi_\\alpha(x) = \\exp(-x^{-\\alpha})$ for $x \u003e 0$ and $\\Phi_\\alpha(x) = 0$ for $x \\le 0$, for some $\\alpha \u003e 0$;\n3. a reversed Weibull distribution function $\\Psi_\\alpha(x) = \\exp(-(-x)^{\\alpha})$ for $x \\le 0$ and $\\Psi_\\alpha(x) = 1$ for $x \u003e 0$, for some $\\alpha \u003e 0$.\n\n## Remarks\n\nThe three families combine into the generalised extreme value distributions $G_\\xi(x) = \\exp\\left(-(1 + \\xi x)^{-1/\\xi}\\right)$ for $1 + \\xi x \u003e 0$, with $G_0 = \\Lambda$ as the limit case $\\xi \\to 0$.\nThe sign of $\\xi$ gives the type: $\\xi \u003e 0$ is Fréchet, $\\xi = 0$ is Gumbel, and $\\xi \u003c 0$ is reversed Weibull.\n\nIdea of the proof.\nIf $F$ is the distribution function of $X_1$, then $P(M_n \\le a_n x + b_n) = F(a_n x + b_n)^n$.\nComparing the limits along $n$ and $nk$ shows that $G$ is max-stable: for each $k \\ge 1$ there are $\\alpha_k \u003e 0$ and $\\beta_k$ with $G(x)^k = G(\\alpha_k x + \\beta_k)$.\nSolving this functional equation gives the three families.\n\nExample: let the $X_k$ be [[exponential|0E99F51GVB]] with rate $1$, so $F(x) = 1 - e^{-x}$ for $x \\ge 0$.\nThen $P(M_n - \\log n \\le x) = (1 - e^{-x}/n)^n \\to \\exp(-e^{-x})$ for every $x$, so the Gumbel law appears with $a_n = 1$ and $b_n = \\log n$.\n\nThe hypothesis that a non-degenerate limit exists is needed.\nFor some laws, such as the geometric and the Poisson distributions, no choice of $a_n$ and $b_n$ gives a non-degenerate limit.\n\n## Explanations\n\n### L1 human\n\nThe largest of many independent measurements has, after a shift and a rescaling, only three possible limit shapes.\nWhich shape appears depends on the tail of the distribution.\nHeavy polynomial tails give the Fréchet law, and light tails such as the exponential or normal give the Gumbel law.\nA finite upper end point can give the Weibull law.\nThis is the basis for modelling floods, record temperatures and other extremes.\n\n### L2 human\n\nThe theorem plays for maxima the role that the [[central limit theorem|0EPZC4Z1FC]] plays for sums.\nThe set of laws whose maxima converge to a given type is its domain of attraction, and Gnedenko characterised these domains through conditions on the tail $1 - F$.\nIn statistics one fits the generalised extreme value family to block maxima of data.\n",
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