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  "body": "\n## Statement\n\nLet $F$ be a [[cumulative distribution function|0MXMS3AE13]] on $\\R$ and let $x_F = \\sup\\set{x : F(x) \u003c 1} \\le \\infty$ be its right endpoint.\nFor $u \u003c x_F$, let $F_u$ be the excess distribution over the threshold $u$:\n$$F_u(y) = \\frac{F(u + y) - F(u)}{1 - F(u)}, \\qquad 0 \\le y \u003c x_F - u,$$\nthe law of $X - u$ given $X \u003e u$ when $X$ has distribution function $F$.\nFor $\\xi \\in \\R$ and $\\beta \u003e 0$, let $G_{\\xi, \\beta}$ be the generalised Pareto distribution function\n$$G_{\\xi, \\beta}(y) = 1 - \\Big(1 + \\frac{\\xi y}{\\beta}\\Big)^{-1/\\xi} \\ (\\xi \\ne 0), \\qquad G_{0, \\beta}(y) = 1 - e^{-y/\\beta},$$\nfor $y \\ge 0$ if $\\xi \\ge 0$ and for $0 \\le y \\le -\\beta/\\xi$ if $\\xi \u003c 0$.\nSay that $F$ is in the maximum domain of attraction of $H_\\xi(x) = \\exp\\big(-(1 + \\xi x)^{-1/\\xi}\\big)$ (with $H_0(x) = \\exp(-e^{-x})$) if, for [[i.i.d.|0AJN7WHBJZ]] $X_1, X_2, \\dots$ with distribution function $F$, there are $a_n \u003e 0$ and $b_n \\in \\R$ such that $(\\max_{i \\le n} X_i - b_n)/a_n$ [[converges in distribution|0JDAQHWA9N]] to $H_\\xi$.\n\nThen the following holds for each $\\xi \\in \\R$.\nThe function $F$ is in the maximum domain of attraction of $H_\\xi$ if and only if there is a positive measurable function $\\beta(u)$ with\n$$\\lim_{u \\uparrow x_F} \\ \\sup_{0 \\le y \u003c x_F - u} \\abs{F_u(y) - G_{\\xi, \\beta(u)}(y)} = 0.$$\n\n## Remarks\n\nIdea of the proof.\nMembership in the domain of attraction of $H_\\xi$ is equivalent to the existence of a positive function $a(u)$ with $\\frac{1 - F(u + a(u) y)}{1 - F(u)} \\to (1 + \\xi y)^{-1/\\xi}$ as $u \\uparrow x_F$, for each admissible $y$.\nThe left side is $1 - F_u(a(u) y)$, so this is pointwise convergence of rescaled excess laws to $G_{\\xi, 1}$, and $\\beta(u) = a(u)$ works.\nUniformity in $y$ follows because the functions are monotone and the limit is continuous.\n\nExamples.\nFor the exponential law $F(x) = 1 - e^{-x}$, $F_u(y) = 1 - e^{-y}$ for every $u$, so $\\xi = 0$ and $\\beta(u) = 1$ give equality.\nFor the [[Pareto law|5N3P64YHQR]] $1 - F(x) = x^{-\\alpha}$ for $x \\ge 1$, $F_u(y) = 1 - (1 + y/u)^{-\\alpha} = G_{1/\\alpha, \\, u/\\alpha}(y)$ for $u \\ge 1$.\nFor the uniform law on $[0, 1]$, $F_u(y) = y/(1 - u) = G_{-1, \\, 1 - u}(y)$.\n\nThe hypothesis is not automatic: some laws lie in no domain of attraction, for example the Poisson and geometric laws.\nTheir excesses over high thresholds have no generalised Pareto limit.\n\n## Explanations\n\n### L1 human\n\nPick a high threshold, such as a flood level, and look only at how far the observations exceed it.\nThe theorem says that, for most continuous distributions met in practice, these excesses follow approximately a generalised Pareto law once the threshold is high enough.\nThis family has just two parameters: a shape $\\xi$, which says how heavy the tail is, and a scale.\nSo one can model rare extreme events without knowing the full distribution.\n\n### L2 human\n\nThe theorem is the threshold counterpart of the [[Fisher–Tippett–Gnedenko theorem|Y3Z99461HJ]], which describes the limit laws of maxima.\nThe same shape parameter $\\xi$ appears in both.\nOne has $\\xi \u003e 0$ for heavy, Pareto-like tails, $\\xi = 0$ for light tails such as the normal and exponential laws, and $\\xi \u003c 0$ for laws with a finite right endpoint.\nIn practice it justifies the peaks-over-threshold method of extreme value statistics.\nOne fits $G_{\\xi, \\beta}$ to the excesses over a high threshold by maximum likelihood and uses the fit to estimate high quantiles and tail probabilities.\n",
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