Fisher–Tippett–Gnedenko theorem

The Fisher–Tippett–Gnedenko theorem characterizes nondegenerate distributional limits of suitably normalized maxima of independent, identically distributed random variables. Such limits belong to the generalized extreme value family.

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Generalized extreme value distribution: A three-parameter family combining the Gumbel, Fréchet, and Weibull distributions for normalized extreme values. The theorem identifies this family as the possible nondegenerate limits of normalized maxima.
Order statistic: A value ranked by size among a sample; the maximum is its largest order statistic. The sample maximum is the top order statistic to which the theorem applies.
Ronald Fisher: A British statistician and geneticist who developed foundational methods in statistical inference. Fisher’s work with Leonard Tippett established an early form of the limit theorem.
Return level: A value expected to be exceeded once within a specified average return period under a probability model. Extreme-value limits help estimate levels associated with rare exceedances.
Central limit theorem: A theorem stating that suitably normalized sums of many independent observations often converge to a normal distribution. It concerns sums rather than maxima, producing a different universal limit family.
Extreme value index: A parameter describing the limiting tail type of an extreme-value distribution. Its sign distinguishes the theorem’s Fréchet, Gumbel, and Weibull limit types.
Convergence in distribution: Convergence of random variables’ cumulative distribution functions at continuity points of the limiting function. The theorem’s limits are distributional, not necessarily pointwise limits of sample values.
Leonard Henry Caleb Tippett: A British statistician whose 1928 study of extreme order statistics identified limiting distributions for maxima. Tippett derived the three classical forms of limiting extreme-value distributions.
Flood frequency analysis: Statistical estimation of the probabilities and magnitudes of extreme river flows or floods. Annual flood maxima are a standard setting for applying extreme-value limit models.
Generalized Pareto distribution: A family of distributions used to model exceedances over a sufficiently high threshold. Threshold exceedances lead to a complementary extreme-value limit result, rather than block-maxima limits.