FisherTippettGnedenko

The Fisher–Tippett–Gnedenko theorem, named after Ronald A. Fisher, L. Tippett, and Boris Gnedenko, is a cornerstone of extreme value theory. It describes the limiting behavior of the maximum of a sequence of independent and identically distributed random variables. Let X1, X2, … be i.i.d. with common distribution F, and set Mn = max{X1, …, Xn}. If there exist constants a_n > 0 and b_n such that (Mn − b_n)/a_n converges in distribution to a non-degenerate limit G, then G must be one of three standard extreme value distributions: Gumbel (Type I), Fréchet (Type II), or Weibull (Type III).

The theorem also clarifies the converse: if F lies in the max-domain of attraction of one of these three limit laws, then suitable sequences a_n and b_n can be chosen so that the normalized maxima converge to the corresponding G.

Tail behavior determines which limit law arises. Gumbel distributions describe light-tailed cases, such as normal or exponential families. Fréchet distributions apply to heavy-tailed laws with infinite right endpoint (e.g., Pareto-type tails). Weibull distributions correspond to bounded upper endpoints (finite maximum). These three families are the canonical limits for maxima and underpin practical extreme value modeling in fields such as hydrology, finance, engineering, and environmental science.

In practice, the theorem provides a framework for fitting and extrapolating the behavior of extreme events, by identifying the relevant type and estimating its parameters.