Gatha Cognition®
Perception, Learning and Reasoning

#### Asymptotics of Univariate Extremes

Statistical Theory of Extremes

11-44

Weibull distributions , Fréchet distributions , Gumbel distributions , Max-stability , LLN for extremes

Here it is stated the basis for the development of the whole book: finite behaviour of sample maxima and minima and behaviour of extremes in large samples; asymptotic distributions of extremes. The laws of large numbers (LLN) for extremes and samples; the additive and the multiplicative laws of large numbers (ALLN and MLLN); right-tail and left-tail behaviour of a distribution function and the Extremal Limit Theorem; domains of attraction for maxima and for minima; max-stability for the class of von Mises-Jenkinson limiting distributions; joint behaviour of sample minimum and sample maximum and the asymptotic distributions of the m-th extremes.

In many cases we do not even know if the observations come from some known parametric family of distributions and, in general, we are not under the independent and identically distributed (i.i.d.) hypothesis.

The non-degenerate limiting distributions of maxima from i.i.d. samples, if they exist, may be one of the following: Weibull, Gumbel and Fréchet.

A general form that integrates in one expression the three limiting distributions of maxima is called von Mises-Jenkinson distribution.

It must be said that the i.i.d conditions are not essential and can be weakened: as a rule the margins of the sequence should not be very different and the correlation/association between two observations must wane out, as the distance between them increases.

The classes of max-stable and limiting distributions for maxima coincide.