# An Extension of the List of Extreme Value Distributions

November 7, 1997

## Abstract

Some applications of extreme value statistics deal with random variables whose distributions have a priori bounded supports. The classical extreme value distributions, however, have unbounded supports. We extend the notion of "maximal value distribution" and prove the existence of a limiting family of distributions, each of which is supported on a bounded interval.

The generalization is based on extending the group of affine transformations of the of the real line, which underlies the classical theory, to the group of fractional linear transformations (of the extended real line). The generalization is, in a certain sense, unique.