Information-theoretic convergence of extreme values to the Gumbel distribution

TL;DR

This paper presents an information-theoretic framework for understanding how extreme value distributions converge to the Gumbel distribution, introducing a new score function and proving convergence in relative entropy.

Contribution

It introduces a novel score function that simplifies analysis of convergence to the Gumbel distribution using information theory.

Findings

Convergence to Gumbel can be characterized via relative entropy.

A new score function behaves well under maximum operations.

Strong convergence results are established under certain conditions.

Abstract

We show how convergence to the Gumbel distribution in an extreme value setting can be understood in an information-theoretic sense. We introduce a new type of score function which behaves well under the maximum operation, and which implies simple expressions for entropy and relative entropy. We show that, assuming certain properties of the von Mises representation, convergence to the Gumbel can be proved in the strong sense of relative entropy.