On convergence of entropy of distribution functions in the max domain of attraction of max stable laws

TL;DR

This paper investigates how the Shannon entropy of normalized maxima of iid random variables converges to the entropy of max stable laws, revealing cases where entropy increases during convergence.

Contribution

It establishes entropy limit theorems for distribution functions in the max domain of attraction, including cases of entropy increase and examples for upper extremes.

Findings

Shannon entropy converges to the limit entropy for normalized maxima.

In some cases, entropy increases during convergence.

Results extend to the k-th upper extremes.

Abstract

Max stable laws are limit laws of linearly normalized partial maxima of indepen- dent, identically distributed (iid) random variables (rvs). These are analogous to stable laws which are limit laws of normalized partial sums of iid rvs. In this paper, we study entropy limit theorems for distribution functions in the max domain of attraction of max stable laws under linear normalization. More specifically, we study the problem of convergence of the Shannon entropy of linearly normalized partial maxima of iid rvs to the corresponding limit entropy when the linearly normalized partial maxima converges to some nondegenerate rv. We are able to show that the Shannon entropy not only converges but, in fact, increases to the limit entropy in some cases. We discuss several examples. We also study analogous results for the k-th upper extremes.