The R\'enyi entropy of L\'evy distribution

TL;DR

This paper explores the Rényi entropy of Lévy distributions, establishing a duality principle and providing asymptotic approximations for large parameters where numerical methods fail.

Contribution

It introduces a novel geometric perspective on Rényi entropy and derives asymptotic formulas for Lévy distributions in challenging parameter regimes.

Findings

Established a duality principle for Rényi entropy.

Derived asymptotic expansions for large Rényi parameters.

Linked entropy concepts to Lebesgue and exotic Lebesgue spaces.

Abstract

The equivalence between non-extensive C. Tsallis entropy and the extensive entropy introduced by Alfr\'ed R\'enyi is discussed. The R\'enyi entropy is studied from the perspective of the geometry of the Lebesgue and generalised, exotic Lebesgue spaces. A duality principle is established. The R\'enyi entropy for the L\'evy distribution, in the domain when the nunerical methods fails, is approximated by asymptotic expansion for the large values of the R\'enyi parameter.