On cumulative Tsallis entropies

TL;DR

This paper explores the properties and bounds of the cumulative Tsallis entropy, a generalization of classical entropy, introducing dual measures and characterizing distribution finiteness, with implications for generalized distributions like q-Gaussian and Logistic.

Contribution

It introduces the dual cumulative Tsallis entropy and two families of risk measures, providing new bounds and characterizations of the entropy's range under various constraints.

Findings

Finiteness of the entropy characterized by ${\mathcal L}_p$-spaces.

Optimal bounds for the entropy range are established.

Maximization leads to a perturbed Logistic distribution.

Abstract

We investigate the cumulative Tsallis entropy, an information measure recently introduced as a cumulative version of the classical Tsallis differential entropy, which is itself a generalization of the Boltzmann-Gibbs statistics. This functional is here considered as a perturbation of the expected mean residual life via some power weight function. This point of view leads to the introduction of the dual cumulative Tsallis entropy and of two families of coherent risk measures generalizing those built on mean residual life. We characterize the finiteness of the cumulative Tsallis entropy in terms of ${\mathcal L}_p$-spaces and show how they determine the underlying distribution. The range of the functional is exactly described under various constraints, with optimal bounds improving on all those previously available in the literature. Whereas the maximization of the Tsallis differential entropy gives rise to the classical $q-$Gaussian distribution which is a generalization of the Gaussian having a finite range or heavy tails, the maximization of the cumulative Tsallis entropy leads to an analogous perturbation of the Logistic distribution.