On the robustness of q-expectation values and Renyi entropy

TL;DR

This paper investigates the robustness of Renyi and nonadditive S_q entropies, along with q-expectation values, across different types of probability distributions, highlighting conditions under which they remain physically robust despite mathematical instabilities.

Contribution

It provides a detailed analysis of the robustness of these functionals for continuous and discrete distributions, including conditions that ensure physical robustness even when mathematical stability is lacking.

Findings

Robustness of entropies and q-expectations in continuous distributions when unbounded and negative entropy cases are excluded.

Robustness of these functionals in finite discrete distributions.

Conditions for physical robustness in infinite discrete distributions despite violations of Lesche-stability.

Abstract

We study the robustness of functionals of probability distributions such as the R\'enyi and nonadditive S_q entropies, as well as the q-expectation values under small variations of the distributions. We focus on three important types of distribution functions, namely (i) continuous bounded (ii) discrete with finite number of states, and (iii) discrete with infinite number of states. The physical concept of robustness is contrasted with the mathematically stronger condition of stability and Lesche-stability for functionals. We explicitly demonstrate that, in the case of continuous distributions, once unbounded distributions and those leading to negative entropy are excluded, both Renyi and nonadditive S_q entropies as well as the q-expectation values are robust. For the discrete finite case, the Renyi and nonadditive S_q entropies and the q-expectation values are robust. For the infinite discrete case, where both Renyi entropy and q-expectations are known to violate Lesche-stability and stability respectively, we show that one can nevertheless state conditions which guarantee physical robustness.