On the robustness of q-expectation values and Rényi entropy

¹  Complex Systems Research Group, HNO, Medical University of Vienna - Währinger Gürtel 18-20, A-1090, Austria, EU
²  Santa Fe Institute - 1399 Hyde Park Road, Santa Fe, NM 87501, USA
³  Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil

Corresponding authors: rudolf.hanel@meduniwien.ac.at thurner@univie.ac.at tsallis@cbpf.br

Received: 12 November 2008
Accepted: 18 December 2008

Abstract

We study the robustness of the functionals of probability distributions such as the Rényi 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 Rényi and nonadditive S_q entropies as well as the q-expectation values are robust. For the discrete finite case, the Rényi and nonadditive S_q entropies and the q-expectation values are robust as well. For the infinite discrete case, where both Rényi 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.