Stability criteria for q-expectation values

TL;DR

This paper investigates the stability of q-expectation values in statistical physics, providing conditions under which these averages are robust to small changes in probability distributions, which is crucial for their practical application.

Contribution

It establishes sufficient conditions for the uniform continuity of q-expectation values, ensuring their robustness under small distribution variations.

Findings

Uniform continuity of q-expectation under certain restrictions

Bounds on permissible distribution variations

Practical implications for robustness estimation

Abstract

In statistical physics lately a specific kind of average, called the q-expectation value, has been extensively used in the context of q-generalized statistics dealing with distributions following power-laws. In this context q-expectation values appear naturally. After it has been recently shown that this non-linear functional is instable, under a very strong notion of stability, it is therefore of high interest to know sufficient conditions for when the results of q-expectations are robust under small variations of the underlying distribution function. We show that reasonable restrictions on the domain of admissible probability distributions restore uniform continuity for the q-expectation. Bounds on the size of admissible variations can be given. The practical usefulness of the theorems for estimating the robustness of the q-expectation value with respect to small variations is discussed.