Stability criteria for q-expectation values

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.