Stability criteria for q-expectation values

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 value appear naturally. Since it was recently shown that this non-linear functional is instable, using 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 and when not. 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.