Is the Tsallis q-mean value instable?

TL;DR

This paper reexamines the claimed instability of the Tsallis q-mean value, demonstrating that previous results stem from discontinuous probability limits and showing stability under proper normalization conditions.

Contribution

It clarifies the conditions under which the Tsallis q-mean value is stable, correcting prior misconceptions by analyzing probability distribution limits.

Findings

The instability claim is due to discontinuous probability limits.

Proper normalization restores the stability of the q-mean value.

Stability holds for large but finite state spaces with appropriate limits.

Abstract

The recent argue about the existence of an instability in the definition of the mean value appearing in the Tsallis non extensive Statistical Mechanic is reconsidered. Here, it is simply underlined that the pair of probability distributions employed in constructing the instability statement have a discontinuous limit when the number of states tends to infinity. That is, although for an arbitrary but finite number of states W, both probability distributions are normalized to the unit, their limits W tending to infinity do not satisfy the normalization condition and thus are not allowed "escort" probabilities for the q-mean value. However, similar distributions converging to the former ones when a parameter W_o is tending to infinity are defined here. They both satisfy the normalization to the unity in the limit W tending to infinity. This simple change allows to show that the stability condition becomes satisfied, for whatever large but fixed value of W_o is chosen.