Generalized-generalized entropies and limit distributions

Limit distributions are not limited to uncorrelated variables but can be constructively derived for a large class of correlated random variables, as was shown e.g. in the context of large deviation theory [1], and recently in a very general setting by Hilhorst and Schehr [2]. At the same time it has been conjectured, based on numerical evidence, that several limit distributions originating from specific correlated random processes follow q-Gaussians. It could be shown that this is not the case for some of these situations, and more complicated limit distributions are necessary. In this work we show the derivation of the analytical form of entropy which -under the maximum entropy principle, imposing ordinary constraints- provides exactly these limit distributions. This is a concrete example for the necessity of more general entropy functionals beyond q statistics