Nonextensive statistical mechanics and central limit theorems II—Convolution of q‐independent random variables

In this article we review recent generalisations of the central limit theorem for the sum of specially correlated (or q‐independent) variables, focusing on q⩾1. Specifically, this kind of correlation turns the probability density function Gq(X) = Aq[1+(q−1)βq(X−μq)2]11−q, which emerges upon maximisation of the entropy Sq = k(1−∫[p(X)]qdX)/(1−q), into an attractor in probability space. Moreover, we also discuss a q‐generalisation of α‐stable Levy distributions which can as well be stable for this special kind of correlation. Within this context, we verify the emergence of a triplet of entropic indices which relate the form of the attractor, the correlation, and the scaling rate, similar to the q‐triplet that connects the entropic indices characterising the sensitivity to initial conditions, the stationary state, and relaxation to the stationary state in anomalous systems.