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In this article we review recent generalisations of the central limit theoremfor the sum of specially correlated (or q-independent) variables, focusing on qgreater or equal than 1. Specifically, this kind of correlation turns theprobability density function known as q-Gaussian, which emerges uponmaximisation of the entropy Sq, into an attractor in probability space.Moreover, we also discuss a q-generalisation of a-stable Levy distributionswhich can as well be stable for this special kind of correlation.Within thiscontext, we verify the emergence of a triplet of entropic indices which relatethe form of the attractor, the correlation, and the scaling rate, similar tothe q-triplet that connects the entropic indices characterising the sensitivityto initial conditions, the stationary state, and relaxation to the stationarystate in anomalous systems.

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