  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
greater or equal than 1. Specifically, this kind of correlation turns the
probability density function known as q-Gaussian, which emerges upon
maximisation of the entropy Sq, into an attractor in probability space.
Moreover, we also discuss a q-generalisation of a-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.
