  We consider probabilistic models of N identical distinguishable, binary
random variables. If these variables are strictly or asymptotically
independent, then, for N>>1, (i) the attractor in distribution space is,
according to the standard central limit theorem, a Gaussian, and (ii) the
Boltzmann-Gibbs-Shannon entropy is extensive, meaning that S_BGS(N) ~ N . If
these variables have any nonvanishing global (i.e., not asymptotically
independent) correlations, then the attractor deviates from the Gaussian. The
entropy appears to be more robust, in the sense that, in some cases, S_BGS
remains extensive even in the presence of strong global correlations. In other
cases, however, even weak global correlations make the entropy deviate from the
normal behavior. More precisely, in such cases the entropic form Sq can become
extensive for some value of q different from unity . This scenario is
illustrated with several new as well as previously described models. The
discussion illuminates recent progress into q-describable nonextensive
probabilistic systems, and the conjectured q-Central Limit Theorem (q-CLT)
which posses a q-Gaussian attractor.
