  Increasing the number $N$ of elements of a system typically makes the entropy
to increase. The question arises on {\it what particular entropic form} we have
in mind and {\it how it increases} with $N$. Thermodynamically speaking it
makes sense to choose an entropy which increases {\it linearly} with $N$ for
large $N$, i.e., which is {\it extensive}. If the $N$ elements are
probabilistically {\it independent} (no interactions) or quasi-independent
(e.g., {\it short}-range interacting), it is known that the entropy which is
extensive is that of Boltzmann-Gibbs-Shannon, $S_{BG} \equiv -k \sum_{i=1}^W
p_i \ln p_i$. If they are however {\it globally correlated} (e.g., through {\it
long}-range interactions), the answer depends on the particular nature of the
correlations. There is a large class of correlations (in one way or another
related to scale-invariance) for which an appropriate entropy is that on which
nonextensive statistical mechanics is based, i.e., $S_q \equiv k
\frac{1-\sum_{i=1}^W p_i^q}{q-1}$ ($S_1=S_{BG}$), where $q$ is determined by
the specific correlations. We briefly review and illustrate these ideas through
simple examples of occupation of phase space. A very similar scenario emerges
with regard to the central limit theorem. We present some numerical indications
along these lines. The full clarification of such a possible connection would
help qualifying the class of systems for which the nonextensive statistical
concepts are applicable, and, concomitantly, it would enlighten the reason for
which $q$-exponentials are ubiquitous in many natural and artificial systems.
