  In this article we review the standard versions of the Central and of the
Levy-Gnedenko Limit Theorems, and illustrate their application to the
convolution of independent random variables associated with the distribution
known as q-Gaussian. This distribution emerges upon extremisation of the
nonadditive entropy, basis of nonextensive statistical mechanics. It has a
finite variance for q < 5/3, and an infinite one for q > 5/3. We exhibit that,
in the case of (standard) independence, the q-Gaussian has either the Gaussian
(if q < 5/3) or the a-stable Levy distributions (if q > 5/3) as its attractor
in probability space. Moreover, we review a generalisation of the product, the
q-product, which plays a central role in the approach of the specially
correlated variables emerging within the nonextensive theory.
