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In this article we review the standard versions of the Central and of theLevy-Gnedenko Limit Theorems, and illustrate their application to theconvolution of independent random variables associated with the distributionknown as q-Gaussian. This distribution emerges upon extremisation of thenonadditive entropy, basis of nonextensive statistical mechanics. It has afinite 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 attractorin probability space. Moreover, we review a generalisation of the product, theq-product, which plays a central role in the approach of the speciallycorrelated variables emerging within the nonextensive theory.

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