1 J un 2 00 6 q - GENERALIZATION OF SYMMETRIC α - STABLE DISTRIBUTIONS . PART I

The classic and the Lévy-Gnedenko central limit theorems play a key role in theory of probabilities, and also in Boltzmann-Gibbs (BG) statistical mechanics. They both concern the paradigmatic case of probabilistic independence of the random variables that are being summed. A generalization of the BG theory, usually referred to as nonextensive statistical mechanics and characterized by the index q (q = 1 recovers the BG theory), introduces global correlations between the random variables, and recovers independence for q = 1. The classic central limit theorem was recently q-generalized by some of us. In the present paper we q-generalize the Lévy-Gnedenko central limit theorem.