q-Generalization of Symmetric Alpha-Stable Distributions: Part II

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. In Part I we described the q-version of the α-stable Lévy distributions. In Part II we study the (q , q, q )−triplet, for which the mapping Fq : Gq → Gq holds. This fact allows us to study the corresponding attractors and to obtain a complete generalization of the q-central limit theorem for random variables with infinite (2q ⎯1)- variance.