] 3 M ay 2 00 8 SYMMETRIC ( q , α ) - STABLE DISTRIBUTIONS . PART I : FIRST REPRESENTATION

The classic central limit theorem and α-stable distributions play a key role in probability theory, 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 special (long range) correlations between the random variables, and recovers independence for q = 1. Recently, a q-central limit theorem consistent with nonextensive statistical mechanics was established [1] which generalizes the classic Central Limit Theorem. In the present paper we introduce and study symmetric (q, α)-stable distributions. The case q = 1 recovers the Lévy α-stable distributions.