2006/06/01 by Umarov, Sabir, Tsallis, Constantino, Gell-Mann, Murray +1
#FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.cond-mat/0606038
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 \citeUmarovTsallisSteinberg 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.