2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160144The 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 \cite{UmarovTsallisSteinberg} 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.17 pages including 3 figuresStatistical MechanicsProbabilitytext