Symmetric $(q,\alpha)$-Stable Distributions. Part I: First Representation

TL;DR

This paper introduces symmetric $(q,\alpha)$-stable distributions, generalizing classical stable laws within the framework of nonextensive statistical mechanics, and explores their properties and representations.

Contribution

It presents the first representation of symmetric $(q,\alpha)$-stable distributions, extending classical stable laws to incorporate nonextensive correlations.

Findings

Generalizes classical stable laws to $(q,\alpha)$-stable distributions.

Recovers Lévy $\alpha$-stable distributions when $q=1$.

Provides foundational representation for these new distributions.

Abstract

The classic central limit theorem and $\alpha$-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 $(q,\alpha)$-stable distributions. The case $q=1$ recovers the L\'evy $\alpha$-stable distributions.