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

Sabir Umarov; Constantino Tsallis; Murray Gell-Mann; Stanly Steinberg

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

Sabir Umarov·Constantino Tsallis·Murray Gell-Mann·Stanly Steinberg

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Abstract

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 \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.

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Sabir Umarov, Constantino Tsallis, Murray Gell-Mann, Stanly Steinberg. 2008-05-03. Symmetric $(q,α)$-Stable Distributions. Part I: First Representation. https://arxiv.org/abs/cond-mat/0606038

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