Symmetric $(q,\alpha)$-Stable Distributions. Part II: Second Representation

TL;DR

This paper introduces a new representation of $(q, ext{alpha})$-stable distributions that extends previous models to all stability and nonextensivity parameters, unifying different descriptions and exploring their properties.

Contribution

It presents a novel second representation of $(q, ext{alpha})$-stable distributions, generalizing earlier results to broader parameter ranges and unifying multiple existing descriptions.

Findings

Generalizes $(q, ext{alpha})$-stable distributions to all $ ext{alpha} ext{ in } (0,2]$ and $Q ext{ in } [1,3)$

Introduces a triplet $(q^{ ext{*}},q,q_{ ext{*}})$ with a specific mapping between distributions

Discusses potential extensions and formulates conjectures for future research.

Abstract

This paper is a continuation of papers \cite{UmarovTsallisSteinberg,UmarovTsallisGellmannSteinberg}. In Part I \cite{UmarovTsallisGellmannSteinberg} a description (representation) of $(q,\alpha)$-stable distributions based on a $F_q$-transform was given. Here, in Part II, we present another description of these distributions. This approach generalizes results of \cite{UmarovTsallisSteinberg} (which corresponds to $\alpha=2, Q\in [1,3)$) to the whole range of stability and nonextensivity parameters $\alpha \in (0,2]$ and $Q \in [1,3),$ respectively. The present case $\alpha=2$ recovers the $q$-Gaussian distributions. Similar to what is discussed in \cite{UmarovTsallisSteinberg}, a triplet $(q^{\ast},q,q_{\ast})$ arises for which the mapping $F_{q^{\ast}}: \mathcal{G}_{q} \to \mathcal{G}_{q_{\ast}}$ holds. Moreover, by unifying the two preceding descriptions, further possible extensions are discussed and some conjectures are formulated.