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

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.

14 pages including 2 figures