Symmetric (q,α)-Stable Distributions. Part II: Second Representation

2006/06/01 by Umarov, Sabir, Tsallis, Constantino, Gell-Mann, Murray +1
#FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)

paper · doi:10.48550/arxiv.cond-mat/0606040

Abstract

This paper is a continuation of papers \citeUmarovTsallisSteinberg,UmarovTsallisGellmannSteinberg. In Part I \citeUmarovTsallisGellmannSteinberg a description (representation) of (q,α)-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 \citeUmarovTsallisSteinberg (which corresponds to α=2, Q∈ [1,3)) to the whole range of stability and nonextensivity parameters α∈ (0,2] and Q ∈ [1,3), respectively. The present case α=2 recovers the q-Gaussian distributions. Similar to what is discussed in \citeUmarovTsallisSteinberg, a triplet (q^∗,q,q_∗) arises for which the mapping F_q^∗: G_q → G_q_∗ holds. Moreover, by unifying the two preceding descriptions, further possible extensions are discussed and some conjectures are formulated.

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