Generalization of symmetric $\alpha$-stable Lévy distributions for $q>1$

Abstract

The $\alpha$-stable distributions introduced by Lévy play an important role in probabilistic theoretical studies and their various applications, e.g., in statistical physics, life sciences, and economics. In the present paper we study sequences of long-range dependent random variables whose distributions have asymptotic power law decay, and which are called $(q,\alpha )$-stable distributions. These sequences are generalizations of i.i.d. $\alpha$-stable distributions, and have not been previously studied. Long-range dependent $(q,\alpha )$-stable distributions might arise in the description of anomalous processes in nonextensive statistical mechanics, cell biology, finance. The parameter $q$ controls dependence. If $q=1$ then they are classical i.i.d. with $\alpha$-stable Lévy distributions. In the present paper we establish basic properties of $(q,\alpha )$-stable distributions, and generalize the result of Umarov, Tsallis and Steinberg (2008), where the particular case $\alpha =2,q\in [1,3),$ was considered, to the whole range of stability and nonextensivity parameters $\alpha \in (0,2]$ and $q\in [1,3),$ respectively. We also discuss possible further extensions of the results that we obtain, and formulate some conjectures.

15 pages including 1 figure. In the new version some new references have been included. To appear in Journal of Mathematical Physics (scheduled for February 2010)

References in corpus (11)

• Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma
• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Analysis of Self-Organized Criticality in the Olami-Feder-Christensen model and in real earthquakes
• Numerical indications of a q-generalised central limit theorem
• Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example
• A note on q-Gaussians and non-Gaussians in statistical mechanics
• A generalization of the central limit theorem consistent with nonextensive statistical mechanics
• On q-Gaussians and Exchangeability
• Nonextensive statistical mechanics and central limit theorems I - Convolution of independent random variables and q-product
• On a representation of the inverse Fq transform
• Nonextensive statistical mechanics and central limit theorems II - Convolution of q-independent random variables

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