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Generalization of symmetric α-stable Lévy distributions for q>1

Umarov, Sabir and Tsallis, Constantino and Gell-Mann, Murray and Steinberg, Stanly (2010) Generalization of symmetric α-stable Lévy distributions for q>1. Journal of Mathematical Physics, 51 (3). Art. No. 033502. ISSN 0022-2488.

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The α-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,α)-stable distributions. These sequences are generalizations of independent and identically distributed α-stable distributions and have not been previously studied. Long-range dependent (q,α)-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 independent and identically distributed with α-stable Lévy distributions. In the present paper we establish basic properties of (q,α)-stable distributions and generalize the result of Umarov et al. [Milan J. Math.76, 307 (2008)], where the particular case α=2, q∊[1,3) was considered, to the whole range of stability and nonextensivity parameters α∊(0,2] and q∊[1,3), respectively. We also discuss possible further extensions of the results that we obtain and formulate some conjectures.

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Additional Information:© 2010 American Institute of Physics. Received 10 November 2009; accepted 4 January 2010; published online 3 March 2010. We acknowledge thoughtful remarks by R. Hersh, E. P. Borges, and S. M. D. Queiros. Financial support by the Fullbright Foundation, SI International and NIH Grant No. P20 GM067594 (USA Agencies), and CNPq and Faperj (Brazilian Agencies) are acknowledged as well.
Funding AgencyGrant Number
Fulbright FoundationUNSPECIFIED
NIHP20 GM067594
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)UNSPECIFIED
Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ)UNSPECIFIED
Issue or Number:3
Record Number:CaltechAUTHORS:20150824-105705142
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Official Citation:Generalization of symmetric α-stable Lévy distributions for q>1 Umarov, Sabir and Tsallis, Constantino and Gell-Mann, Murray and Steinberg, Stanly, Journal of Mathematical Physics, 51, 033502 (2010), DOI:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:59843
Deposited By: George Porter
Deposited On:24 Aug 2015 18:14
Last Modified:03 Oct 2019 08:50

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