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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Asymptotic estimates of multi-dimensional stable densities and their applications


Author: Toshiro Watanabe
Journal: Trans. Amer. Math. Soc. 359 (2007), 2851-2879
MSC (2000): Primary 60E07, 60G52; Secondary 60G51, 60J45
Published electronically: January 26, 2007
MathSciNet review: 2286060
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Abstract: The relation between the upper and lower asymptotic estimates of the density and the fractal dimensions on the sphere of the spectral measure for a multivariate stable distribution is discussed. In particular, the problem and the conjecture on the asymptotic estimates of multivariate stable densities in the work of Pruitt and Taylor in 1969 are solved. The proper asymptotic orders of the stable densities in the case where the spectral measure is absolutely continuous on the sphere, or discrete with the support being a finite set, or a mixture of such cases are obtained. Those results are applied to the moment of the last exit time from a ball and the Spitzer type limit theorem involving capacity for a multi-dimensional transient stable process.


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Additional Information

Toshiro Watanabe
Affiliation: Center for Mathematical Sciences, The University of Aizu, Aizu-Wakamatsu Fukushima, 965-8580 Japan
Email: t-watanb@u-aizu.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04152-9
PII: S 0002-9947(07)04152-9
Keywords: Stable density, spectral measure, transient L\'evy process, last exit time
Received by editor(s): September 13, 2004
Received by editor(s) in revised form: June 13, 2005
Published electronically: January 26, 2007
Dedicated: Dedicated to Minoru Motoo on his 77th birthday
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.