## Asymptotic estimates of multi-dimensional stable densities and their applications

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- by Toshiro Watanabe PDF
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**359**(2007), 2851-2879 Request permission

## 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.## References

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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
- Received by editor(s): September 13, 2004
- Received by editor(s) in revised form: June 13, 2005
- Published electronically: January 26, 2007
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**359**(2007), 2851-2879 - MSC (2000): Primary 60E07, 60G52; Secondary 60G51, 60J45
- DOI: https://doi.org/10.1090/S0002-9947-07-04152-9
- MathSciNet review: 2286060

Dedicated: Dedicated to Minoru Motoo on his 77th birthday