Asymptotic estimates of multi-dimensional stable densities and their applications

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 densitie...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2007-06, Vol.359 (6), p.2851-2879
Main Author: Watanabe, Toshiro
Format: Article
Language:English
Subjects:
Borel sets
Density
Density estimation
Distribution theory
Exact sciences and technology
Integers
Mathematical moments
Mathematical theorems
Mathematics
Multivariate analysis
Nonparametric inference
Parametric inference
Probability and statistics
Probability theory and stochastic processes
Random walk
Sciences and techniques of general use
Semigroups
Spectral energy distribution
Statistics
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-07-04152-9