Asymptotic estimates of multi-dimensional stable densities and their applications

By Toshiro Watanabe. 2007

79 citations recorded by the source corpus.

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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/16c4df704ed779ccedc3e95381d0b0f50f94c447

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Citations in the corpus

Estimates of Tempered Stable Densities (2008, background): “G lowacki and W. Hebisch proved in [8] and [9] that if µ has a bounded density, gµ, with respect to the surface measure on S then p1(x) ≤ c(1 + |x|)−d−α.”
Estimates of the potential kernel and Harnack's inequality for the anisotropic fractional Laplacian (2005, background): “Here we only need isotropic estimates of {Pt} from above, and our considerations become simpler than those of [53] and [43].”
Density and tails of unimodal convolution semigroups (2013, background): “In particular the directions which are not charged by the Lévy measure see in general lighter asymptotics of pt(dx) [13, 31].”
Positive definite metric spaces (2010, result): “[38]) it appears that known results do not suffice for this purpose.”
Estimates of transition densities and their derivatives for jump Lévy processes (2013, background): “More recent asymptotic results for stable Lévy processes are given in papers [32] and [4].”
Regularity theory for general stable operators (2014, background): “The decay of the heat kernel has been studied in [12] and [16] in case that dμ(θ) = a(θ)dθ (see also [7, 40]).”
Transition density estimates for jump Lévy processes (2010, methodology): “Our methods are different than those of [36].”