Average densities of the image and zero set of stable processes
type: Item
instance of: scholarly article
title: Average densities of the image and zero set of stable processes
zbMATH Open document ID: 0819.60038
DOI: 10.1016/0304-4149(94)00050-4
author: Y. Xiao
author: Kenneth J. Falconer
published in: Stochastic Processes and their Applications
publication date: 1995-08-15T00:00:00Z
review text: The ``order-two'' or ``average'' density of a measure \(\mu\) at a point \(x\) is defined as \[ \lim_{T \to \infty} (1/T) \int^ T_ 0 \mu (B(x,e^{-s})) e^{\alpha s} ds \] for appropriate \(\alpha\), where \(B(x,r)\) is the closed ball of centre \(x\) and radius \(r\). The authors study the existence of order-two densities of the image and zero sets of the sample paths of a wide class of stable processes. They prove the existence of a constant \(\eta\) (which depends on the process) such that, with probability one, the order-two density of the natural measure on the image or zero set exists and equals \(\eta\) almost everywhere on the set.
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 60J99
zbMATH DE Number: 735485
zbMATH Keywords: stable processes
zbMATH Keywords: zero sets
zbMATH Keywords: order-two density
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: image
reviewed by: Barbara Szyszkowicz
MaRDI profile type: Publication
full work available at URL: 0304-4149(94)00050-4
OpenAlex ID: W2045881702
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description: scientific article; zbMATH DE number 735485