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April 2003 Measuring the range of an additive Lévy process
Davar Khoshnevisan, Yimin Xiao, Yuquan Zhong
Ann. Probab. 31(2): 1097-1141 (April 2003). DOI: 10.1214/aop/1048516547

Abstract

The primary goal of this paper is to study the range of the random field $X(t) = \sum_{j=1}^N X_j(t_j)$, where $X_1,\ldots, X_N$\vspace*{-1pt} are independent Lévy processes in $\R^d$.

To cite a typical result of this paper, let us suppose that $\Psi_i$ denotes the Lévy exponent of $X_i$ for each $i=1,\ldots,N$. Then, under certain mild conditions, we show that a necessary and sufficient condition for $X(\R^N_+)$ to have positive $d$-dimensional Lebesgue measure is the integrability of the function $\R^d \ni \xi \mapsto \prod_{j=1}^N \Re \{ 1+ \Psi_j(\xi)\}^{-1}$. This extends a celebrated result of Kesten and of Bretagnolle in the one-parameter setting. Furthermore, we show that the existence of square integrable local times is yet another equivalent condition for the mentioned integrability criterion. This extends a theorem of Hawkes to the present random fields setting and completes the analysis of local times for additive Lévy processes initiated in a companion by paper Khoshnevisan, Xiao and Zhong.

Citation

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Davar Khoshnevisan. Yimin Xiao. Yuquan Zhong. "Measuring the range of an additive Lévy process." Ann. Probab. 31 (2) 1097 - 1141, April 2003. https://doi.org/10.1214/aop/1048516547

Information

Published: April 2003
First available in Project Euclid: 24 March 2003

zbMATH: 1039.60048
MathSciNet: MR1964960
Digital Object Identifier: 10.1214/aop/1048516547

Subjects:
Primary: 60G60 , 60J45 , 60J55

Keywords: Additive Lévy processes , capacity , energy , Hausdorff dimension , Local times , strictly stable processes

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2003
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