The Annals of Probability

Measuring the range of an additive Lévy process

Davar Khoshnevisan, Yimin Xiao, and Yuquan Zhong

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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.

Article information

Source
Ann. Probab. Volume 31, Number 2 (2003), 1097-1141.

Dates
First available in Project Euclid: 24 March 2003

Permanent link to this document
http://projecteuclid.org/euclid.aop/1048516547

Digital Object Identifier
doi:10.1214/aop/1048516547

Mathematical Reviews number (MathSciNet)
MR1964960

Zentralblatt MATH identifier
1039.60048

Subjects
Primary: 60G60: Random fields 60J55: Local time and additive functionals 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Keywords
Additive Lévy processes strictly stable processes capacity energy local times Hausdorff dimension

Citation

Khoshnevisan, Davar; Xiao, Yimin; Zhong, Yuquan. Measuring the range of an additive Lévy process. The Annals of Probability 31 (2003), no. 2, 1097--1141. doi:10.1214/aop/1048516547. http://projecteuclid.org/euclid.aop/1048516547.


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  • SALT LAKE CITY, UTAH 84112-0090 E-MAIL: davar@math.utah.edu URL: http://www.math.utah.edu/ davar Y. XIAO DEPARTMENT OF STATISTICS AND PROBABILITY A-413 WELLS HALL MICHIGAN STATE UNIVERSITY
  • EAST LANSING, MICHIGAN 48824 E-MAIL: xiao@stt.msu.edu URL: http://www.stt.msu.edu/ xiaoy imi Y. ZHONG INSTITUTE OF APPLIED MATHEMATICS ACADEMIA SINICA
  • BEIJING, 100080 PEOPLE'S REPUBLIC OF CHINA