Measuring the range of an additive Lévy process

• Davar Khoshnevisan ^[1] ; Yimin Xiao ^[2] ; Yuquan Zhong ^[3]
  1. [1] University of Utah

     University of Utah

     Estados Unidos

     
  2. [2] Michigan State University

     Michigan State University

     City of East Lansing, Estados Unidos

     
  3. [3] Academia Sinica

     Academia Sinica

     Taiwán

     

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 2, 2003, págs. 1097-1141
• Idioma: inglés
• DOI: 10.1214/aop/1048516547
• Texto completo no disponible (Saber más ...)
• Resumen

  • The primary goal of this paper is to study the range of the random field X(t)=∑Nj=1Xj(tj), where X1,…,XN\vspace*{-1pt} are independent Lévy processes in \Rd.

    To cite a typical result of this paper, let us suppose that Ψi denotes the Lévy exponent of Xi for each i=1,…,N. Then, under certain mild conditions, we show that a necessary and sufficient condition for X(\RN+) to have positive d-dimensional Lebesgue measure is the integrability of the function \Rd∋ξ↦∏Nj=1R{1+Ψj(ξ)}−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.