Moment measures of heavy-tailed renewal point processes: asymptotics and applications

¹ Laboratoire LMA, Université de Poitiers, Téléport 2, BP 30179, 86962 Futuroscope-Chasseneuil Cedex, France
clement.dombry@math.univ-poitiers.fr
² Department of Mathematics, Uppsala University, Box 480 SE 75106 Uppsala, Sweden
ikaj@math.uu.se

Received: 21 February 2012

Abstract

We study higher-order moment measures of heavy-tailed renewal models, including a renewal point process with heavy-tailed inter-renewal distribution and its continuous analog, the occupation measure of a heavy-tailed Lévy subordinator. Our results reveal that the asymptotic structure of such moment measures are given by explicit power-law density functions. The same power-law densities appear naturally as cumulant measures of certain Poisson and Gaussian stochastic integrals. This correspondence provides new and extended results regarding the asymptotic fluctuations of heavy-tailed sources under aggregation, and clarifies existing links between renewal models and fractional random processes.