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

Published online by Cambridge University Press:  01 August 2013

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

Keywords

Heavy-tailed renewal processmoment measuresfractional Brownian motionfractional Poisson motion
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 567 - 591
Copyright
© EDP Sciences, SMAI, 2013

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