Geodesic
Moment measures of heavy-tailed renewal point processes: asymptotics and applications
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 567-591

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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.

DOI : 10.1051/ps/2012010
Classification : 60K05, 60G22, 60F05
Keywords: Heavy-tailed renewal process, moment measures, fractional brownian motion, fractional Poisson motion 
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     title = {Moment measures of heavy-tailed renewal point processes: asymptotics and applications},
     journal = {ESAIM: Probability and Statistics},
     pages = {567--591},
     publisher = {EDP-Sciences},
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     year = {2013},
     doi = {10.1051/ps/2012010},
     mrnumber = {3085633},
     zbl = {1291.60178},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2012010/}
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Dombry, Clément; Kaj, Ingemar. Moment measures of heavy-tailed renewal point processes: asymptotics and applications. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 567-591. doi: 10.1051/ps/2012010

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