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 process, moment measures, fractional brownian motion, fractional Poisson motion
@article{PS_2013__17__567_0,
author = {Dombry, Cl\'ement and Kaj, Ingemar},
title = {Moment measures of heavy-tailed renewal point processes: asymptotics and applications},
journal = {ESAIM: Probability and Statistics},
pages = {567--591},
year = {2013},
publisher = {EDP Sciences},
volume = {17},
doi = {10.1051/ps/2012010},
mrnumber = {3085633},
zbl = {1291.60178},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2012010/}
}
TY - JOUR AU - Dombry, Clément AU - Kaj, Ingemar TI - Moment measures of heavy-tailed renewal point processes: asymptotics and applications JO - ESAIM: Probability and Statistics PY - 2013 SP - 567 EP - 591 VL - 17 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ps/2012010/ DO - 10.1051/ps/2012010 LA - en ID - PS_2013__17__567_0 ER -
%0 Journal Article %A Dombry, Clément %A Kaj, Ingemar %T Moment measures of heavy-tailed renewal point processes: asymptotics and applications %J ESAIM: Probability and Statistics %D 2013 %P 567-591 %V 17 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/ps/2012010/ %R 10.1051/ps/2012010 %G en %F PS_2013__17__567_0
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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