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

Clément Dombry; Ingemar Kaj

ESAIM: Probability and Statistics (2013)

  • Volume: 17, page 567-591
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Dombry, Clément, and Kaj, Ingemar. "Moment measures of heavy-tailed renewal point processes: asymptotics and applications." ESAIM: Probability and Statistics 17 (2013): 567-591. <http://eudml.org/doc/274393>.

@article{Dombry2013,
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.},
author = {Dombry, Clément, Kaj, Ingemar},
journal = {ESAIM: Probability and Statistics},
keywords = {Heavy-tailed renewal process; moment measures; fractional brownian motion; fractional Poisson motion; heavy-tails; renewal process; fractional Brownian motion; Poisson process},
language = {eng},
pages = {567-591},
publisher = {EDP-Sciences},
title = {Moment measures of heavy-tailed renewal point processes: asymptotics and applications},
url = {http://eudml.org/doc/274393},
volume = {17},
year = {2013},
}

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
PB - EDP-Sciences
VL - 17
SP - 567
EP - 591
AB - 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.
LA - eng
KW - Heavy-tailed renewal process; moment measures; fractional brownian motion; fractional Poisson motion; heavy-tails; renewal process; fractional Brownian motion; Poisson process
UR - http://eudml.org/doc/274393
ER -

References

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