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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 renewalpoint process with heavy-tailed inter-renewal distribution and its continuous analog, theoccupation measure of a heavy-tailed Lévy subordinator. Our results reveal that theasymptotic structure of such moment measures are given by explicit power-law densityfunctions. The same power-law densities appear naturally as cumulant measures of certainPoisson and Gaussian stochastic integrals. This correspondence provides new and extendedresults 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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References

J. Bertoin, Lévy Processes. Cambridge University Press, Cambridge (1996).
Biermé, H., Estrade, A. and Kaj, I., Self-similar random fields and rescaled random balls models. J. Theor. Prob. 23 (2010) 1110–1141. Google Scholar
N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation. Cambridge University Press, Cambridge (1987).
D.J. Daley and D. Vere-Jones, An introduction to the theory of point processes, in Elementary theory and methods I. Probab. Appl. 2nd edition. Springer-Verlag (2003).
Dombry, C. and Kaj, I., The on-off network traffic model under intermediate scaling. Queuing Syst. 69 (2011) 29–44. Google Scholar
Gaigalas, R.. A Poisson bridge between fractional Brownian motion and stable Lévy motion. Stoch. Proc. Appl. 116 (2006) 447–462. Google Scholar
Gaigalas, R. and Kaj, I., Convergence of scaled renewal processes and a packet arrival model. Bernoulli 9 (2003) 671–703. Google Scholar
Grimmett, G., Weak convergence using higher-order cumulants. J. Theor. Prob. 5 (1992) 767–773. Google Scholar
I. Kaj and A. Martin-Löf, Scaling limit results for the sum of many inverse Lévy subordinators. arXiv:1203.6831 [math.PR] (2012).
I. Kaj and M.S. Taqqu, Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach, in: In an Out of Equilibrium 2, Progress Probability, vol. 60 edited by M.E. Vares and V. Sidoravicius. Birkhauser (2008) 383–427.
O. Kallenberg. Foundations of Modern Probability, 2nd edition. Springer-Verlag, New York (2002).
Lévy, J.B. and Taqqu, M.S., Renewal reward processes with heavy-tailed interrenewal times and heavy-tailed rewards. Bernoulli 6 (2000) 23–44. Google Scholar
Mikosch, T., Resnick, S., H. Rootzen and A. Stegeman,Is network traffic approximated by stable Lévy motion or fractional Brownian motion. Ann. Appl. Probab. 12 (2002) 23–68. Google Scholar
Pipiras, V. and Taqqu, M.S., The limit of a renewal-reward process with heavy-tailed rewards is not a linear fractional stable motion. Bernoulli 6 (2000) 607–614. Google Scholar
Pipiras, V., Taqqu, M.S. and Lévy, J.B., Slow, fast and arbitrary growth conditions for renewal reward processes when the renewals and the rewards are heavy-tailed. Bernoulli 10 (2004) 121–163. Google Scholar
S.I. Resnick, Extreme values, regular variation, and point processes. Springer, New York (1987).
S.I. Resnick, Heavy-tail phenomena, Probabilistic and statistical modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007).
G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes. Chapman and Hall (1994).
Taqqu, M.S., Willinger, W. and Sherman, R., Proof of a fundamental result in self-similar traffic modeling. Comput. Commun. Rev. 27 (1997) 5–23. Google Scholar
Teugels, J.L., Renewal theorems when the first and the second moment is infinite. Ann. Math. Statist. 39 (1968) 1210–1219. Google Scholar