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

ESAIM: Probability and Statistics (2013)

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

## Access Full Article

top## Abstract

top## How to cite

topDombry, 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

top- [1] J. Bertoin, Lévy Processes. Cambridge University Press, Cambridge (1996). Zbl0938.60005MR1406564
- [2] H. Biermé, A. Estrade and I. Kaj, Self-similar random fields and rescaled random balls models. J. Theor. Prob.23 (2010) 1110–1141. Zbl1213.60096MR2735739
- [3] N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation. Cambridge University Press, Cambridge (1987). Zbl0667.26003MR898871
- [4] 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). Zbl0657.60069MR1950431
- [5] C. Dombry and I. Kaj, The on-off network traffic model under intermediate scaling. Queuing Syst.69 (2011) 29–44. Zbl1235.60121MR2835229
- [6] R. Gaigalas. A Poisson bridge between fractional Brownian motion and stable Lévy motion. Stoch. Proc. Appl.116 (2006) 447–462. Zbl1087.60080MR2199558
- [7] R. Gaigalas and I. Kaj, Convergence of scaled renewal processes and a packet arrival model. Bernoulli9 (2003) 671–703. Zbl1043.60077MR1996275
- [8] G. Grimmett, Weak convergence using higher-order cumulants. J. Theor. Prob.5 (1992) 767–773. Zbl0759.60018MR1182679
- [9] I. Kaj and A. Martin-Löf, Scaling limit results for the sum of many inverse Lévy subordinators. arXiv:1203.6831 [math.PR] (2012).
- [10] 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. Zbl1154.60020MR2477392
- [11] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Springer-Verlag, New York (2002). Zbl0892.60001MR1876169
- [12] J.B. Lévy and M.S. Taqqu, Renewal reward processes with heavy-tailed interrenewal times and heavy-tailed rewards. Bernoulli6 (2000) 23–44. Zbl0954.60071MR1781180
- [13] T. Mikosch, S. Resnick, H. Rootzen and A. Stegeman,Is network traffic approximated by stable Lévy motion or fractional Brownian motion. Ann. Appl. Probab.12 (2002) 23–68. Zbl1021.60076MR1890056
- [14] V. Pipiras and M.S. Taqqu, The limit of a renewal-reward process with heavy-tailed rewards is not a linear fractional stable motion. Bernoulli6 (2000) 607–614. Zbl0963.60032MR1777685
- [15] V. Pipiras, M.S. Taqqu and J.B. Lévy, Slow, fast and arbitrary growth conditions for renewal reward processes when the renewals and the rewards are heavy-tailed. Bernoulli10 (2004) 121–163. Zbl1043.60040MR2044596
- [16] S.I. Resnick, Extreme values, regular variation, and point processes. Springer, New York (1987). Zbl1136.60004MR900810
- [17] S.I. Resnick, Heavy-tail phenomena, Probabilistic and statistical modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007). Zbl1152.62029MR2271424
- [18] G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes. Chapman and Hall (1994). Zbl0925.60027MR1280932
- [19] M.S. Taqqu, W. Willinger and R. Sherman, Proof of a fundamental result in self-similar traffic modeling. Comput. Commun. Rev.27 (1997) 5–23.
- [20] J.L. Teugels, Renewal theorems when the first and the second moment is infinite. Ann. Math. Statist.39 (1968) 1210–1219. Zbl0164.19103MR230390

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.