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Small and Large Scale Behavior of the Poissonized Telecom Process

Author

Listed:
  • Serge Cohen

    (Université Paul Sabatier)

  • Murad S. Taqqu

    (Boston University)

Abstract

The stable Telecom process has infinite variance and appears as a limit of renormalized renewal reward processes. We study its Poissonized version where the infinite variance stable measure is replaced by a Poisson point measure. We show that this Poissonized version converges to the stable Telecom process at small scales and to the Gaussian fractional Brownian motion at large scales. This process is therefore locally as well as asymptotically self-similar. The value of the self-similarity parameter at large scales, namely the self-similarity parameter of the limit fractional Brownian motion, depends on the form the Poissonized Telecom process. The Poissonized Telecom process is a Poissonized mixed moving average. We investigate more general Poissonized mixed moving averages as well.

Suggested Citation

  • Serge Cohen & Murad S. Taqqu, 2004. "Small and Large Scale Behavior of the Poissonized Telecom Process," Methodology and Computing in Applied Probability, Springer, vol. 6(4), pages 363-379, December.
    • Handle: RePEc:spr:metcap:v:6:y:2004:i:4:d:10.1023_b:mcap.0000045085.17224.82
    DOI: 10.1023/B:MCAP.0000045085.17224.82
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    References listed on IDEAS

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    1. Cioczek-Georges, R. & Mandelbrot, B. B., 1996. "Alternative micropulses and fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 64(2), pages 143-152, November.
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    Cited by:

    1. Cohen, Serge & Lacaux, Céline & Ledoux, Michel, 2008. "A general framework for simulation of fractional fields," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1489-1517, September.
    2. Houdré, C. & Kawai, R., 2006. "On fractional tempered stable motion," Stochastic Processes and their Applications, Elsevier, vol. 116(8), pages 1161-1184, August.
    3. repec:jss:jstsof:14:i18 is not listed on IDEAS
    4. Vladas Pipiras & Murad S. Taqqu, 2008. "Small and Large Scale Asymptotics of some Lévy Stochastic Integrals," Methodology and Computing in Applied Probability, Springer, vol. 10(2), pages 299-314, June.

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