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Small and Large Scale Asymptotics of some Lévy Stochastic Integrals

Author

Listed:
  • Vladas Pipiras

    (University of North Carolina at Chapel Hill)

  • Murad S. Taqqu

    (Boston University)

Abstract

We provide general conditions for normalized, time-scaled stochastic integrals of independently scattered, Lévy random measures to converge to a limit. These integrals appear in many applied problems, for example, in connection to models for Internet traffic, where both large scale and small scale asymptotics are considered. Our result is a handy tool for checking such convergence. Numerous examples are provided as illustration. Somewhat surprisingly, there are examples where rescaling towards large times scales yields a Gaussian limit and where rescaling towards small time scales yields an infinite variance stable limit, and there are examples where the opposite occurs: a Gaussian limit appears when one converges towards small time scales and an infinite variance stable limit occurs when one converges towards large time scales.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:10:y:2008:i:2:d:10.1007_s11009-007-9052-4
    DOI: 10.1007/s11009-007-9052-4
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    References listed on IDEAS

    as
    1. 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.
    2. Houdré, C. & Kawai, R., 2006. "On fractional tempered stable motion," Stochastic Processes and their Applications, Elsevier, vol. 116(8), pages 1161-1184, August.
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