Small and Large Scale Asymptotics of some Lévy Stochastic Integrals

Vladas Pipiras () and Murad S. Taqqu ()
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Vladas Pipiras: University of North Carolina at Chapel Hill
Murad S. Taqqu: Boston University

Methodology and Computing in Applied Probability, 2008, vol. 10, issue 2, 299-314

Abstract: 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.

Keywords: Poisson and Gaussian integrals; Small and large scales; Convergence; Self-similarity; Local self-similarity; Primary 60F15, 60G18; Secondary 60E07 (search for similar items in EconPapers)
Date: 2008
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DOI: 10.1007/s11009-007-9052-4

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