Small and Large Scale Asymptotics of some Lévy Stochastic Integrals |
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Authors: | Vladas Pipiras Murad S Taqqu |
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Institution: | (1) Department of Statistics & OR, University of North Carolina at Chapel Hill, Smith Bldg., CB3260, Chapel Hill, NC 27599, USA;(2) Department of Mathematics, Boston University, 111 Cummington St., Boston, MA 02215, USA |
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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.
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Keywords: | Poisson and Gaussian integrals Small and large scales Convergence Self-similarity Local self-similarity |
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