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A functional limit theorem for general shot noise processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  04 May 2020

Alexander Iksanov Open the ORCID record for Alexander Iksanov [Opens in a new window]  and
Bohdan Rashytov
Alexander Iksanov*
Affiliation:
Taras Shevchenko National University of Kyiv
Bohdan Rashytov*
Affiliation:
Taras Shevchenko National University of Kyiv
*
*Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine.
*Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine.
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Abstract

By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally Hölder continuous Gaussian limit process, and that the response function is regularly varying at infinity, we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann–Liouville process. We specialize our result for five particular counting processes. Also, we investigate Hölder continuity of the limit processes for general shot noise processes.

Keywords

Hölder continuityshot noise processweak convergence in the Skorokhod space

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 280 - 294
Copyright
© Applied Probability Trust 2020

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