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Long range dependence of heavy-tailed random functions

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  16 September 2021

Rafal Kulik  and
Evgeny Spodarev
Rafal Kulik*
Affiliation:
University of Ottawa
Evgeny Spodarev*
Affiliation:
Ulm University
*
*Postal address: Department of Mathematics and Statistics, 150 Louis Pasteur Private, K1N 6N5 Ottawa, Ontario, Canada.
**Postal address: Institute of Stochastics, Helmholtzstrasse 18, D-89069 Ulm, Germany. Email address: evgeny.spodarev@uni-ulm.de
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Abstract

We introduce a definition of long range dependence of random processes and fields on an (unbounded) index space $T\subseteq \mathbb{R}^d$ in terms of integrability of the covariance of indicators that a random function exceeds any given level. This definition is specifically designed to cover the case of random functions with infinite variance. We show the value of this new definition and its connection to limit theorems via some examples including subordinated Gaussian as well as random volatility fields and time series.

Keywords

Long memoryrandom processrandom fieldtime serieslevel setheavy tailinfinite varianceshort range dependencemixingsubordinated Gaussian processstochastic volatility processlimit theorem

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60G60: Random fields 60G15: Gaussian processes 60F05: Central limit and other weak theorems
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 3 , September 2021 , pp. 569 - 593
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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