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Recurrence properties of autoregressive processes with super-heavy-tailed innovations

Part of: Computer system organization Markov processes

Published online by Cambridge University Press:  14 July 2016

Assaf Zeevi  and
Peter W. Glynn
Assaf Zeevi*
Affiliation:
Columbia University
Peter W. Glynn*
Affiliation:
Stanford University
*
∗ Postal address: Graduate School of Business, Columbia University, 3022 Broadway, New York, NY 10027-6902, USA. Email address: assaf@gsb.columbia.edu
∗∗ Postal address: Department of Management Science and Engineering, Stanford University, Stanford, CA 94305-4026, USA. Email address: glynn@stanford.edu
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Abstract

This paper studies recurrence properties of autoregressive (AR) processes with ‘super-heavy-tailed’ innovations. Specifically, we study the case where the innovations are distributed, roughly speaking, as log-Pareto random variables (i.e. the tail decay is essentially a logarithm raised to some power). We show that these processes exhibit interesting and somewhat surprising behaviour. In particular, we show that AR(1) processes, with the usual root assumption that is necessary for stability, can exhibit null-recurrent as well as transient dynamics when the innovations follow a log-Cauchy-type distribution. In this regime, the recurrence classification of the process depends, somewhat surprisingly, on the value of the constant pre-multiplier of this distribution. More generally, for log-Pareto innovations, we provide a positive-recurrence/null-recurrence/transience classification of the corresponding AR processes.

Keywords

Autoregressive processheavy tailMarkov processrecurrencestochastic stabilitytransience

MSC classification

Primary: 68M20: Performance evaluation; queueing; scheduling 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 639 - 653
Copyright
Copyright © Applied Probability Trust 2004 

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