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On the Distribution of the Nearly Unstable AR(1) Process with Heavy Tails

Part of: Stochastic processes Limit theorems Distribution theory Approximations and expansions Approximation methods and numerical treatment of dynamical systems

Published online by Cambridge University Press:  01 July 2016

Mariana Olvera-Cravioto
Mariana Olvera-Cravioto*
Affiliation:
Columbia University
*
∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, 306 S. W. Mudd Building, 500 W. 120th Street, New York, NY 10027, USA. Email address: molvera@ieor.columbia.edu
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Abstract

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We consider a nearly unstable, or near unit root, AR(1) process with regularly varying innovations. Two different approximations for the stationary distribution of such processes exist: a Gaussian approximation arising from the nearly unstable nature of the process and a heavy-tail approximation related to the tail asymptotics of the innovations. We combine these two approximations to obtain a new uniform approximation that is valid on the entire real line. As a corollary, we obtain a precise description of the regions where each of the Gaussian and heavy-tail approximations should be used.

Keywords

AR processnear unit rootheavy tailweighted sumuniform approximationlarge deviation

MSC classification

Primary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 37M10: Time series analysis
Secondary: 60F10: Large deviations 60G10: Stationary processes 60G50: Sums of independent random variables; random walks 60G70: Extreme value theory; extremal processes 62E17: Approximations to distributions (nonasymptotic)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 1 , March 2010 , pp. 106 - 136
Copyright
Copyright © Applied Probability Trust 2010 

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