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Survival probabilities of autoregressiveprocesses

Published online by Cambridge University Press:  28 November 2013

Christoph Baumgarten
Christoph Baumgarten*
Affiliation:
Technische Universität Braunschweig, Institut für Mathematische Stochastik, Pockelsstrasse 14, 38106 Braunschweig, Germany. baumgart@math.tu-berlin.de
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Abstract

Given an autoregressive process X of order p(i.e.Xn = a1Xn−1 + ··· + apXn−p + Ynwhere the random variables Y1, Y2,... are i.i.d.), we study the asymptotic behaviour of the probability that the processdoes not exceed a constant barrier up to time N (survival or persistenceprobability). Depending on the coefficients a1,...,ap and the distribution ofY1, we state conditions under which the survival probabilitydecays polynomially, faster than polynomially or converges to a positive constant. Specialemphasis is put on AR(2) processes.

Keywords

Autoregressive processautoregressive moving averageboundary crossing probabilityone-sided exit problempersistence probablitysurvival probability

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 145 - 170
Copyright
© EDP Sciences, SMAI, 2013

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