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Extremes of autoregressive threshold processes

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Claudia Brachner ,
Vicky Fasen  and
Alexander Lindner
Claudia Brachner*
Affiliation:
Allianz Investment Management SE
Vicky Fasen*
Affiliation:
Technische Universität München
Alexander Lindner*
Affiliation:
Technische Universität Braunschweig
*
∗ Postal address: Allianz Investment Management SE, Königinstrasse 28, 80802 München, Germany.
∗∗ Postal address: Center for Mathematical Sciences, Technische Universität München, Boltzmannstrasse 3, D-85747 Garching, Germany. Email address: fasen@ma.tum.de
∗∗∗ Postal address: Institute for Mathematical Stochastics, Technische Universität Braunschweig, Pockelsstrasse 14, D-38106 Braunschweig, Germany. Email address: a.lindner@tu-bs.de
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Abstract

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In this paper we study the tail and the extremal behaviors of stationary solutions of threshold autoregressive (TAR) models. It is shown that a regularly varying noise sequence leads in general to only an O-regularly varying tail of the stationary solution. Under further conditions on the partition, it is shown however that TAR(S,1) models of order 1 with S regimes have regularly varying tails, provided that the noise sequence is regularly varying. In these cases, the finite-dimensional distribution of the stationary solution is even multivariate regularly varying and its extremal behavior is studied via point process convergence. In particular, a TAR model with regularly varying noise can exhibit extremal clusters. This is in contrast to TAR models with noise in the maximum domain of attraction of the Gumbel distribution and which is either subexponential or in ℒ(γ) with γ > 0. In this case it turns out that the tail of the stationary solution behaves like a constant times that of the noise sequence, regardless of the order and the specific partition of the TAR model, and that the process cannot exhibit clusters on high levels.

Keywords

Ergodicexponential tailextreme value theoryO-regular variationpoint processregular variationSETAR processsubexponential distributiontail behaviorTAR process

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G10: Stationary processes 60G55: Point processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 428 - 451
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Financial support from the Deutsche Forschungsgemeinschaft through a research grant is gratefully acknowledged.

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