Advances in Applied Probability

Extremes of autoregressive threshold processes

Claudia Brachner, Vicky Fasen, and Alexander Lindner

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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.

Article information

Adv. in Appl. Probab. Volume 41, Number 2 (2009), 428-451.

First available in Project Euclid: 6 July 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G10: Stationary processes 60G55: Point processes

Ergodic exponential tail extreme value theory O-regular variation point process regular variation SETAR process subexponential distribution tail behavior TAR process


Brachner, Claudia; Fasen, Vicky; Lindner, Alexander. Extremes of autoregressive threshold processes. Adv. in Appl. Probab. 41 (2009), no. 2, 428--451. doi:10.1239/aap/1246886618.

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