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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: [email protected]
∗∗∗ Postal address: Institute for Mathematical Stochastics, Technische Universität Braunschweig, Pockelsstrasse 14, D-38106 Braunschweig, Germany. Email address: [email protected]
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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.

References

An, H. Z. and Huang, F. C. (1996). The geometric ergodicity of nonlinear autoregressive models. Statistica Sinica 6, 943–956.Google Scholar
Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115.Google Scholar
Brachner, C. (2004). Tailverhalten autoregressiver Thresholdmodelle. , Technische Universität München.Google Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sine law. Theory Prob. Appl. 10, 351–360.Google Scholar
Chan, K. and Tong, H. (1985). On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations. Adv. Appl. Prob. 17, 666–678.Google Scholar
Chen, R. and Tsay, R. (1991). On the ergodicity of TAR(1) processes. Ann. Appl. Prob. 1, 613–634.CrossRefGoogle Scholar
Davis, R. A. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Prob. 23, 879–917.Google Scholar
Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.Google Scholar
Davis, R. and Resnick, S. (1988). Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution. Stoch. Process. Appl. 30, 41–68.Google Scholar
Diop, A. and Guegan, D. (2004). Tail behavior of a threshold autoregressive stochastic volatility model. Extremes 7, 367–375.CrossRefGoogle Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google Scholar
Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York.Google Scholar
Fasen, V. (2005). Extremes of regularly varying Lévy-driven mixed moving average processes. Adv. Appl. Prob. 37, 993–1014.Google Scholar
Fasen, V. (2006). Extremes of subexponential Lévy driven moving average processes. Stoch. Process. Appl. 116, 1066–1087.Google Scholar
Fasen, V., Klüppelberg, C. and Lindner, A. (2006). Extremal behavior of stochastic volatility models. In Stochastic Finance, eds Shiryaev, A. et al., Springer, New York, pp. 107–155.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Hsing, T. (1993). On some estimates based on sample behavior near high level excursions. Prob. Theory Relat. Fields 95, 331–356.Google Scholar
Klüppelberg, C. and Lindner, A. (2005). Extreme value theory for moving average processes with light-tailed innovations. Bernoulli 11, 381–410.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.Google Scholar
Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41, 407–424. (Correction: 44 (2007), 295–305.)CrossRefGoogle Scholar
Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. Appl. Prob. 18, 66–138.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
Resnick, S. I. (2007). Heavy-Tail Phenomena. Springer, New York.Google Scholar
Rootzén, H. (1986). Extreme value theory for moving average processes. Ann. Prob. 14, 612–652.Google Scholar
Rootzén, H. (1987). A ratio limit theorem for the tails of weighted sums. Ann. Prob. 15, 728–747.Google Scholar
Tong, H. (1977). Contribution to the discussion of the paper entitled ‘Stochastic modelling of riverflow time series’ by A. J. Lawrance and N. T. Kottegoda. J. R. Statist. Soc. A 140, 34–35.Google Scholar
Tong, H. (1990). Nonlinear Time Series. Oxford University Press.Google Scholar
Tong, H. and Lim, K. S. (1980). Threshold autoregression, limit cycles and cyclical data. J. R. Statist. Soc. B 42, 245–292.Google Scholar