Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-23T00:29:20.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Extreme events of Markov chains

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  17 March 2017

I. Papastathopoulos ,
K. Strokorb ,
J. A. Tawn  and
A. Butler
I. Papastathopoulos*
Affiliation:
University of Edinburgh and the Alan Turing Institute
K. Strokorb*
Affiliation:
University of Mannheim
J. A. Tawn*
Affiliation:
Lancaster University
A. Butler*
Affiliation:
Biomathematics and Statistics Scotland
*
* Postal address: School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK. Email address: [email protected]
** Postal address: Institute of Mathematics, University of Mannheim, 68131 Mannheim, Germany. Email address: [email protected]
*** Postal address: Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK. Email address: [email protected]
**** Postal address: Biomathematics and Statistics Scotland, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The extremal behaviour of a Markov chain is typically characterised by its tail chain. For asymptotically dependent Markov chains, existing formulations fail to capture the full evolution of the extreme event when the chain moves out of the extreme tail region, and, for asymptotically independent chains, recent results fail to cover well-known asymptotically independent processes, such as Markov processes with a Gaussian copula between consecutive values. We use more sophisticated limiting mechanisms that cover a broader class of asymptotically independent processes than current methods, including an extension of the canonical Heffernan‒Tawn normalisation scheme, and reveal features which existing methods reduce to a degenerate form associated with nonextreme states.

Keywords

Asymptotic independenceconditional extremesextreme value theoryMarkov chainhidden tail chaintail chain

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60G10: Stationary processes
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 1 , March 2017 , pp. 134 - 161
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Basrak, B. and Segers, J. (2009).Regularly varying multivariate time series.Stoch. Process. Appl. 119,1055–1080.Google Scholar
Billingsley, P. (1999).Convergence of Probability Measures,2nd edn.John Wiley,New York.Google Scholar
Butler, A. (2005).Statistical modelling of synthetic oceanographic extremes.Doctoral Thesis, Lancaster University.Google Scholar
Coles, S. G.,Heffernan, J. E. and Tawn, J. A. (1999).Dependence measures for extreme value analyses.Extremes 2,339–365.Google Scholar
Davis, R. A. and Mikosch, T. (2009).Extreme value theory for GARCH processes.In Handbook of Financial Time Series,eds T. G. Andersen et al.,Springer,Berlin,pp.187–200.Google Scholar
De Haan, L.,Resnick, S. I.,Rootzén, H. and de Vries, C. G. (1989).Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes.Stoch. Process. Appl. 32,213–224.Google Scholar
Dieudonné, J. (1960).Foundations of Modern Analysis(Pure Appl. Math. X).Academic Press,New York.Google Scholar
Drees, H.,Segers, J. and Warchoł, M. (2015).Statistics for tail processes of Markov chains.Extremes 18,369–402.CrossRefGoogle Scholar
Eastoe, E. F.and Tawn, J. A. (2012).The distribution for the cluster maxima of exceedances of subasymptotic thresholds.Biometrika 99,43–55.CrossRefGoogle Scholar
Heffernan, J. E. (2000).A directory of coefficients of tail dependence.Extremes 3,279–290.Google Scholar
Heffernan, J. E. and Resnick, S. I. (2007).Limit laws for random vectors with an extreme component.Ann. Appl. Prob. 17,537–571.Google Scholar
Heffernan, J. E. and Tawn, J. A. (2004).A conditional approach for multivariate extreme values.J. R. Statist. Soc. B 66,497–546.Google Scholar
Hüsler, J. and Reiss, R.-D. (1989).Maxima of normal random vectors: between independence and complete dependence.Statist. Prob. Lett. 7,283–286.CrossRefGoogle Scholar
Janssen, A. and Segers, J. (2014).Markov tail chains.J. Appl. Prob. 51,1133–1153.Google Scholar
Joe, H. (2015).Dependence Modeling with Copulas(Monogr. Statist. Appl. Prob. 134).CRC Press,Boca Raton, FL.Google Scholar
Kulik, R. and Soulier, P. (2015).Heavy tailed time series with extremal independence.Extremes 18,273–299.Google Scholar
Leadbetter, M. R. (1983).Extremes and local dependence in stationary sequences.Z. Wahrscheinlichkeitsth. 65,291–306.Google Scholar
Ledford, A. W. and Tawn, J. A. (1997).Modelling dependence within joint tail regions.J. R. Statist. Soc. B 59,475–499.CrossRefGoogle Scholar
Ledford, A. W. and Tawn, J. A. (2003).Diagnostics for dependence within time series extremes.J. R. Statist. Soc. B 65,521–543.Google Scholar
Mikosch, T. (2003).Modeling dependence and tails of financial time series.In Extreme Values in Finance, Telecommunications, and the Environment,Chapman & Hall/CRC Press,Boca Raton, FL,pp.185–286.Google Scholar
Mikosch, T. and Starica, C. (2000).Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process.Ann. Statist. 28,1427–1451.Google Scholar
Nelsen, R. B. (2006).An Introduction to Copulas,2nd edn.Springer,New York.Google Scholar
O'Brien, G. L. (1987).Extreme values for stationary and Markov sequences.Ann. Prob. 15,281–291.Google Scholar
Papastathopoulos, I. and Tawn, J. A. (2016).Conditioned limit laws for inverted max-stable processes.J. Multivariate Anal. 150,214–228.Google Scholar
Perfekt, R. (1994).Extremal behaviour of stationary Markov chains with applications.Ann. Appl. Prob. 4,529–548.Google Scholar
Perfekt, R. (1997).Extreme value theory for a class of Markov chains with values in ℝ d .Adv. Appl. Prob. 29,138–164.Google Scholar
Reich, B. J. and Shaby, B. A. (2012).A hierarchical max-stable spatial model for extreme precipitation.Ann. Appl. Statist. 6,1430–1451.Google Scholar
Resnick, S. I. and Zeber, D. (2013).Asymptotics of Markov kernels and the tail chain.Adv. Appl. Prob. 45,186–213.Google Scholar
Resnick, S. I. and Zeber, D. (2014).Transition kernels and the conditional extreme value model.Extremes 17,263–287.CrossRefGoogle Scholar
Rootzén, H. (1988).Maxima and exceedances of stationary Markov chains.Adv. Appl. Prob. 20,371–390.Google Scholar
Segers, J. (2007).Multivariate regular variation of heavy-tailed Markov chains.Preprint. Available at https://arxiv.org/abs/math/0701411v1.Google Scholar
Smith, R. L. (1990).Max-stable processes and spatial extremes.Tech. Rep., University of North Carolina.Google Scholar
Smith, R. L. (1992).The extremal index for a Markov chain.J. Appl. Prob. 29,37–45.Google Scholar
Smith, R. L.,Tawn, J. A. and Coles, S. G. (1997).Markov chain models for threshold exceedances.Biometrika 84,249–268.Google Scholar
Tawn, J. A. (1988).Bivariate extreme value theory: models and estimation.Biometrika 75,397–415.CrossRefGoogle Scholar
Winter, H. C. and Tawn, J. A. (2016).Modelling heatwaves in central France: a case-study in extremal dependence.J. R. Statist. Soc. Ser. C 65,345–365.CrossRefGoogle Scholar
Yun, S. (1998).The extremal index of a higher-order stationary Markov chain.Ann. Appl. Prob. 8,408–437.Google Scholar