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Asymptotics of Markov Kernels and the Tail Chain

Part of: Stochastic processes Markov processes Applications

Published online by Cambridge University Press:  04 January 2016

Sidney I. Resnick  and
David Zeber
Sidney I. Resnick*
Affiliation:
Cornell University
David Zeber*
Affiliation:
Cornell University
*
∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, 284 Rhodes Hall, Ithaca, NY 14853, USA. Email address: sir1@cornell.edu
∗∗ Postal address: Department of Statistical Science, Cornell University, 301 Malott Hall, Ithaca, NY 14853, USA. Email address: dsz5@cornell.edu
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Abstract

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An asymptotic model for the extreme behavior of certain Markov chains is the ‘tail chain’. Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics, such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and nonextreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular variation property of the finite-dimensional distributions under assumptions on the marginal tails alone.

Keywords

Extreme valuesMarkov chainmultivariate regular variationtransition kerneltail chainheavy tail

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 62P05: Applications to actuarial sciences and financial mathematics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 1 , March 2013 , pp. 186 - 213
Copyright
© Applied Probability Trust 

References

Basrak, B. and Segers, J. (2009). Regularly varying multivariate time series. Stoch. Process. Appl. 119, 1055–1080.CrossRefGoogle Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Billingsley, P. (1971). Weak Convergence of Measures: Applications in Probability. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.Google Scholar
Bortot, P. and Coles, S. (2000). A sufficiency property arising from the characterization of extremes of Markov chains. Bernoulli 6, 183–190.CrossRefGoogle Scholar
Bortot, P. and Coles, S. (2003). Extremes of Markov chains with tail switching potential. J. R. Statist. Soc. B 65, 851–867.Google Scholar
Bortot, P. and Tawn, J. A. (1998). Models for the extremes of Markov chains. Biometrika 85, 851–867.Google Scholar
Das, B. and Resnick, S. I. (2011). Conditioning on an extreme component: model consistency with regular variation on cones. Bernoulli 17, 226–252.CrossRefGoogle Scholar
Das, B. and Resnick, S. I. (2011). Detecting a conditional extreme value model. Extremes 14, 29–61.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
Feigin, P. D., Kratz, M. F. and Resnick, S. I. (1996). Parameter estimation for moving averages with positive innovations. Ann. Appl. Prob. 6, 1157–1190.CrossRefGoogle 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.CrossRefGoogle Scholar
Hsing, T. (1989). Extreme value theory for multivariate stationary sequences. J. Multivariate Anal. 29, 274–291.Google Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.CrossRefGoogle 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
Resnick, S. I. (2007). 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. (1988). Maxima and exceedances of stationary Markov chains. Adv. Appl. Prob. 20, 371–390.CrossRefGoogle Scholar
Segers, J. (2007). Multivariate regular variation of heavy-tailed Markov chains. Preprint. Available at http://arxiv.org/abs/math/0701411v1.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
Yun, S. (1998). The extremal index of a higher-order stationary Markov chain. Ann. Appl. Prob. 8, 408–437.CrossRefGoogle Scholar
Yun, S. (2000). The distributions of cluster functionals of extreme events in a dth-order Markov chain. J. Appl. Prob. 37, 29–44.Google Scholar