Skip to main content Accessibility help
×
Home

Asymptotics of Markov Kernels and the Tail Chain

  • Sidney I. Resnick (a1) and David Zeber (a1)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Asymptotics of Markov Kernels and the Tail Chain
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Asymptotics of Markov Kernels and the Tail Chain
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Asymptotics of Markov Kernels and the Tail Chain
      Available formats
      ×

Copyright

Corresponding author

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

References

Hide All
[1] Basrak, B. and Segers, J. (2009). Regularly varying multivariate time series. Stoch. Process. Appl. 119, 10551080.
[2] Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.
[3] Billingsley, P. (1971). Weak Convergence of Measures: Applications in Probability. Society for Industrial and Applied Mathematics, Philadelphia, PA.
[4] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
[5] Bortot, P. and Coles, S. (2000). A sufficiency property arising from the characterization of extremes of Markov chains. Bernoulli 6, 183190.
[6] Bortot, P. and Coles, S. (2003). Extremes of Markov chains with tail switching potential. J. R. Statist. Soc. B 65, 851867.
[7] Bortot, P. and Tawn, J. A. (1998). Models for the extremes of Markov chains. Biometrika 85, 851867.
[8] Das, B. and Resnick, S. I. (2011). Conditioning on an extreme component: model consistency with regular variation on cones. Bernoulli 17, 226252.
[9] Das, B. and Resnick, S. I. (2011). Detecting a conditional extreme value model. Extremes 14, 2961.
[10] Davis, R. A. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Prob. 23, 879917.
[11] Feigin, P. D., Kratz, M. F. and Resnick, S. I. (1996). Parameter estimation for moving averages with positive innovations. Ann. Appl. Prob. 6, 11571190.
[12] Heffernan, J. E. and Resnick, S. I. (2007). Limit laws for random vectors with an extreme component. Ann. Appl. Prob. 17, 537571.
[13] Heffernan, J. E. and Tawn, J. A. (2004). A conditional approach for multivariate extreme values. J. R. Statist. Soc. B 66, 497546.
[14] Hsing, T. (1989). Extreme value theory for multivariate stationary sequences. J. Multivariate Anal. 29, 274291.
[15] Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
[16] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
[17] Perfekt, R. (1994). Extremal behaviour of stationary Markov chains with applications. Ann. Appl. Prob. 4, 529548.
[18] Perfekt, R. (1997). Extreme value theory for a class of Markov chains with values in R d . Adv. Appl. Prob. 29, 138164.
[19] Resnick, S. I. (2007). Extreme Values, Regular Variation and Point Processes. Springer, New York.
[20] Resnick, S. I. (2007). Heavy-Tail Phenomena. Springer, New York.
[21] Rootzén, H. (1988). Maxima and exceedances of stationary Markov chains. Adv. Appl. Prob. 20, 371390.
[22] Segers, J. (2007). Multivariate regular variation of heavy-tailed Markov chains. Preprint. Available at http://arxiv.org/abs/math/0701411v1.
[23] Smith, R. L. (1992). The extremal index for a Markov chain. J. Appl. Prob. 29, 3745.
[24] Smith, R. L., Tawn, J. A. and Coles, S. G. (1997). Markov chain models for threshold exceedances. Biometrika 84, 249268.
[25] Yun, S. (1998). The extremal index of a higher-order stationary Markov chain. Ann. Appl. Prob. 8, 408437.
[26] Yun, S. (2000). The distributions of cluster functionals of extreme events in a dth-order Markov chain. J. Appl. Prob. 37, 2944.

Keywords

MSC classification

Related content

Powered by UNSILO

Asymptotics of Markov Kernels and the Tail Chain

  • Sidney I. Resnick (a1) and David Zeber (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.