Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-24T18:30:22.426Z Has data issue: false hasContentIssue false
  • English
  • Français

Markov Tail Chains

Part of: Stochastic analysis Stochastic processes Applications Markov processes

Published online by Cambridge University Press:  30 January 2018

A. Janssen  and
J. Segers
A. Janssen*
Affiliation:
University of Hamburg
J. Segers*
Affiliation:
Université Catholique de Louvain
*
∗ Postal address: Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany. Email address: anja.janssen@math.uni-hamburg.de
∗∗ Postal address: Institut de Statistique, Université Catholique de Louvain, Voie du Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium. Email address: johan.segers@uclouvain.be
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The extremes of a univariate Markov chain with regularly varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper we extend this fact to Markov chains with multivariate regularly varying marginal distributions in Rd. We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, we will show that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is also Markovian. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.

Keywords

Autoregressive conditional heteroskedasticityextreme value distribution(multivariate) Markov chainmultivariate regular variationrandom walkstochastic difference equationtail chaintail-switching potential

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60G10: Stationary processes 60H25: Random operators and equations 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 1133 - 1153
Copyright
© Applied Probability Trust 

References

Basrak, B. and Segers, J. (2009). Regularly varying multivariate time series. Stoch. Process. Appl. 119, 1055–1080. (Erratum: 121 (2011), 896–898.)CrossRefGoogle Scholar
Basrak, B., Davis, R. A. and Mikosch, T. (2002a). A characterization of multivariate regular variation. Ann. Appl. Prob. 12, 908–920.CrossRefGoogle Scholar
Basrak, B., Davis, R. A. and Mikosch, T. (2002b). Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Boman, J. and Lindskog, F. (2009). Support theorems for the Radon transform and Cramér–Wold theorems. J. Theoret. Prob. 22, 683–710.CrossRefGoogle 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.CrossRefGoogle Scholar
Buraczewski, D., Damek, E. and Mirek, M. (2012). Asymptotics of stationary solutions of multivariate stochastic recursions with heavy tailed inputs and related limit theorems. Stoch. Process. Appl. 122, 42–67.Google Scholar
Buraczewski, D. et al. (2009). Tail-homogeneity of stationary measures for some multidimensional stochastic recursions. Prob. Theory Relat. Fields 145, 385–420.Google Scholar
Coles, S. G., Tawn, J. A. and Smith, R. L. (1994). A seasonal Markov model for extremely low temperatures. Environmetrics 5, 221–239.Google Scholar
Collamore, J. F. and Vidyashankar, A. N. (2013). Tail estimates for stochastic fixed point equations via nonlinear renewal theory. Stoch. Process. Appl. 123, 3378–3429.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
De Saporta, B., Guivarc'h, Y. and Le Page, E. (2004). On the multidimensional stochastic equation Y{n+1} = AnYn+Bn . C. R. Math Acad. Sci. Paris 339, 499–502.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.CrossRefGoogle Scholar
Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.CrossRefGoogle Scholar
Gomes, I. M., De Haan, L. and Pestana, D. (2004). Joint exceedances of the ARCH process. J. Appl. Prob. 41, 919–926. (Correction: 41 (2004), 919–926.)Google Scholar
Hult, H. and Samorodnitsky, G. (2008). Tail probabilities for infinite series of regularly varying random vectors. Bernoulli 14, 838–864.CrossRefGoogle Scholar
Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.CrossRefGoogle Scholar
Kifer, Y. (1986). Ergodic Theory of Random Transformations. Birkhäuser, Boston, MA.Google Scholar
Klüppelberg, C. and Pergamenchtchikov, S. (2003). Renewal theory for functionals of a Markov chain with compact state space. Ann. Prob. 31, 2270–2300.CrossRefGoogle Scholar
Klüppelberg, C. and Pergamenchtchikov, S. (2004). The tail of the stationary distribution of a random coefficient AR(q) model. Ann. Appl. Prob. 14, 971–1005.Google Scholar
Letac, G. (1986). A contraction principle for certain Markov chains and its applications. In Random Matrices and Their Applications (Brunswick, Maine, 1984; Contemp. Math. 50), American Mathematical Society, Providence, RI, pp. 263–273.Google Scholar
Meinguet, T. and Segers, J. (2010). Regularly varying time series in Banach spaces. Preprint. Available at http://arxiv.org/abs/1001.3262.Google Scholar
Mirek, M. (2011). Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps. Prob. Theory Relat. Fields 151, 705–734.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 {\BBR}^d. Adv. Appl. Prob. 29, 138–164.Google Scholar
Resnick, S. I. (2007). Heavy-Tail Phenomena. Probabilistic and Statistical Modeling. Springer, New York.Google Scholar
Resnick, S. I. and Zeber, D. (2013). Asymptotics of Markov kernels and the tail chain. Adv. Appl. Prob. 45, 186–213.CrossRefGoogle Scholar
Segers, J. (2007). Multivariate regular variation of heavy-tailed Markov chains. Discussion Paper 0703, Institut de statistique, Université catholique de Louvain. Available at http://uk.arxiv.org/abs/math/0701411.Google Scholar
Smith, R. L. (1992). The extremal index for a Markov chain. J. Appl. Prob. 29, 37–45.CrossRefGoogle Scholar
Smith, R. L., Tawn, J. A. and Coles, S. G. (1997). Markov chain models for threshold exceedances. Biometrika 84, 249–268.CrossRefGoogle Scholar
Van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press.Google Scholar
Yun, S. (1998). The extremal index of a higher-order stationary Markov chain. Ann. Appl. Prob. 8, 408–437.Google 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
Zivot, E. (2009). Practical issues in the analysis of univariate GARCH models. In Handbook of Financial Time Series, Springer, Berlin, pp. 113–155.Google Scholar