Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-24T16:43:50.902Z Has data issue: false hasContentIssue false
  • English
  • Français

Extreme Value Theory for a Class of Markov Chains with Values in ℝd

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Roland Perfekt
Roland Perfekt*
Affiliation:
University of Lund
*
∗ Postal address: Department of Mathematical Statistics, University of Lund, Box 118, S-22100, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider extreme value theory for a class of stationary Markov chains with values in ℝd. The asymptotic distribution of Mn, the vector of componentwise maxima, is determined under mild dependence restrictions and suitable assumptions on the marginal distribution and the transition probabilities of the chain. This is achieved through computation of a multivariate extremal index of the sequence, extending results of Smith [26] and Perfekt [21] to a multivariate setting. As a by-product, we obtain results on extremes of higher-order, real-valued Markov chains. The results are applied to a frequently studied random difference equation.

Keywords

MULTIVARIATE EXTREME VALUESMULTIVARIATE EXTREMAL INDEXSTATIONARY MARKOV CHAIN

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60G10: Stationary processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 138 - 164
Copyright
Copyright © Applied Probability Trust 1997 

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

[1] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[2] Bougerol, P. and Picard, N. (1992) Strict stationarity of generalized autogressive processes. Ann. Prob. 20, 1714–1730.CrossRefGoogle Scholar
[3] Chernick, M. R., Hsing, T. and Mccormick, W. P. (1991) Calculating the extremal index for a class of stationary sequences. Adv. Appl. Prob. 23, 835–850.Google Scholar
[4] Davis, R. A. (1979) Maxima and minima of stationary sequences. Ann. Prob. 7, 453–460.Google Scholar
[5] Davis, R. A. (1982) Limit laws for the maximum and minimum of stationary sequences. Z. Wahrscheinlichkeitsth. 61, 31–42.CrossRefGoogle Scholar
[6] Davis, R. A., Marengo, J. and Resnick, S. I. (1985) Extremal properties of a class of multivariate moving averages. In Proc. 45th Session of the ISI (Amsterdam) L1–4, 1–14.Google Scholar
[7] Davis, R. A. and Resnick, S. I. (1985) Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.Google Scholar
[8] Galambos, J. (1987) The Asymptotic Theory of Extreme Order Statistics. Krieger, Malabar, FL.Google Scholar
[9] Goldie, C. M. (1991) Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.Google Scholar
[10] Grincevicius, A. (1981) A random difference equation. Lithuanian Math. J. 21, 302–306.Google Scholar
[11] Haan, L. De and Resnick, S. I. (1977) Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitsth. 40, 317–337.Google Scholar
[12] Haan, L. De, Resnick, S. I., Rootzen, H. and Vries, C. G. De (1989) Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. Stoch. Proc. Appl. 32, 213–224.Google Scholar
[13] Hsing, T. (1989) Extreme value theory for multivariate stationary sequences. J. Multivar. Anal. 29, 274–291.Google Scholar
[14] Hsing, T. (1984) Point processes associated with extreme value theory. Technical Report 83. University of North Carolina.Google Scholar
[15] Hsing, T., Hüsler, J. and Leadbetter, M. R. (1988) On the exceedance point process for a stationary sequence. Prob. Theory Rel. Fields 78, 97–112.Google Scholar
[16] Hüsler, J. (1990) Multivariate extreme values in stationary random sequences. Stoch. Proc. Appl. 35, 99–108.Google Scholar
[17] Kesten, H. (1973) Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.Google Scholar
[18] Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer, New York.CrossRefGoogle Scholar
[19] Nandagopalan, S. (1993) On the multivariate extremal index. Technical Report 391. University of North Carolina.Google Scholar
[20] O'Brien, G. L. (1987) Extreme values for stationary and Markov sequences. Ann. Prob. 15, 281–291.Google Scholar
[21] Perfekt, R. (1994) Extremal behaviour of stationary Markov chains with applications. Ann. Appl. Prob. 4, 529–548.Google Scholar
[22] Resnick, S. I. (1987) Extreme Values, Regular Variation and Point Processes. Springer, New York.Google Scholar
[23] Resnick, S. I. and Willekens, E. (1991) Moving averages with random coefficients and random coefficient autoregressive models. Commun. Statist.-Stoch. Models 7, 511–525.CrossRefGoogle Scholar
[24] Rootzen, H. (1978) Extremes of moving averages of stable processes. Ann. Prob. 6, 847–869.Google Scholar
[25] Rootzen, H. (1988) Maxima and exceedances of stationary Markov chains. Adv. Appl. Prob. 20, 371–390.Google Scholar
[26] Smith, R. L. (1992) The extremal index for a Markov chain. J. Appl. Prob. 29, 37–45.Google Scholar
[27] Vervaat, W. (1979) On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.CrossRefGoogle Scholar