Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-20T03:08:17.079Z Has data issue: false hasContentIssue false
  • English
  • Français

Maxima and exceedances of stationary Markov chains

Published online by Cambridge University Press:  01 July 2016

Holger Rootzén
Holger Rootzén*
Affiliation:
University of Copenhagen
*
∗Present address: Department of Mathematical Statistics, University of Lund and Lund Institute of Technology, Box 118, S-221 00 Lund, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent work by Athreya and Ney and by Nummelin on the limit theory for Markov chains shows that the close connection with regeneration theory holds also for chains on a general state space. Here this is used to study extremal behaviour of stationary (or asymptotically stationary) Markov chains. Many of the results center on the ‘clustering’ of extremes of adjacent values of the chains. In addition one criterion for convergence of extremes of general stationary sequences is derived. The results are applied to waiting times in the GI/G/1 queue and to autoregressive processes.

Keywords

REGENERATIVE PROCESSESCLUSTERING OF EXTREME VALUESPOINT PROCESS OF EXCEEDANCESSTATIONARY SEQUENCESGI/G/1 QUEUEAUTOREGRESSIVE PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 2 , June 1988 , pp. 371 - 390
Copyright
Copyright © Applied Probability Trust 1988 

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] Anderson, C. W. (1970) Extreme value theory for a class of discrete distributions with applications to some stochastic processes. J. Appl. Prob. 7, 99–113.CrossRefGoogle Scholar
[2] Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
[3] Barnporff-Nielsen, O. (1964) On the limit distribution of the maximum of a random number of random variables. Acta Math. Acad. Sci. Hungar. 15, 399–403.Google Scholar
[4] Berman, S. M. (1962) Limiting distribution of the maximum term in sequences of dependent random variables. Ann. Math. Statist. 33, 894–908.CrossRefGoogle Scholar
[5] Berman, S. M. (1964) Limiting distribution of the maximum of a diffusion process. Ann. Math. Statist. 35, 319–329.Google Scholar
[6] Berman, S. M. (1964) Limit theorems for the maximum term in stationary sequences. Ann. Math. Statist. 35, 502–516.Google Scholar
[7] Berman, S. M. (1982) Sojurns and extremes of stationary processes. Ann. Prob. 10, 1–46.CrossRefGoogle Scholar
[8] Blake, I. F. and Lindsey, W. C. (1973) Level-crossing problems for random processes. IEEE Trans. Inf. Theory 19, 295–315.Google Scholar
[9] Darling, D. A. and Siegert, A. J. F. (1953) The first passage problem for continuous Markov processes. Ann. Math. Statist. 24, 624–639.Google Scholar
[10] Denzel, G. E. and O&Brien, G. L. (1975) Limit theorems for extreme values of chain-dependent processes. Ann. Prob. 3, 773–779.Google Scholar
[11] Hsing, T., Hüsler, J., and Leadbetter, M. R. (1986) Limits for exceedance point processes. Report #150, Center for Stochastic processes, University of North Carolina, Chapel Hill.Google Scholar
[12] Iglehart, D. L. (1972) Extreme values in the GI/G/1 queue. Ann. Math. Statist. 43, 627–635.CrossRefGoogle Scholar
[13] Kallenberg, O. (1976) Random Measures. Academic Press, London.Google Scholar
[14] Leadbetter, M. R. (1983) Extremes and local dependence in stationary sequences. Z. Wahrscheinlichkeitsth. 65, 291–306.Google Scholar
[15] Leadbetter, M. R., Lindgren, G., and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
[16] Lindvall, T. (1976) An invariance principle for thinned random measures. Stud. Scient. Hungar. 11, 269–275.Google Scholar
[17] Rootzen, H. (1976) Fluctuations of sequences which converge in distribution. Ann. Prob. 4, 456–463.Google Scholar
[18] Rootzen, H. (1978) Extremes of moving averages of stable processes. Ann. Prob. 6, 847–869.CrossRefGoogle Scholar
[19] Rootzen, H. (1986) Extreme value theory for moving average processes. Ann. Prob. 14, 612–652.Google Scholar
[20] Serfozo, R. (1980) High-level exceedances of regenerative and semi-stationary processes. J. Appl. Prob. 17, 423–431.Google Scholar
[21] Tweedie, R. L. (1983) Criteria for rates of convergence of Markov chains, with application to queueing and storage theory. In Probability, Statistics and Analysis, London Mathematical Lecture Notes 79:; Ed. Kingman, J. F. C., and Reuter, C. E. H., Cambridge University Press.Google Scholar