Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-18T01:14:27.787Z Has data issue: false hasContentIssue false
  • English
  • Français

The extremal index and clustering of high values for derived stationary sequences

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Ishay Weissman  and
Uri Cohen
Ishay Weissman*
Affiliation:
Technion — Israel Institute of Technology
Uri Cohen*
Affiliation:
Technion — Israel Institute of Technology
*
∗Postal address: Faculty of Industrial Engineering and Management, Technion — Israel Institute of Technology, Technion City, Haifa 32000, Israel.
∗Postal address: Faculty of Industrial Engineering and Management, Technion — Israel Institute of Technology, Technion City, Haifa 32000, Israel.
Get access Rights & Permissions [Opens in a new window]

Abstract

Given a sequence of independent identically distributed random variables, we derive a moving-maximum sequence (with random translations). The extremal index of the derived sequence is computed and the limiting behaviour of clusters of high values is studied. We are then given two or more independent stationary sequences whose extremal indices are known. We derive a new stationary sequence by taking either a pointwise maximum or by a mixture of the original sequences. In each case, we compute the extremal index of the derived sequence.

Keywords

MIXTURESMOVING MAXIMUMPOINTWISE MAXIMUMSTRONG-MIXING

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60G55: Point processes 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 4 , December 1995 , pp. 972 - 981
Copyright
Copyright © Applied Probability Trust 1995 

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

Alpuim, M.T. (1989) An extremal Markovian sequence. J. Appl. Prob. 26, 219–232.Google Scholar
Davis, R. 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
Hsing, T. (1988) On the extreme order statistics for a stationary sequence. Stoch. Proc. Appl. 29, 155–169.Google Scholar
Hsing, T. (1990) Estimating the extremal index under m-dependence and tail balancing. Preprint.Google Scholar
Hsing, T. (1991) Estimating the parameters of rare events. Stoch. Proc. Appl. 37, 117–139.Google Scholar
Hsing, T. (1993) Extremal index estimation for a weakly dependent stationary sequence. Ann. Statist. 21, 2043–2071.Google Scholar
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
Joe, H. (1991) Estimation of extremal index function and the distribution of the maximum of a stationary. dependent sequence. Preprint.Google Scholar
Leadbetter, M.R. (1983) Extremes and local dependence in stationary sequences. Z. Wahrscheinlichkeitsth. 65, 291–306.CrossRefGoogle Scholar
Leadbetter, M.R. and Rootzén, H. (1988) Extremal theory for stochastic processes. Ann. Prob. 16, 431–478.Google Scholar
Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
Leadbetter, M.R., Weissman, I., De Haan, L. and Rootzen, H. (1989) On clustering of high values in stationary series. International Meeting on Statistical Climatology, Rotorua, New Zealand.Google Scholar
Loynes, R.M. (1965) Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36, 993–999.Google Scholar
Nandagopalan, S. (1994) On the multivariate extremal index. J. Res. Natn. Inst. Stand. Technol. 99, 543–550.Google Scholar
Newell, G.F. (1964) Asymptotic extremes for m-dependent random variables. Ann. Math. Statist. 35, 1322–1325.Google Scholar
O'Brien, G.L. (1974) The maximum term of uniformly mixing stationary processes. Z. Wahrscheinlichkeitsth. 30, 57–63.Google Scholar
O'Brien, G.L. (1987) Extreme value analysis for stationary and Markov sequences. Ann. Prob. 15, 281–291.CrossRefGoogle Scholar
Rootzen, H. (1988) Maxima and exceedances of stationary Markov chain. Adv. Appl. Prob. 20, 371–390.Google Scholar
Smith, R.L. The extremal index for a Markov chain. J. Appl. Prob. 29, 37–45.CrossRefGoogle Scholar
Smith, R.L. and Weissman, I. (1994) Estimating the extremal index. J. R. Statist. Soc. B. 56, 515–528.Google Scholar