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Extremes: Limit Results for Univariate and Multivariate Nonstationary Sequences

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Abstract

We review the limiting behaviour of extremes and exceedances of univariate and multivariate nonstationary random sequences. The approach we present is based on an extension of the methods in the stationary case. It is extended also for any visits of the random sequence to some rare set, instead of the usual set (u n , ∞). We discuss some special cases as normal, periodic and independent sequences and review also the dependence structure of the components of the multivariate maxima.

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References

  • Alpuim, M.T., Catkan, N. and Hüsler, J. (1992) Nonstationary max-autoregressive random sequences. Techn. Report. To be published.

    Google Scholar 

  • Balkema, A. and Resnick, S. (1977) Max-infinite divisibility. J. Appl. Probab. 14, 309–319.

    Article  MATH  MathSciNet  Google Scholar 

  • Ballerini, R. and McCormick, W.P. (1989) Extreme value theory for processes with periodic variances. Comm. in Statist., Stoch. Models 5, 45–61.

    Google Scholar 

  • Coles, S.G. and Tawn, J. (1994) Statistical methods for multivariate extremes: an application to structural design. Appl. Statist. 43.

    Google Scholar 

  • Daley, D.J. and Hall, P. (1984) Limit laws for the maximum of weighted and shifted iid. random variables. Ann. Probab. 12, 571–587.

    Article  MATH  MathSciNet  Google Scholar 

  • Davison A.C. and Smith, R.L. (1990) Models for exceedances over high thresholds. J. R. Statist. Soc. B, 52, 393–442.

    MATH  MathSciNet  Google Scholar 

  • Falk, M., Hüsler, J. and Reiss, R.D. (1994) Extreme Values and Rare Events, DMV-Seminar 1991, Birkhauser, Basel and Boston.

    Google Scholar 

  • Galambos, J. (1987) The Asymptotic Theory of Extreme Order Statistics. (2nd. ed.) Krieger, Florida.

    MATH  Google Scholar 

  • Hüsler, J. (1979) The limiting behaviour of the last exit time for sequences of independent, identically distributed random variables. Z. Wahrscheinlichkeitstheorie verw. Geb. 50, 159–164.

    Article  MATH  Google Scholar 

  • Hüsler, J. (1983) Asymptotic approximations of crossing probabilities of random sequences. Z. Wahrscheinlichkeitstheorie verw. Geb. 63, 257–270.

    Article  MATH  Google Scholar 

  • Hüsler, J. (1986a) Extreme values of noil-stationary random sequences. J. Appl. Probab. 23, 937–950.

    Article  MATH  MathSciNet  Google Scholar 

  • Hüsler, J. (1986b) Extreme values and rare events of non-stationary random sequences. In Dependence in Probability and Statistics, edited by Eberlein, E. and Taqqu, M.S., Birkhauser, Boston.

    Google Scholar 

  • Hüsler, J. (1989a) Limit properties for multivariate extremes in sequences of independent, non-identically distributed random vectors. Stoch. Proc. Appl. 31 (1989) 105–116.

    Article  MATH  Google Scholar 

  • Hüsler, J. (1989b) Limit distributions of multivariate extreme values in non- stationary sequences of random vectors. In Extreme Value Theory, Lecture Notes in Statistics 51, J. Hüsler and R.-D. Reiss, Eds., Springer, Berlin, 234–245.

    Chapter  Google Scholar 

  • Hüsler, J. (1993) A note on exceedances and rare events of non-stationary sequences. J. Appl. Probab. 30 Dezember.

    Google Scholar 

  • Hüsler, J. and Schiipbach, M. (1988) Limit results for maxima in non- stationary multivariate Gaussian sequences. Stoch. Proc. Appl. 28, 91–99.

    Article  MATH  Google Scholar 

  • Hüsler, J. and Reiss, R.D. (1989) Maxima of normal random vectors: between independence and total dependence. Statistics Probab. Letters 7, 283–286.

    Google Scholar 

  • Juncosa, M.L. (1949) The asymptotic behaviour of the minimum in a sequence of random variables. Duke Math. J. 16, 609–618.

    MATH  MathSciNet  Google Scholar 

  • Leadbetter, M.R. (1983) Extremes and local dependence in stationary sequences. Z. Wahrscheinlichkeitstheorie verw. Geb. 65, 291–306.

    Article  MATH  MathSciNet  Google Scholar 

  • Leadbetter, M.R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics, Springer, New York.

    Google Scholar 

  • Leadbetter, M.R. and Nandagopalan, S. (1989) On exceedance point processes for stationary sequences under mild oscillation restrictions. In Extreme Value Theory, Lecture Notes in Statistics 51, J. Hüsler and R.-D. Reiss, Eds., Springer, Berlin, p. 69–80.

    Google Scholar 

  • Leadbetter, M.R. and Rootzen, H. (1988) Extremal theory for stochastic processes. Ann. Probab. 16, 431–478.

    Article  MATH  MathSciNet  Google Scholar 

  • Meijzler, D.G. (1950) On the limit distribution of the maximal term of a variational series. Dopovidi Akad. Nauk Ukrain. SSR 1, 3–10.

    Google Scholar 

  • Nandagopalan, S. (1992) On the multivariate extremal index. Techn. Report.

    Google Scholar 

  • Nandagopalan, S., Leadbetter, M.R. and Hüsler, J. (1992) Limit theorems of multi-dimensional random measures. To be published.

    Google Scholar 

  • Pancheva E.I. (1985) Limit theorems for extreme order statistics under nonlinear normalization. In Stability Problems for Stochastic Models, Lecture Notes in Math. 1155, 248–309, Springer, Berlin.

    Google Scholar 

  • Resnick, S. (1987) Extreme Values, Regular Variation, and Point Processes, Springer, New York.

    MATH  Google Scholar 

  • Rootzen, H. (1986) Extreme value theory for moving average processes. Ann. Probab. 14, 612–652.

    Article  MATH  MathSciNet  Google Scholar 

  • Serfozo, R. (1980) High level exceedances of regenerative and semi-stationary processes. J. Appl. Probab. 17, 432–431.

    Article  MathSciNet  Google Scholar 

  • Serfozo, R. (1988) Extreme values of birth and death processes and queues. Stoch. Proc. Appl. 27, 291–306.

    Article  MATH  MathSciNet  Google Scholar 

  • Turkman, K.F. and Walker, A.M. (1983) Limit laws for the maxima of a class of quasi-stationary sequences. J. Appl. Probab. 20, 814–821.

    Article  MATH  MathSciNet  Google Scholar 

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© 1994 Kluwer Academic Publishers

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Hüsler, J. (1994). Extremes: Limit Results for Univariate and Multivariate Nonstationary Sequences. In: Galambos, J., Lechner, J., Simiu, E. (eds) Extreme Value Theory and Applications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3638-9_17

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  • DOI: https://doi.org/10.1007/978-1-4613-3638-9_17

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-3640-2

  • Online ISBN: 978-1-4613-3638-9

  • eBook Packages: Springer Book Archive

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