On extreme values in stationary sequences

  • M. R. Leadbetter
Article

Summary

In this paper, extreme value theory is considered for stationary sequences ζn satisfying dependence restrictions significantly weaker than strong mixing. The aims of the paper are:
  1. (i)

    To prove the basic theorem of Gnedenko concerning the existence of three possible non-degenerate asymptotic forms for the distribution of the maximum Mn = max(ξ1...ξn), for such sequences.

     
  2. (ii)
    To obtain limiting laws of the form
    $$\mathop {\lim }\limits_{n \to \infty } \Pr \{ M_n^{(r)} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } u\} = e^{ - \tau } \sum\limits_{s = 0}^{r - 1} {\tau ^s /S!} $$
    where Mn(r)is the r-th largest of ξ1...ξn, and Prξ1>unΤ/n. Poisson properties (akin to those known for the upcrossings of a high level by a stationary normal process) are developed and used to obtain these results.
     
  3. (iii)

    As a consequence of (ii), to show that the asymptotic distribution of Mn(r)(normalized) is the same as if the {ξn} were i.i.d.

     
  4. (iv)

    To show that the assumptions used are satisfied, in particular by stationary normal sequences, under mild covariance conditions.

     

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References

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Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • M. R. Leadbetter
    • 1
  1. 1.University of North CarolinaUSA

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