On extreme values in stationary sequences

Summary

In this paper, extreme value theory is considered for stationary sequences ζ n 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 M n = max(ξ 1...ξ n), 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 M (r)n is the r-th largest of ξ 1...ξ n, and Prξ 1>u nΤ/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 M (r)n (normalized) is the same as if the {ξ n} 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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Work done while visiting Cambridge University.

Research supported by Office of Naval Research, under Contract N00014-67-A-0321-0002.

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Leadbetter, M.R. On extreme values in stationary sequences. Z. Wahrscheinlichkeitstheorie verw Gebiete 28, 289–303 (1974). https://doi.org/10.1007/BF00532947

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Keywords

  • Covariance
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Normal Process