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:
-
(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.
-
(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 n∼Τ/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.
-
(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.
-
(iv)
To show that the assumptions used are satisfied, in particular by stationary normal sequences, under mild covariance conditions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Berman, S.M.: Limit theorems for the maximum term in stationary sequences. Ann. Math. Statist. 35, 502–516 (1964)
Berman, S.M.: Asymptotic independence of the numbers of high and low level crossings of stationary Gaussian Processes. Ann. Math. Statist. 42, 927–945 (1971)
Cramér, H.: On the intersections between the trajectories of a normal stationary stochastic process and a high level. Ark. Mat. 6, 337–349 (1966)
Deo, C.M.: A note on strong mixing Gaussian sequences. Ann. Prob. 1, 186–187 (1973)
Gnedenko, B.V.: Sur la distribution limite du terme maximum d'une série aléatoire. Ann. of Math. 44, 423–453 (1943)
Gnedenko, B.V., Kolmogorov, A.V.: Limit distributions for sums of independent random variables. New York: Addison Wesley 1954
Gumbel, E.J.: Statistics of externes. New York: Columbia Univ. Press 1958
Loynes, R.M.: Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36, 993–999 (1965)
Watson, G.S.: Extreme values in samples from m-dependent stationary stochastic processes. Ann. Math. Statist. 25, 798–800 (1954)
Welsch, R.E.: A weak convergence theorem for order statistics from strong-mixing processes. Ann. Math. Statist. 42, 1637–1646 (1971)
Author information
Authors and Affiliations
Additional information
Work done while visiting Cambridge University.
Research supported by Office of Naval Research, under Contract N00014-67-A-0321-0002.
Rights and permissions
About this article
Cite this article
Leadbetter, M.R. On extreme values in stationary sequences. Z. Wahrscheinlichkeitstheorie verw Gebiete 28, 289–303 (1974). https://doi.org/10.1007/BF00532947
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00532947