Electronic Resource
Springer
Probability theory and related fields
28 (1974), S. 289-303
ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 n (r) 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 n (r) (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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00532947
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