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Extreme value theory for multivariate stationary sequences. (English) Zbl 0679.62039
The author considers multivariate extremal type theorems and some related problems in a setting where the assumptions of independence and linear normalization in the classical setting are weakened. For a stationary sequence of random vectors he introduces a distributional mixing condition which is an obvious extension of M. R. Leadbetter’s condition \(D(u_ n)\) in the one-dimensional case [Z. Wahrscheinlichkeitstheorie verw. Gebiete 28, 289-303 (1974; Zbl 0265.60019)]. For a sequence satisfying this condition the following topics are studied.
(1) To obtain characterizations of the weak limit F of properly normalized partial maxima. (2) To study a condition under which the partial maxima behave as they would if the sequence were i.i.d. (3) To consider problems in connection with the independence of the margins of F.
Reviewer: T.Mori

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60F05 Central limit and other weak theorems
60F99 Limit theorems in probability theory
60G10 Stationary stochastic processes
62G30 Order statistics; empirical distribution functions
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