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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 9, Fasc. 2,
pages 51 - 66
 

EXTREME VALUE THEORY FOR ASYMPTOTIC STATIONARY SEQUENCES

Władysław Szczotka

Abstract: The problem of behaviour of a (max X - b ) n 1<k<n k n is considered when a > 0, n |b |< oo n and the sequence X = (X ,k > 1) k is asymptotically stationary in variation.

X is said to be asymptotically stationary in variation if ||L(X )- L(X0)||--> 0, n where X = (X ,k > 1), n n+k while L(X ) n and L(X0) denote the distributions of the sequences X n and X0 = (X0,k > 1), k respectively. The sequence X0 of random variables X0 k is stationary and it is said to be a stationary representation of X.

The main result states: under ||L(X )- L(X0) ||--> 0 n and some natural conditions concerned X and X0, the sequence of distributions L(a (max X - b )) n 1<k<n k n weakly converges provided the sequence of L(a (max X0 - b )) n 1<k<n k n weakly converges and the limits are the same. An analogous result is also formulated for the processes of exceedances.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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