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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 38, Fasc. 10,
pages 191 - 207
DOI: 10.19195/0208-4147.38.1.10
 

EXTREMES OF MULTIDIMENSIONAL STATIONARY GAUSSIAN RANDOM FIELDS

Natalia Soja-KukieŁa

Abstract: Let (X (t) : t = (t,t ,...,t) ∈ [0,∞ )d) 1 2 d be a centered stationary Gaussian field with almost surely continuous sample paths, unit variance and correlation function r satisfying r(t) < 1 for every t ⁄= 0 and ∑d αi ∑d αi r(t) = 1- i=1 |ti| + o( i=1|ti| ) , as t → 0 , with some α1,α2,...,αd ∈ (0,2] . The main result of this contribution is the description of the asymptotic behaviour of x P( sup(X (t) : t ∈ Jm ) ≤ u ) , as u → ∞ , for some Jordan-measurable sets x J m of volume proportional to d -1 P(sup(X (t) : t ∈ [0,1] ) > u) (1+ o(1)) .

2010 AMS Mathematics Subject Classification: Primary: 60G15; Secondary: 60G70, 60G60.

Keywords and phrases: Gaussian random field, supremum, limit theorem, asymptotics, Berman condition, strong dependence.

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