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ON THE EXTREMES OF STATIONARY GAUSSIAN RANDOM FIELDS UNDER STRONG DEPENDENCE
MARIA DA GRAC¸A TEMIDO
Dedicated to Professor Maria de Nazar´e Mendes Lopes
Abstract. In this work a stationary standard gaussian random field {Xn,m,(n,m) 2 N2}, with correlations rn,m satisfying rn,m ln(nm) ! 0, n, m ! +1, is considered. We prove that the limit in distribution of the sequence of point processes of exceedances, for some normalized level, is a Cox process. Then, the maximum of n ⇥ m variables of the random field converges in distribution to a convolution of the Gumbel with the gaussian distribution. Our results extend the ones presented by Choi [2], where = 0 is considered, as well as the results of Leadbetter, Lindgren, and Rootz´en [5], where the setup of real gaussian sequences is regarded.
Pour m’avoir aid´e Une amie De m’avoir enseign´e Merci
1. Introduction
Extreme value theory for random fields has been recently object of inten- sive research. Without being exhaustive, we mention the works of Choi [2], of Pereira and Ferreira [7], of Pereira [6], and references therein. In fact, a several results of the extreme values theory in the real line, that is for sequences of real random variables, stationary or not, have been extended when a random field is regarded. It should be pointed out that, traditionally, extreme values the- ory of gaussian processes has always received many esteem, being many times
Accepted: 14 February 2015.
2010 Mathematics Subject Classification. 60G70, 60G60, 60G55, 60F05.
Key words and phrases. Extremes, limit in distribution, point process, gaussian random
fields.
This work was partially supported by the Centre for Mathematics of the Univer-
sity of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Part- nership Agreement PT2020.
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