Page 134 - Textos de Matemática Vol. 47

124 M. G. TEMIDO
the pioneers when some new theory appears, due both to its mathematical tractability either by its numerous applications.
Let {Xn,m} be a stationary standard gaussian random field on N2, with correlations ri,j = E(X`,mX`+i,m+j), such that
ri,j = r|i|,|j| (1.1)
for each i and j in Z. We note that this condition is satisfied if the random field is isotropic, that is, if ri,j, for (i,j) 2 Z2, is a function only of the euclidean norm k(i, j)k (cf. Adler [1]).
Let {Nn,m} be the sequence, doubly indexed, of the point process of ex- ceedances of a real high level un,m by the random variables X1,1, . . . , Xn,m, defined by
Nn,m(B)=]⇢(i,j)2N20 :✓i,j◆2BandXi,j>un,m , 8B✓]0,1]2. (1.2) nm
In this paper we study the convergence in distribution of the sequence of point processes of exceedances, {Nn,m}, when the correlations of the random field {Xn,m} satisfy a strong dependence condition, specified by
rn,m ln(nm) ! 0, n,m ! +1. (1.3)
The case = 0 is studied in [2] for stationary gaussian random fields and in [6] for non stationary ones. In fact, considering that {Xn,m} is a non stationary gaussian random field, in [6] is studied its extremal behaviour, assuming that the correlations r(i1 ,i2 ),(j1 ,j2 ) satisfy |r(i1 ,i2 ),(j1 ,j2 ) | < ⇢|i1 j1 |,|i2 j2 | , for some sequence {⇢n,m} such that ⇢n,m ln(nm) ! 0, n,m ! +1, ⇢n,0 lnn ! 0, n ! +1and⇢0,mlnm!0, m!+1.
We end this section recalling the underlying main results for gaussian se- quences of real random variables.
Throughout this work and denote, respectively, the standard gaussian distribution function and its density.
Consider that {Xn} is a stationary standard gaussian sequence, with cor- relations sequence {rn } satisfying rn log n ! 0, n ! +1. The sequence of point process of exceedances of the high level un by X1, X2, . . . , Xn is given by Nn(B)=#{j 1:j/n2B,Xj >un},foreachBorelsubsetBof]0,1].Itis well known that (cf. Leadbetter, Lindgren, and Rootz´en [5]), if un is such that n(1 (un)) ! e x, n ! +1, x 2 R, the sequence {Nn} converges in distribu- tion to a Cox process with stochastic intensity exp( x +p2 Y ), where Y is a standard gaussian random variable. If = 0, the limit in distribution of {Nn} is, obviously, a Poisson process with intensity = e x. As a trivial corollary, the sequence of maxima {Mn }, with Mn = max{X1 , . . . , Xn }, under the clas- sicalnormalizationun :=un(x)=x/p2lnn+p2lnn ln(4⇡lnn)/(2p2lnn),