Page 135 - Textos de Matemática Vol. 47

STONGLY DEPENDENT GAUSSIAN RANDOM FIELDS 125
converges in distribution to a random variable whose distribution is a mixture of Gumbel distributions regulated by Y . For = 0, the classical Gumbel limit is obtained. In the scope of non stationary gaussian sequence, extensions of these results can be found in Hu¨sler [3] and in Temido [8].
2. Main lemmas
When dealing with the distribution function of gaussian vectors, the Normal Comparison Lemma ([5]) appears as a widely useful result. This lemma esta- blishes bounds for the di↵erence between two multivariate gaussian distribution functions, which are convenient functions of their covariances. Our next lemma is a particular version of that result, with a specific approach for the random fields.
Lemma 2.1. Let {Xn,m} and {Yn,m} be stationary standard gaussian random
fields with correlations ri,j and vi,j, respectively, satisfying (1.1). Consider that
un,m,n,m 1arerealnumbers.Then,forQn,m ={(i,j):1in,1j
(2.1)
m}andIn,m ={(i,j):(i,j)6=(0,0),0in,0jm},itholdsthat 0 10 1
\ X \ ! P @i,j 2Qn,m{Xi,j  un,m}A P @i,j 2Qn,m{Yi,j  un,m}A
Knm
with K > 0 and wi,j = max{|ri,j |, |vi,j |}.
(i,j )2In,m
|ri,j vi,j|exp u2n,m , 1+wi,j
In order to specify an approximation for the distribution of the vector {Xi,j : (i, j ) 2 Qn,m } of the gaussian random field, in the next lemma we give conditions which imply that the upper bounds in (2.1) are asymptotically zero. This lemma is an extension of Lemma 5.2.1 of [2], where only the assump- tion rn,m ln(nm) ! 0, n, m ! +1, is considered. It should be remarked that suitable conditions on the behaviour of the sequences {r0,j}j and {ri,0}i are also needed.
Lemma 2.2. Consider that {rn,m} and {un,m} are real sequences, is a non negative real number and ⇢n,m = /ln(n m). Assume that {n m(1 (un,m ))}n,m , {rn,0 ln n}n Xand {r0,m ln m}m are bounded sequ!ences and that (1.3) holds. Then
nm |ri,j ⇢n,m|exp u2n,m !0, n,m!+1, (2.2)
1+wi,j
with wi,j = max{|ri,j|,⇢n,m} and In,m defined in Lemma 2.1.
(i,j )2In,m