Page 141 - Textos de Matemática Vol. 47

STONGLY DEPENDENT GAUSSIAN RANDOM FIELDS 131
The following theorem, the main result of this work, states that a Cox process is obtained as the limit in distribution of the sequence of point processes of the exceedances defined by (1.2). Is then established the expected extension of the classical results for gaussian sequences presented in [5], when 0 is regarded, as well as the ones for random fields due to [2] for = 0.
Theorem 3.1. Let {Xn,m} be a stationary standard gaussian random field with correlations satisfying (1.1) and (1.3) and suppose that the sequences {rn,0 ln n} and {r0,m ln m} are bounded. Consider un,m = x/an,m + bn,m with x 2 R and an,m and bn,m defined by (2.3). Then the sequence {Nn,m} of point processes of exceedances of the high level un,m by the random field {Xn,m} converges in dis- tribution to the Cox process with stochastic intensity exp x + p2 Y , that is, the point process with distribution characterized by (3.1).
Proof. We use a Kallenberg’s theorem (see [4, Theorem 4.7] or [5, Theo- rem A.1]). Since for each ]a, b]⇥]c, d] ✓ ]0, 1]2 we get
E (Nn,m (]a, b]⇥]c, d])) = n(b a)m(d c) (1 (un,m )) ! (b a)(d c)e x, n,m!+1,
it remains to prove that
P ({Nn,m(B) = 0}) ! P ({N(B) = 0}), n,m ! +1,
for all B of the form Sq`=1 B`, with q 2 N and B1,...,Bq disjoint subsets of ]0, 1]2.
We use the notations
Mn,m (]a,b]⇥]c,d]) = max{Xi,j : [an] < i  [bn] , [cn] < j  [dn]}
and
Mn,m (]a,b]⇥]c,d],⇢n,m) = max{Yi,j : [an] < i  [bn] , [cn] < j  [dn]}
where {Yi,j} represents a standard gaussian random field with all the covari- ances equal to ⇢n,m = / ln(n m) (except ⇢0,0 = 1).
Let B1, · · · , Bq be disjoint subsets of ]0, 1]2 and let Y be a standard gaussian randpom variable independent of {Mn,m(Bi,0),i=1,2,···q}. Consequently, for each n and m and for each B`, the random variables Mn,m (B`, ⇢n,m) and 1 ⇢n,mMn,m (B`, 0) + p⇢n,mY have the same distribution. Then, with