Page 140 - Textos de Matemática Vol. 47

130 M. G. TEMIDO
Write ⇤⇤ :=supwi,0.Sincethesequence{ ⇤⇤lnn}isboundedand ⇤⇤lnnk
i p
implies ⇤⇤ < 1 for n large, we obtain
2
nm
u2n,m ! |ri,0 ⇢n,m|exp !
Xn
i=p+1 1 + wi,0
2 ⇤⇤ u2n,m 2nm exp 1+ ⇤⇤
(n m)2
= K ⇤⇤ ln n + ⇤⇤ ln m exp (2 ⇤⇤ ln n)
!
 2 n2 m ⇤⇤ exp u2n,m exp
 K n2m ⇤⇤ ln(n m) exp (2 ⇤⇤ ln n + 2 ⇤⇤ ln m)
u2n,m ⇤⇤ 1+ ⇤⇤
✓ m1 2 ⇤⇤ ◆
K 1+lnm!0,m!+1.
1 m1 2 ⇤⇤ m1 2 ⇤⇤
For the sums concerned the set {0} ⇥ (B1 [ B2) we use the same arguments. Suppose now that {un,m} is a real sequence satisfying n m(1 (un,m))  ⌧ and that {vn,m} is another real sequence such that n m(1 (vn,m)) ! ⌧, as
n,m!+1.Thus,formandnwithpn2 +m2>M,withMlargeenough, we get vn,m  un,m. Then, since (2.2) holds for {vn,m} it follows the same for
{un,m }.
⇤
3. Point processes of exceedances
Consider the Cox process N defined over ]0,1]2, with stochastic intensity givenbyexp x +p2 Y ,wherex2R, 2R+0 andY isastandard gaussian random variable. The distribution of N is defined, in terms of its probability mass function, by
P
\q ! N(Bi) = si
i=1
Z +1 Yq (Bi)e x +zp2
= s ! 1 i=1 i
!
x +zp2
⇥ exp( (Bi)e ) (z)dz
(3.1)
where, for q 1, s1,...,sq are non negative integers, B1,··· ,Bq are disjoint subsets of ]0, 1]2 and represents the Lebesgue measure.