Page 142 - Textos de Matemática Vol. 47

132 M. G. TEMIDO
wn,m = p1 ⇢n,m(un,m zp⇢n,m) we obtain
P =P
\q ! {Mn,m (B`, ⇢n,m)  un,m}
`=1 !
\q p\
1 ⇢n,mMn,m(B`,0)+p⇢n,m Y un,m
`=1 Z+1 q
=
Moreover, due to Lemma 2.2, we get
\q ! \q P {Mn,m(B`)  un,m} P
`=1 `=1
!
1
`=1
P
{Mn,m(B`, 0)  wn,m} (z)dz.
(3.2)
when n,m ! 1.
Let {Xn⇤,m} be a standard gaussian random field of independent random
variables and denote by {Nn⇤,m} the sequence of point process of exceedances, of a normalized level, by the random variables X1⇤,1, . . . , Xn⇤,m.
Mutatis mutandis, we can establish the expected generalization of Corol- lary 5.2.2 of [5]. Indeed, for each B ⇢]0,1]2 and for all s 2 N0, we have
⇤ e ⌧ (B)(⌧ (B))s
P(Nn,m(B) = s) ! s! , n,m ! +1, (3.3)
where ⌧ = lim nm(1 (x/an,m + bn,m)). Furthermore, for disjoint sub- n,m!+1
sets of ]0, 1]2, B1, · · · , Bk, the vector Nn⇤,m(B1), · · · , Nn⇤,m(Bk) , converges in distribution to the product of limits in (3.3).
! {Mn,m(B`, ⇢n,m)  un,m} ! 0,
Now, taking into account that p1 ⇢n,m(un,m zp⇢n,m)= 1 1⇢n,m+on,m(⇢n,m) ✓ x +bn,m zp⇢n,m◆
2 an,m = x+ zp2 +b +o(a 1 )
an,m n,m n,m
and then ⌧ := ⌧(x) = x + ⌧ zp2 , it follows that