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A. C. ROSA AND M. E. NOGUEIRA
Since
FT(y/x) F(y/x) = BT (x,y)+ 1 NT (x,y) NT (x,y) +
+ 1 DT (x)
y2E
DT (x)
(BT (x,y)+F(y/x)) DT (x) DT (x) ,
we may write
sup|FT (y/x) F (y/x)|
1 A2,T (x) + DT (x)

with A1,T(x) = sup|BT (x,y)|, A2,T(x) = sup NT (x,y) NT (x,y) and
A1,T (x) + + 1
(A1,T (x) + 1) A3,T (x), y2E y2E
Taking into account Lemma 4.1 and the fact that g(x) > 0, in order to prove Theorem 3.1 it is enough to establish the almost sure convergence to zero of Ai,T (x), for i = 1, 2, 3.
(↵) We firstly analyse the behavior of A1,T (x).
Note that A1,T (x) = 1 sup |CT (x, y)|, where
DT (x) y2E
CT (x,y)=NT (x,y) F(y/x)DT (x).
Equivalently, CT (x, y) is given by
1 ZT ✓ ✓x Xt◆ ✓ ✓y Yt◆. ◆ . ◆
d E K E G St , F(y/x) Ft dt. ThT 0 hT hT
DT (x) A3,T (x) = DT (x) DT (x) .
Then, (H2) implies that
1ZT✓✓x X◆ ◆
CT (x,y)= d E K t (Xt)/Ft dt, ThT 0 hT
with (Xt)=E✓G✓y Yt ◆/Xt◆ F(y/x). hT
But, for every w 2 ZRd such that g(w) > 0, we have +1
(w) = K0 (u) (F (y hT u/w) F (y/x)) du 1