Page 112 - Textos de Matemática Vol. 47

102 A. C. ROSA AND M. E. NOGUEIRA as well as A(3) (x)  DT (x) O h 1 .
2,T T
On the other hand, recalling the notation and the result established in
Lemma 4.2, we conclude that, for every ✏ > 0,
⇣ (2) ⌘
P A2,Tn(x)>✏ 

=
pTn max P j2{1,...,pTn}
8(b a) h exp Tn
Z ! Tn (1)
8(b a)h T C1✏2 Tn n
Tn h2d
Tn .
Wt,Tn(x,yj,Tn)dt > ✏ 0
C1 ✏2 Tn h2d Tn
log Tn
Then, for T large enough, P ⇣A(2) (x) > ✏⌘ is bounded by C T C1 ✏2L1 ,
2 n2d under assumption (H6), for some positive constants C2 and L1.
n
2,Tn
A suitable choice of and L1 assures that Tn 2d < +1, implying n=1
2,Tn
( ) To conclude the a.s. convergence to zero of A3,Tn(x) it su ces to apply
Lemma.
Lemma 4.2 with i = 0 and to use a similar reasoning. ⇤ We now state our main result.
Theorem 3.2. Under the assumptions of Theorem 3.1, assume that f(·/x) is bounded. For all ↵ 2]0, 1[, we have
|q↵,T (x) q↵ (x)| ! 0 a.s.. T !+1
Proof. Observe that, for each ↵ 2 ]0, 1[, we may choose a, b 2 R such that q↵(x) 2 [a,b]. Using this fact and the boundness of f(·/x), we get
|q↵,T (x) q↵ (x)| = O( sup |FT (y/x) F (y/x)|) a.s., y2[a,b]
by the argument presented in [7, Lemma 6.7].
4. Auxiliary results
⇤
Lemma 4.1. Suppose that conditions (H1) (i), (iv), (H4) (ii), (H5) (ii) hold. If K has compact support, we have
|DT (x) g(x)| ! 0 a.s. and DT (x) g(x) ! 0 a.s.. T !+1 T !+1
Proof. Please see [13, p. 78]. ⇤
P 2 +1 C1 ✏ L1
the a.s. convergence of A(2) (x) to zero, as n ! +1, via the Borel-Cantelli