Page 111 - Textos de Matemática Vol. 47

ESTIMATION OF THE CONDITIONAL QUANTILE FUNCTION
and so, using (H3), |CT (x, y)| is almost surely bounded by
101
dt.
CZT ✓✓x Xt◆⇣ ⌘ ◆ d E K hT μ + kXt xk Ft
ThT 0 hT Finally, under (H4), we get
|CT (x,y)|DT (x)O⇣h T +h T⌘ a.s..
The a.s. convergence to zero of A1,T (x) follows now from assumption (H6).
( ) In order to prove the a.s. convergence to zero of A2,T (x), define, for > 1,
8>< b a b a
> if 2N p = > h T h T
T >b a b a
: h +1if h 2/N
TT
where [u] denotes the integer part of the real number u. Dividing E = [a, b] into the intervals
Ij,T =⇥a+(j 1)h T,a+jh T⇥, j2{1,...,pT 1}, and
we may writeA2,T (x)
IpT,T =⇥a+(pT 1)h T,b⇤,
max sup NT (x, y) N T (x, y)
j2{1,...,pT } y2Ij,T
max NT (x,yj,T) NT (x,yj,T) ,
j2{1,...,pT }
max sup NT (x,yj,T) NT (x,y) ,
j2{1,...,pT } y2Ij,T
with
A(1) (x)= 2,T
A(2)(x)= 2,T
A(3)(x)= 2,T
2,T 2,T 2,T
max sup |NT (x,y) NT (x,yj,T)|,
=

j2{1,...,pT } y2Ij,T
A(1) (x) + A(2) (x) + A(3) (x),
yj,T beinganarbitrarypointinIj,T,j2{1,...,pT}.
The fact that G is absolutely continuous and (H5) (i) yield
!
(1)
A2,T (x)  max sup
j2{1,...,pT } y2Ij,T
= DT(x)O h 1 , T
1
d T hT
Z T 0
K
✓ x Xt ◆ | y yj,T | hT hT
k0 dt