Page 109 - Textos de Matemática Vol. 47

ESTIMATION OF THE CONDITIONAL QUANTILE FUNCTION 99
(H5) The kernel K0 verifies:
(i) K0 is bounded, k0 = sup |K0(y)|;
y2E (ii) K0 haZs compact support;
+1
(iii) μ = |u| K0(u) du < +1.
1
T h2d
(H6) lim hT =0, lim T =+1.
T!+1 T!+1 logT
Notice that hypothesis (H1) relies on the ergodic character of the data and was introduced by Didi and Louani [14, 15], as far as we know. The arguments presented by the authors show that it consists of a natural continuous-time version of the conditions considered by Delecroix et al. [11]. The interested reader is referred to Bergelson et al. [2] for some recent asymptotic results concerning continuous-time ergodic processes. Conditions (H4) to (H6) are quite common in kernel estimation of conditional functionals. The regularity assumption on the conditional distribution function imposed in (H3) is weaker than hypothesis (A3)(ii) of Chaouch and Khardani [7]. Assumption (H2) is a Markov-type condition similar to the one required in the previous paper (cf. (A4), p. 69) and to condition (H3) (i) of Didi and Louani [15].
3. Main results
Next we state the uniform almost sure convergence of FT (·/x) to the distri-
bution function F(·/x).
Theorem 3.1. Under the general conditions of the previous section and as-
sumptions (H1) to (H6), we have
sup |FT (y/x) F (y/x)| ! 0 a.s..
y2E T !+1
Proof. Let us consider:
1 ZT ✓ ✓x Xt◆. ◆
DT(x)=Thd EKhT Ft dt; T0
1ZT ✓✓y Yt◆✓x Xt◆. ◆ NT(x,y)= d EG K Ft dt;
BT (x,y)= NT (x,y) F(y/x). DT (x)
T hT 0 hT hT