Page 82 - Textos de Matemática Vol. 47

72 P. E. OLIVEIRA
We start by describing the asymptotic behaviour of the expectation and variance of fb (x) and gb (x).
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Lemma 3.1. Assume that (SB1), (SB2) and (K) hold. Then
Efb(x)=f (x)+ (K,x,h), n0
◆ .
Proof. The proof is straightforward after writing the mathematical expecta-
Var fb (x) = f1(x) + (K2, x, h) + o ✓ 1
n
nF (x, h) nF (x, h)
tions as integrals over [0, 1]. ⇤
Note that if lim (K,x,h) = 0 the behaviour of the variance has a h!0b f1(x) ⇣1⌘
simpler description: Var fn(x) = nF (x,h) +o nF (x,h) . Similar comments apply throughout this paper but, to avoid repetitions, we will not be referring to such
variations at each place.
Lemma 3.2. Assume that (SB1), (SB2) and (K) hold. Then E gb n ( x ) = ( r ( x ) + w r ( x , h ) ) ( f 0 ( x ) + ( K , x , h ) ) ,
(s(x) + ws(x, h))(f1(x) + (K2, x, h)) ✓ 1 ◆ Vargbn(x)= Z nF(x,h) +o nF(x,h) .
Proof. The proof follows as for the previous lemma, by writing
E gb n ( x ) = 1 ( r ( x ) + r ( u ) r ( x ) ) K ( k x u k ) P X ( d u )
F(x,h) h
1 Z (r(x) r(u))K(kx uk)P (du) w (x,h)(f (x)+ (K,x,h)).
and remarking that
F(x,h) hX r 0
⇤
The following result is now obvious.
Corollary 3.3. Assume that wr(x,·), ws(x,·), (K,x.·) and (K2,x,·) are
bounded, (SB1), (SB2), (K) hold and
nF (x, h) ! +1. (3.2)
Then Varfb (x) ! 0 and Vargb (x) ! 0. nn