18
Denis Belomestny and Markus Reiβ
Making use of R0U 2uecu2 du = —1 = E (cU2) U2 for any c > 0, we esti-
mate the last integral by
U2
e eTσru +2TЫь 1 ∣u∣s+2 du 6 e2TkμkL1 Us+3E(T'.U2)
—U
and derive from ∣∣μ∣∣^ 1 = Fμ(0) 6 2R for the bias part in the linear term
∣L>∣ . ∆2E(T2U2).
(38)
For the variance part of the linear error term we use the support proper-
ties supp(wσU) ∈ [-U, U] and supp(bk) = [xk—1, xk+1]. Several applications
of the Plancherel identity, the Cauchy-Schwarz inequality and estimate (35)
then yield
=
—U —
Cov
ʌ'
(u — i)uFO(u) (v — i)vFO(v)
φT ( u — i ) ’ φT ( v — i )
wσU (u)wσU (v)
du dv
x I Γ
=X δ2∣
k=1 —
ψτ ( u
2
— i)—1(u — i)uF bk (u)wσU (u) duI
N I∞
2π δk2 II
k=1 - '
F 1∖Tτ ( u
— i)—1(u — i)uwσU (u) (x)bk (—x) dx
6 2π
N xk+1
Xδk2 II
k=1 xk-1
F—
φτ ( u
i)—1(u
i)uw
dx ∣bk ∣2L2
∆∣δ∣l2∞ ∞ III
J-∞a
F—
φτ ( u
i)—1(u
i)uw
dx
U
^HI2~ / ∣ψτ(u
—U
i)∣ 2(u4 + u2)wσ(u)2 du
∆U-1
E(Tσ2U2
)∣δ∣l2∞ .
Altogether we obtain for the linear error term
E
2
L(u)wσ(u)du∣ ] . E(Tσ2U2) (∆4 + U-1 ∆∣∣δ∣∣2^).
(39)
It remains to estimate the quadratic remainder term. Due to Lemma 1
and Proposition 2 we have
E
2
R(u)wσU (u) duII
(40)
U U I I2 u4wU (u)v4wU (v)
< E E F ( O-O )( u ) F ( O-O )( v )∣ ---σ. ( ) , σ2 ( ) d u d v.
-U -U κ(u)2κ(v)2
The independence of (εk ) and the finiteness of their fourth order moments
entail the inequality