THÈSE Estimation, validation et identification des modèles ARMA ...

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 105
In view of (3.15), and using A2 and the compactness of Θ, we have
et(θ) = Xt +
2
i≥1
∞
i=1
Ci(θ)Xt−i, supCi(θ)
≤ Kρ
θ∈Θ
i .
We thus have
Esup et(θ)
θ∈Θ
2 < ∞, (3.16)
by Assumption A7. Now, we will consider the norm defined by : Z2 = EZ 2 ,
where Z is a d1 random vector. In view of Proposition 1, (3.16) and supθ∈Θλh(θ) ≤
Kρh , we have
sup ∂et(θ)
θ∈Θ ∂θ ′
≤
supλi(θ)×
θ∈Θ
sup
Id2 (p+q) ⊗et(θ)
θ∈Θ
2 < ∞. (3.17)
Let et = (e1t,...,edt) ′ . The non zero components of the vector vecMn(θ) are of the
form n −1 n
t=1 eit(θ)ejt(θ), for (i,j) ∈ {1,...,d} 2 . We deduce that the elements of the
matrix ∂vecMn(θ)/∂θ ′ are linear combinations of
2
n
n
t=1
eit(θ) ∂ejt(θ)
.
∂θ
By the Cauchy-Schwartz inequality we have
n
2
sup eit(θ)
n
t=1
θ∈Θ
∂ejt(θ)
∂θ
≤ 2 n
sup{e
n
t=1
θ∈Θ
2 n
2
it (θ)}×
n
t=1
sup
2
∂ejt(θ)
θ∈Θ ∂θ
≤ 21
n
supet(θ)
n θ∈Θ
2 × 1
n
n sup ∂et(θ)
θ∈Θ ∂θ ′
The ergodic theorem shows that almost surely
n
2
lim
n→∞ n sup
eit(θ)
θ∈Θ
∂ejt(θ)
≤ 2 Esup et(θ)
∂θ θ∈Θ
2
×E
sup ∂et(θ)
θ∈Θ ∂θ ′
2
.
t=1
Now using (3.16) and (3.17), the right-hand side of the inequality is bounded. We then
deduce that
sup
∂vecMn(θ)
θ∈Θ ∂θ ′
= Oa.s(1). (3.18)
A Taylor expansion of vec ˆ Mn about θ0 gives
t=1
vec ˆ Mn = vecMn + ∂vecMn(θ ∗ n )
∂θ ′ ( ˆ θn −θ0),
t=1
2
.