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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 104
Lemma 3.5. Under Assumptions A1–A8, we have
vec( ˆ
λi −λi(θ0)) ≤ Kρ i ×oa.s(1) a.s as n → ∞,
where ρ is a constant belonging to [0,1[, and K > 0.
Proof of Lemma 3.5. Boubacar Mainassara and Francq (2009) showed the strong
consistency of ˆ θn ( ˆ θn → θ0 a.s. as n → ∞), which entails
ˆ
θn −θ0
= oa.s(1). (3.13)
We have A∗ ij,h = O(ρh ) and B∗ ij,h = O(ρh ) uniformly in θ ∈ Θ for some ρ ∈ [0,1[. In
view of (3.4), we thus have supθ∈Θλh(θ) ≤ Kρh . Similarly for any m ∈ {1,...,k0},
we have
sup
∂λh(θ)
θ∈Θ ∂θm
≤ Kρh . (3.14)
Using a Taylor expansion of vec ˆ λi about θ0, we obtain
vec ˆ λi = vecλi + ∂vecλi(θ ∗ n )
∂θ ′ ( ˆ θn −θ0),
where θ ∗ n is between ˆ θn and θ0. For any multiplicative norm, we have
vec( ˆ
λi −λi) ≤
∂vecλi(θ ∗ n )
∂θ ′
In view of (3.13) and (3.14), the proof is complete. ✷
Lemma 3.6. Under Assumptions A1–A8, we have
Proof of Lemma 3.6. We have
p
et(θ) = Xt −
For any θ ∈ Θ, let
Mn(θ) := 1
n
n
t=1
i=1
ˆMn → M a.s as n → ∞.
A −1
0 AiXt−i +
q
i=1
Id 2 (p+q) ⊗e ′ t(θ) ⊗2
Now the ergodic theorem shows that almost surely
ˆ
θn −θ0.
A −1 −1
0 BiB0 A0et−i(θ) ∀t ∈ Z. (3.15)
and M(θ) := E
Mn(θ) → M(θ).
Id 2 (p+q) ⊗e ′ t(θ) ⊗2
.