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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 108
Lemma 3.9. Under Assumptions A1–A8, we have
sup
i,j
ˆ
Γn(i,j)−Γ(i,j) → 0 in probability as n → ∞.
Proof of Lemma 3.9. For any θ ∈ Θ, let
Mn ij,h(θ) := 1
n
By the ergodic theorem, we have
n−|h|
t=1
e ′ t−h(θ)⊗ Id 2 (p+q) ⊗e ′ t−j−h(θ)
⊗ e ′ t (θ)⊗ I d 2 (p+q) ⊗e ′ t−i (θ) .
Mn ij,h(θ) → Mij,h(θ) := E e ′ t−h (θ)⊗ I d 2 (p+q) ⊗e ′ t−j−h (θ)
⊗ e ′ t(θ)⊗ Id 2 (p+q) ⊗e ′ t−i(θ) a.s.
A Taylor expansion of vec ˆ Mn ij,h around θ0 and (3.2) give
vec ˆ Mn ij,h = vecMn ij,h +
In view of (3.19), we then deduce that
lim
n→∞ sup
sup
i,j≥1 |h|