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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 107
The proof is complete. ✷
Proof of Theorem 3.2. Let the matrices
ˆΛij = Id ⊗ ˆ λ ′
i ⊗ Id ⊗ ˆ λ ′
j , Λij = {Id ⊗λ ′ i}⊗ Id ⊗λ ′
j ,
ˆ∆e0 =
vec ˆ Σ −1
e0
vec ˆ Σ −1
e0
′
and ∆e0 =
vecΣ −1
e0
−1 ′
vecΣe0 .
For any multiplicative norm, we have
vecI −vec În
≤ 4
Γ(i,j)−
Γn(i,j) ˆ
Λijvec∆e0
i,j≥1
+ ˆ
Γn(i,j) Λij
− ˆ
Λijvec∆e0
+ ˆ
Γn(i,j) ˆ
Λijvec
∆e0 − ˆ
∆e0 .
Lemma 3.5 and λi = O(ρi ) entail
vec ˆ
Λij −vecΛij ≤ vec Id ⊗( ˆ λi −λi)⊗(Id ⊗ ˆ
λj)
+ vec (Id ⊗λi)⊗ Id ⊗( ˆ
λj −λj)
We also have Λij = O(ρ i+j ). In view of Lemma 3.8 and Σ −1
e0
ˆ ∆e0 −∆e0
≤ Kρ i+j ×oa.s(1).
< ∞, we have
≤ vec ˆΣ −1
e0 −Σ−1
e0 vec ˆ Σ −1
′
e0
+
−1
vecΣ
e0 vec ˆΣ −1
e0 −Σ −1
′
e0 → 0 a.s.
In the univariate case Francq, Roy and Zakoïan (2005) showed (see the proofs of their
Lemmas A.1 and A.3) that sup ℓ,ℓ ′ >0|Γ(ℓ,ℓ ′ )| < ∞. This can be directly extended in the
multivariate case. The non zero elements of Γ(i,j) are of the form
∞
h=−∞
We thus have
sup
i,j≥1
E(ei1 tei2 t−iei3 t−hei4 t−h−j), for (i1,i2,i3,i4) ∈ {1,...,d} 4 .
∞
h=−∞
E(ei1 tei2 t−iei3 t−hei4 t−h−j)
≤ sup Γ(i,j) < ∞. (3.19)
i,j≥1
We then deduce that sup i,j≥1Γ(i,j) = O(1). The proof will thus follow from Lemma
3.9 below, in which we show the consistency of ˆ Γn(i,j) uniformly in i and j. ✷