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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 109
where
g1 = sup sup ˆ
Mn ij,h −Mij,h
|f(hbn)|,
i,j≥1 |h|Tn
Mij,h.
The non zero elements of Mij,h are of the form E(ei1 tei2 t−iei3 t−hei4 t−h−j), with
(i1,i2,i3,i4) ∈ {1,...,d} 4 . Now, using the covariance inequality obtained by Davydov
(1968), it is easy to show that
We then deduce that
|E(ei1 tei2 t−iei3 t−hei4 t−h−j)| = |Cov(ei1 tei2 t−i,ei3 t−hei4 t−h−j)|
≤ Kα ν/(2+ν)
ǫ
(h).
Mij,h ≤ Kα ν/(2+ν)
ǫ (h). (3.22)
In view of A7, we thus have g3 → 0 as n → ∞. Let m be a fixed integer and write
g2 ≤ s1 +s2, where
s1 =
|f(hbn)−1|Mij,h and s2 =
|f(hbn)−1|Mij,h.
|h|≤m
bn i,j≥1 |h|