Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Asymptotic property for a Lévy parametric model

Auteur(s):

Code(s) de Classification MSC:

• 62A10 The likelihood approach
• 60F99 None of the above but in this section
• 60J30 Processes with independent increments

Résumé: We consider a Lévy process $\th Z$ depending on an unknown parameter $\th,$ which is observed at times $i/n$ over $[0,1].$ We know that for an $\al$-stable Lévy process $Z$, the associated parametric models satisfy the LAN property with rate $\sqrt{n}$. In this paper, we show that this result does not persist if $Z$ is the sum of a symmetric stable and a Poisson process. For $0<\al<2$ we prove that the limiting model is a non-Gaussian shift and that the optimal rate for estimating $\th$ is $n^{1/\al}$. We show also that we cannot construct an estimator converging with this rate, the best we can achieve is a random rate between $\sqrt{n}$ and $n^{1/\al}$.

Mots Clés: convergence of likelihoods ; stable convergence in law ; stable processes

Date: 2002-03-27

Prépublication numéro: PMA-716