Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**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*