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

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

Asymptotic property for a LÚvy parametric model


Code(s) de Classification MSC:

RÚsumÚ: We consider a LÚvy 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 LÚvy 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 ClÚs: convergence of likelihoods ; stable convergence in law ; stable processes

Date: 2002-03-27

Prépublication numéro: PMA-716