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:**

- 60F17 Functional limit theorems; invariance principles
- 60H99 None of the above but in this section

**Résumé:** In this paper we study the semiamrtingales $X$ which are defined on an
extension of a basic filtered probability space
$\Ba=(\Om,\Fa,\fit_{t\geq0},P)$ and which, conditionally on $\Fa$,
have independent increments. We first give a general characterization
for such processes. Then we prove that if all martingales of the basis
$\Ba$ can be written as a sum of stochastic integrals w.r.t. the
continuous martingale part and the compensated jump measure of $Y$,
then a process $X$ has $\Fa$-conditional independent increments if and
only if the characteristics of the pair $(X,Y)$, on the extended
space, are indeed predictable w.r.t. the filtration $\fit$. Finally we
prove a functional convergence result toward a process $X$ of this kind.

**Mots Clés:** *Stable convergence ; Lévy processes*

**Date:** 2001-05-21

**Prépublication numéro:** *PMA-662*