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

### Weak convergence of positive self similar Markov processes and overshoots of Lévy processes

Auteur(s):

Code(s) de Classification MSC:

• 60G18 Self-similar processes
• 60G51 Processes with independent increments
• 60B10 Convergence of probability measures

Résumé: We give necessary and sufficient conditions for the law of a positive self-similar Markov process to converge weakly as its initial state tends to 0 and we describe the limit law. Our proof is based on Lamperti's representation which relates any positive self-similar process to a unique L\'evy process. Then we show that the convergence mentioned above holds if and only if the process of the overshoots of the underlying L\'evy process $\xi$ in the Lamperti's representation converges weakly at infinity and $E\left(\log^+\int_0^{T_1}\exp\xi_s\,ds\right)<\infty$, where $T_1=\inf\{t:\xi_t\ge1\}$. Under these conditions, we give a pathwise construction of the limit law.

Mots Clés: Self-similar process ; Lévy process ; Lamperti's representation ; overshoot ; weak convergence ; first passage time

Date: 2004-04-02

Prépublication numéro: PMA-899