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:**

- 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*