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
- 60G17 Sample path properties
- 60F15 Strong theorems

**Résumé:** We consider increasing self--similar Markov processes $\{X_t, t\geq 0\}$ on $]0,\infty[.$ By using the Lamperti's bijection between self--similar Markov processes and L\'evy processes, we determine the functions $f$ for which there exists a constant $c\in \re_+\setminus\{0\}$ such that $\liminf_{t\to\infty} X_t/f(t) = c $ with probability $1$. The determination of such functions depends on the subordinator $\xi$ associated to $X$ through the distribution of the L\'evy exponential functional and the Laplace exponent of $\xi.$ We provide an analogous result for the self--similar Markov process associated to the negative of a subordinator.

**Mots Clés:** *Self-similar Markov processes ; Subordinators ; Exponential functional of Lévy process ; Weak duality of Markov processes
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**Date:** 2002-09-19

**Prépublication numéro:** *PMA-755*

**Pdf file : **PMA-755.pdf