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

### A law of iterated logarithm for increasing self-similar Markov processes

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

Date: 2002-09-19

Prépublication numéro: PMA-755

Pdf file : PMA-755.pdf