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

### Fractional Brownian motions as "higher-order" fractional derivatives of Brownian local times

Auteur(s):

Code(s) de Classification MSC:

• 60J55 Local time and additive functionals
• 60F05 Central limit and other weak theorems

Résumé: Fractional derivatives ${\cal D}^\gamma$ of Brownian local times are well defined for all $\gamma<3/2$. We show that, in the weak convergence sense, these fractional derivatives admit themselves derivatives which feature all fractional Brownian motions. Strong approximation results are also developed as counterparts of limit theorems for Brownian additive functionals which feature the fractional derivatives of Brownian local times.

Mots Clés: Local time ; additive functional ; principal value ; Brownian sheet ; fractional Brownian motion ; Hilbert transform ; fractional derivative

Date: 1999-09-03

Prépublication numéro: PMA-524