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

Strong approximations of additive functionals of a planar Brownian motion

Auteur(s):

Code(s) de Classification MSC:

• 60F15 Strong theorems
Résumé: This paper is devoted to the study of the additive functional $t \to\int_0^t f(W(s)) ds$, where $f$ denotes a measurable function and $W$ is a planar Brownian motion. Kasahara and Kotani [19]have obtained its second-order asymptotic behaviors, by using the skew-product representation of $W$ and the ergodicity of the angular part. We prove that the vector $(\int_0^\cdot f_j(W(s)) d s)_{1\le j \le n}$ can be strongly approximated by a multi-dimensional Brownian motion time changed by an independent inhomogeneous L\'evy process. This strong approximation yields central limit theorems and almost sure behaviors for additive functionals. We also give their applications to winding numbers and to symmetric Cauchy process.