Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Strong approximations of additive functionals of a planar Brownian motion


Code(s) de Classification MSC:

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.

Mots Clés: Additive functionals ; strong approximation

Date: 2002-12-11

Prépublication numéro: PMA-779

Pdf file : PMA-779.pdf