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:**

- 60F15 Strong theorems
- 60J65 Brownian motion, See also {58G32}

**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*