"Strong approximations of additive functionals of a planar Brownian motion"

Endre Csáki, Antónia Földes and Yueyun Hu

Summary: This paper is devoted to the study of the additive functional of a planar Brownian motion. Kasahara and Kotani have obtained its second-order asymptotic behaviors, by using the skew-product representation and the ergodicity of the angular part. We prove that the vector of additive functionals 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.

Keywords: Additive functionals; strong approximations.

AMS 2000 subject classification: 60F15; 60J65.