Strong approximations of additive functionals of a planar Brownian motion

This paper is devoted to the study of the additive functional t→∫0tf(W(s)) ds, where f denotes a measurable function and W is a planar Brownian motion. Kasahara and Kotani (Z. Wahrsch. Verw. Gebiete 49(2) (1979) 133) have obtained its second-order asymptotic behavior, by using the skew-product representation of W and the ergodicity of the angular part. We prove that the vector ∫0·fj(W(s)) ds1⩽j⩽n can be strongly approximated by a multi-dimensional Brownian motion time changed by an independent inhomogeneous Lévy 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.