## Some divergent integrals of Brownian motion.(English)Zbl 0618.60074

Let $$(X_ t$$, $$t\geq 0)$$ denote a two-dimensional Brownian motion starting from 0. If $$f: {\mathbb{R}}^ 2\to {\mathbb{R}}_+$$ is a measurable function which is integrable with respect to Lebesgue measure, then, for each $$\epsilon\in (0,1)$$, the integral $$\int^{1}_{\epsilon}dsf(X_ s)$$ is almost surely finite. The asymptotic behaviour of the integral as $$\epsilon\to 0$$ is studied. For some particular values of f, unusual limits in law are obtained. Also the authors derive necessary and sufficient criteria for $$\int^{1}_{0}dsf(X_ s)<\infty$$ almost surely.
Reviewer: G.Kersting

### MSC:

 60J55 Local time and additive functionals 60F05 Central limit and other weak theorems

### Keywords:

ergodic theorems; asymptotic laws; asymptotic behaviour