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

Pitman, J. W.; Yor, M.

Adv. Appl. Probab., Spec. Suppl. 1986, 109-116 (1986).

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