Brownian local time approximated by a Wiener sheet. (English) Zbl 0674.60072

Csáki, E.; Csörgö, M.; Földes, A.; Révész, P.

Ann. Probab. 17, No. 2, 516-537 (1989).

Let L(a,t) be a jointly continuous Brownian local time. The authors prove the existence of a pair (W,\^L) of independent processes, where W is a Wiener sheet and \(\hat L\) is distributed as L(0,\(\cdot)\), such that a.s. as \(t\to \infty:\) \(\alpha)\quad \sup_{0\leq a\leq a^*t^{\delta /2}}| L(a,t)-L(0,t)-2W(a,\hat L(0,t))| =O(t^{(1+\delta)/4- \epsilon /2}),\) \[ \beta)\quad | \hat L(0,t)-L(0,t)| =O(t^{15/32}\log^ 2t), \] where \(a^*\) is a positive constant and \(\delta\) and \(\epsilon\) are small enough.
This strong approximation property explains nicely a weaker result obtained by M. Yor [Sémin. de Probabilités XVII, Proc. 1981/82, Lect. Notes Math. 986, 89-105 (1983; Zbl 0514.60075)] and yields asymptotic properties for the Brownian local times.

Reviewer: J.Bertoin

MSC:

60J55	Local time and additive functionals
60J65	Brownian motion
60G57	Random measures
60J60	Diffusion processes

Keywords:

Wiener sheet; Brownian local time; strong approximation property; asymptotic properties for the Brownian local times

Citations:

Zbl 0514.60075