Asymptotics for an arcsin type result. (English) Zbl 0796.60046

Hobson, David

Ann. Inst. Henri Poincaré, Probab. Stat. 30, No. 2, 235-243 (1994).

Summary: Let \(A_ t\) be the amount of time that a Brownian motion spends above 0 before time \(t\). For fixed \(t\) the ratio \(A_ t/t\) has distribution independent of \(t\); viewed as a function of time \(A_ t/t\) can become arbitrarily small. We consider the effect of modifying the denominator. In particular, if \(f\) is monotonic, then \(\liminf A_ t/tf(t)=0\) or \(\infty\) according as \(\int^ \infty\sqrt{f(t)} (dt/t)\) diverges or converges. The proof considers \(A_ t\) at the ends of “long” negative excursions and involves showing the existence of infinitely many such excursions.

MSC:

60G17	Sample path properties
60J65	Brownian motion

Keywords:

Brownian sample path; arcsin law; excursion theory; Brownian motion