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

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
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