## The Annals of Probability

#### Abstract

Let $W(t), 0 \leq t < \infty$, be a Wiener process. This paper proves that $\lim \sup_{T \rightarrow \infty} \sup_{0 < t \leq T} \frac{|W(T) - W(T - t)|}{\{2t(\log(T/t) + \log\log t)\}^{1/2}} = 1, a.s.,$ $\lim_{T \rightarrow \infty} \sup_{0 < t \leq T} \sup_{t \leq s \leq T} \frac{|W(T) - W(s - t)|}{\{2t(\log(T/t) + \log \log t)\}^{1/2}} = 1, a.s.$ These results give an affirmative answer to the questions posed by Hanson and Russo without additional assumptions.

#### Article information

Source
Ann. Probab., Volume 14, Number 4 (1986), 1252-1261.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176992366

Digital Object Identifier
doi:10.1214/aop/1176992366

Mathematical Reviews number (MathSciNet)
MR866346

Zentralblatt MATH identifier
0613.60027

JSTOR