Answers to Some Questions About Increments of a Wiener Process

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.