Some "LIM INF" Results for Increments of a Wiener Process

Abstract

Let $W(t)$ for $0 \leq t < \infty$ be a standard Wiener process, suppose $0 < a_T \leq T$ for $T > 0$, and let $d(T, t) = \{2t\lbrack\log(T/t) + \log \log t \rbrack\}^{1/2}$. Quantities such as $\lim, \inf_{T \rightarrow \infty} \sup_{a_T \leq t \leq T} \frac{W(T) - W(T - t)}{d(T,t)},$ $\lim, \inf_{T \rightarrow \infty} \sup_{\substack{0 \leq t \leq T - a_T\\0 \leq s \leq a_T}} \frac{|W(t + s) - W(t)|}{d(t + a_T, a_T)}$ and $\lim, \inf_{T \rightarrow \infty} \sup_{\substack{0 \leq u < v \leq T\\a_T \leq v - u}} \frac{|W(v) - W(u)|}{d(v, v - u)}$ are investigated.