Some More Results on Increments of the Wiener Process

Abstract

Let $W(T)$ for $0 \leq T < \infty$ be a standard Weiner process and suppose that $c_k$ and $b_k$ are fixed sequences of real numbers satisfying $0 \leq c_k < b_k < \infty$. Let $K(\omega)$ be the set of limit points (as $T \rightarrow \infty$) of $\frac{W(b_k;\omega) - W(c_k;\omega)}{\{2(b_k - c_k)\lbrack\log(b_k/(b_k - c_k)) + \log\log b_k\rbrack\}^{1/2}}$ where $\omega$ is a point in the probability space on which $W(T)$ is defined. We give necessary conditions on $b_k$ and $c_k$ to have $K(\omega) = \lbrack -1, 1\rbrack$ a.s. We also give some related results and discuss sharpness.