On the Increments of Wiener and Related Processes

Abstract

Let $\{W(t), 0 \leq t < +\infty\}$ be a standard Wiener process and $0 < b_t \leq t$ be a nondecreasing function of $t$. The properties of the process $Y_1(t) = b^{-1/2}_t \sup_{0\leq s \leq t - b_t}(W(s + b_t) - W(s))$ are investigated. One of the results says that $\lim_{t\rightarrow\infty}(Y_1(t) - (2 \log tb^{-1}_t)^{1/2}) = 0$ a.s. if $b_t$ is "much less" than $t$. Analogous properties of similar processes are studied.