Brownian Motion with Lower Class Moving Boundaries Which Grow Faster Than $t^{1/2}$

Abstract

Upper and lower bounds are obtained for $P(|W(t)| \leq f(t), t \leq u)$ and $P(|S(n)| \leq f(n), n \leq N), u, N$ large, where $W(t)$ is a Brownian motion, $S(n)$ is a random walk with $ES(1) = 0, E|S(1)|^{2+2\eta} < \infty$, and $f(t)$ is a deterministic function growing faster than $t^{1/2}$ but slower than $(2t \ln \ln t)^{1/2}$.