Boundary Crossing Probabilities Associated with Motoo's Law of the Iterated Logarithm

Abstract

For certain recurrent diffusion processes, Motoo has given an integral test which allows one to determine whether an increasing function belongs to the upper or lower class relative to the process at hand. We show that a refinement of his methods yields asymptotic estimates for the tail probabilities of the time of last crossing of an upper class function $g$ in cases where the speed measure of the process has sufficiently thin tails and the curve $g$ increases sufficiently slowly. Similar results are derived for certain extremal processes and for non-decreasing stable processes.