Lower functions for processes with stationary independent increments (Q1093253)

Let \(\{X_ t\}\) be an \(R^ 1\)-valued process with stationary independent increments and \(A_ t=\sup _{s\leq t}| X_ s|\). In this paper we find a sufficient condition for the existence of a nonnegative and nondecreasing function h(t) such that lim \(A_ t/h(t)=C\) a.s. as \(t\to 0\) and \(t\to \infty\), for some positive finite constant C when h(t) takes a particular form. Also two analytic conditions are considered as application.