On a maximum of stable Lévy processes

Let $G$ and $S$ be the location and the maximal value of a stable Lévy process $X$ on an interval $[0,a]$. It is shown that the dimensionless $S/G^h$, where h is the self-similarity parameter of $X$, is independent of $G$. This fact allows us to analyze $G$ for the trajectories of $X$ with high and low maxima.