On the Range of Subordinators

In this note we look into detail at the box-counting dimension of subordinators. Given that $X$ is a non-decreasing Levy process, which is not a compound Poisson process, we show that in the limit, the minimum number of boxes of size $\delta$ that cover the range of $(X_s)_{s\leq t}$ is a.s. of order $t/U(\delta)$, where $U$ is the potential function of $X$. This is a more refined result than the lower and upper index of the box-counting dimension computed in the literature which deals with the asymptotic number of boxes at logarithmic scale.