Fractal-Dimensional Properties of Subordinators

Abstract

This work looks at the box-counting dimension of sets related to subordinators (non-decreasing L\'evy processes). It was recently shown in [Savov, 2014] that almost surely $\lim_{\delta\to0}U(\delta)N(t,\delta) = t$, where $N(t,\delta)$ is the minimal number of boxes of size at most $\delta$ needed to cover a subordinator's range up to time $t$, and $U(\delta)$ is the subordinator's renewal function. Our main result is a central limit theorem (CLT) for $N(t,\delta)$, complementing and refining work in [Savov, 2014]. Box-counting dimension is defined in terms of $N(t,\delta)$, but for subordinators we prove that it can also be defined using a new process obtained by shortening the original subordinator's jumps of size greater than $\delta$. This new process can be manipulated with remarkable ease in comparison to $N(t,\delta)$, and allows better understanding of the box-counting dimension of a subordinator's range in terms of its L\'evy measure, improving upon [Corollary 1, Savov, 2014]. Further, we shall prove corresponding CLT and almost sure convergence results for the new process.