  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.

Probability