# Fractal-Dimensional Properties of Subordinators

## TL;DR

This paper investigates the fractal properties of subordinators, establishing a central limit theorem for the box-counting covering number, and introduces a new process to better understand the subordinator's dimension in relation to its Lévy measure.

## Contribution

It provides a CLT for the box-counting number of subordinators and introduces a new process simplifying the analysis of their fractal dimensions.

## Key findings

- Established a CLT for the number of boxes needed to cover a subordinator's range.
- Proposed a new process by truncating jumps, facilitating easier analysis of fractal properties.
- Connected the box-counting dimension to the Lévy measure of the subordinator.

## 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.