On the Range of Subordinators

TL;DR

This paper investigates the box-counting dimension of subordinators, revealing that the minimal number of boxes needed to cover their range scales with the potential function, offering a refined understanding beyond previous logarithmic scale results.

Contribution

It provides a detailed analysis of the box-counting dimension of subordinators, establishing a precise asymptotic relation involving the potential function.

Findings

The minimal number of boxes scales as t/U(δ) almost surely.

Refines previous results by providing a non-logarithmic scale asymptotic.

Offers a deeper understanding of the geometric properties of subordinators.

Abstract

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.