Fractal-Dimensional Properties of Subordinators

TLDR: In this paper, the authors studied the box-counting dimension of sets related to subordinators (non-decreasing Levy processes) and proved a central limit theorem for this dimension.

Abstract: This work looks at the box-counting dimension of sets related to subordinators (non-decreasing Levy processes). It was recently shown in Savov (Electron Commun Probab 19:1–10, 2014) that almost surely $$\lim _{\delta \rightarrow 0}U(\delta )N(t,\delta ) = t$$ , 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 with $$N(t,\delta )$$ , and allows better understanding of the box-counting dimension of a subordinator’s range in terms of its Levy measure, improving upon Savov (2014, Corollary 1). Further, we shall prove corresponding CLT and almost sure convergence results for the new process.