Fractal-Dimensional Properties of Subordinators

This work looks at the box-counting dimension of sets related to subordinators (non-decreasing Lévy processes). It was recently shown in Savov (Electron Commun Probab 19:1–10, 2014) that almost surely $$\lim _{\delta \rightarrow 0}U(\delta )N(t,\delta ) = t$$ lim δ → 0 U ( δ ) N ( t , δ ) = t , where $$N(t,\delta )$$ N ( t , δ ) is the minimal number of boxes of size at most $$ \delta $$ δ needed to cover a subordinator’s range up to time t, and $$U(\delta )$$ U ( δ ) is the subordinator’s renewal function. Our main result is a central limit theorem (CLT) for $$N(t,\delta )$$ N ( t , δ ) , complementing and refining work in Savov (2014). Box-counting dimension is defined in terms of $$N(t,\delta )$$ N ( t , δ ) , 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 $$\delta $$ δ . This new process can be manipulated with remarkable ease in comparison with $$N(t,\delta )$$ N ( t , δ ) , and allows better understanding of the box-counting dimension of a subordinator’s range in terms of its Lévy measure, improving upon Savov (2014, Corollary 1). Further, we shall prove corresponding CLT and almost sure convergence results for the new process.