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Fractal-dimensional properties of subordinators

Barker, A. (2019) Fractal-dimensional properties of subordinators. Journal of Theoretical Probability, 32 (3). pp. 1202-1219. ISSN 0894-9840

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To link to this item DOI: 10.1007/s10959-018-0813-5


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 limδ→0U(δ)N(t,δ)=t , where N(t,δ) is the minimal number of boxes of size at most δ needed to cover a subordinator’s range up to time t, and U(δ) is the subordinator’s renewal function. Our main result is a central limit theorem (CLT) for N(t,δ) , complementing and refining work in Savov (2014). Box-counting dimension is defined in terms of 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 δ . This new process can be manipulated with remarkable ease in comparison with 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.

Item Type:Article
Divisions:No Reading authors. Back catalogue items
ID Code:77131


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