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 doi: 10.1007/s10959-018-0813-5

Abstract/Summary

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
URI	https://centaur.reading.ac.uk/id/eprint/77131
Identification Number/DOI	10.1007/s10959-018-0813-5
Refereed	Yes
Divisions	No Reading authors. Back catalogue items
Publisher	Springer
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