Fractal-Dimensional Properties of Subordinators

Adam Barker ()
Additional contact information
Adam Barker: University of Reading

Journal of Theoretical Probability, 2019, vol. 32, issue 3, 1202-1219

Abstract: Abstract 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.

Keywords: Lévy processes; Subordinators; Fractal dimension; Box-counting dimension; Primary 60G51; 28A80; Secondary 60G75; 60F05; 60F15 (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-018-0813-5 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:32:y:2019:i:3:d:10.1007_s10959-018-0813-5

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-018-0813-5

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().