Electronic Communications in Probability

A note on the series representation for the density of the supremum of a stable process

Daniel Hackmann and Alexey Kuznetsov

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Abstract

An absolutely convergent double series representation for the density of the supremum of $\alpha$-stable Lévy process was obtained by Hubalek and Kuznetsov for almost all irrational $\alpha$. This result cannot be made stronger in the following sense: the series does not converge absolutely when $\alpha$ belongs to a certain subset of irrational numbers of Lebesgue measure zero. Our main result in this note shows that for every irrational $\alpha$ there is a way to rearrange the terms of the double series, so that it converges to the density of the supremum. We show how one can establish this stronger result by introducing a simple yet non-trivial modification in the original proof of Hubalek and Kuznetsov.

Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 42, 5 pp.

Dates
Accepted: 6 June 2013
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465315581

Digital Object Identifier
doi:10.1214/ECP.v18-2757

Mathematical Reviews number (MathSciNet)
MR3070908

Zentralblatt MATH identifier
1323.60065

Subjects
Primary: 60G52: Stable processes

Keywords
stable processes supremum Mellin transform continued fractions

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Hackmann, Daniel; Kuznetsov, Alexey. A note on the series representation for the density of the supremum of a stable process. Electron. Commun. Probab. 18 (2013), paper no. 42, 5 pp. doi:10.1214/ECP.v18-2757. https://projecteuclid.org/euclid.ecp/1465315581


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References

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