Electronic Communications in Probability
- Electron. Commun. Probab.
- Volume 18 (2013), paper no. 42, 5 pp.
A note on the series representation for the density of the supremum of a stable process
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.
Electron. Commun. Probab., Volume 18 (2013), paper no. 42, 5 pp.
Accepted: 6 June 2013
First available in Project Euclid: 7 June 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60G52: Stable processes
This work is licensed under a Creative Commons Attribution 3.0 License.
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