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


 
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1. Title Title of document A note on the series representation for the density of the supremum of a stable process
 
2. Creator Author's name, affiliation, country Daniel Hackmann; York University; Canada
 
2. Creator Author's name, affiliation, country Alexey Kuznetsov; York University; Canada
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) stable processes, supremum, Mellin transform, continued fractions
 
3. Subject Subject classification 60G52
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Natural Sciences and Engineering Research Council of Canada
 
7. Date (YYYY-MM-DD) 2013-06-06
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/2757
 
10. Identifier Digital Object Identifier 10.1214/ECP.v18-2757
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 18
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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