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

**Daniel Hackmann**

*(York University)*

**Alexey Kuznetsov**

*(York University)*

#### 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.

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Pages: 1-5

Publication Date: June 6, 2013

DOI: 10.1214/ECP.v18-2757

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