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A note on the series representation for the density of the supremum of a stable process. (English) Zbl 1323.60065
Summary: An absolutely convergent double series representation for the density of the supremum of an \(\alpha\)-stable Lévy process was obtained by F. Hubalek and A. Kuznetsov [Electron. Commun. Probab. 16, 84–95, electronic only (2011; Zbl 1231.60040)] 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 the 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 [loc. cit.].

MSC:
60G52 Stable stochastic processes
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