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

2013/05/03 by Hackmann, Daniel, Kuznetsov, Alexey
#60G52 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1305.0722

Abstract

An absolutely convergent double series representation for the density of the supremum of α-stable Levy process is given in [3, Theorem 2] for almost all irrational α. This result cannot be made stronger in the following sense: the series does not converge absolutely when α belongs to a certain subset of irrational numbers of Lebesgue measure zero (see [6, Theorem 2]). Our main result in this note shows that for every irrational α 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 [3,Theorem 2].

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