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

TL;DR

This paper demonstrates that the series representation for the density of the supremum of an alpha-stable Levy process can be rearranged to converge for all irrational alpha, extending previous results that only held for almost all irrationals.

Contribution

It introduces a method to rearrange the series to ensure convergence for all irrational alpha, strengthening earlier convergence results.

Findings

Series can be rearranged to converge for all irrational alpha.

The modification in the proof extends the applicability of the series representation.

The result clarifies the convergence behavior of the series for the supremum density.

Abstract

An absolutely convergent double series representation for the density of the supremum of $\alpha$-stable Levy process is given in [3, Theorem 2] 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 (see [6, Theorem 2]). 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 [3,Theorem 2].