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.
Highlights
Consider an α-stable Lévy process X, defined by the characteristic exponentΨ(z) = − ln E eizX1 = |z|α × eπiα( −ρ)1{z>0} + e−πiα( −ρ) 1{z
Our main result shows that the right way to compute the partial sums in (1.4), (1.5) is over triangles {m + αn < C : m ≥ 0, n ≥ 0} ⊂ Z2, and that one can always find an increasing sequence Ck → +∞, such that the partial sums will converge to p(x)
The following theorem is our main result in this note
Summary
In this case an absolutely convergent series representation for p(x) is given in [5, Theorem 10]. (iv) Theorem 2 in [6] states that the previous result cannot be substantially improved: There exists an uncountable dense subset L ⊂ L, such that for all α ∈ Land almost all ρ the series in (1.4), (1.5) do not converge absolutely for all x > 0.
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