Positive stable densities and the bell-shape

TL;DR

This paper proves that positive stable densities are bell-shaped, with derivatives that vanish exactly n times and alternate in sign, confirming earlier graphical predictions.

Contribution

It establishes the bell-shape property for positive stable densities, a previously conjectured characteristic, providing a rigorous mathematical proof.

Findings

Positive stable densities are bell-shaped.

Derivatives of these densities vanish exactly n times.

Sign of derivatives alternates with order.

Abstract

We show that positive stable densities are bell-shaped, that is their n-th derivatives vanish exactly n times on (0,+oo) and have an alternating sign sequence. This confirms the graphic predictions of Holt and Crow (1973) in the positive case.