Unimodality of the Lévy spectral function

Carol Alf and Thomas Alfonso O’Connor

Vol. 69 (1977), No. 2, 285–290

DOI: 10.2140/pjm.1977.69.285

Abstract

A. Ya. Khinchin proved that if Φ and Ψ are characteristic functions and Φ(t) = t⁻¹ ∫ ₀^tΨ(u)du, then the distribution function of Φ is convex on (−∞,0) and concave on (0,+∞). A similar theorem is proved here for logarithms of infinitely divisible characteristic functions and their Lévy spectral functions.

Mathematical Subject Classification 2000

Primary: 60E05

Milestones

Received: 20 April 1976

Revised: 11 October 1976

Published: 1 April 1977