Singular infinitely divisible distributions whose characteristic functions vanish at infinity

Abstract

In the course of discussing dynamical systems which enjoy strong mixing but have singular spectrum, E. Hewitt and the author, (2), recently constructed families of symmetric random variables which satisfy inter alia the following properties:(i) Zt is purely singular and has full support,(ii) χ(t)(x) → 0 as ± x → ∞, where χ(t) is the characteristic function of Zt,(iii)′ Zt+s, Zt + Zs have the same null events,(iv) whenever s ≠ t, Zt and Zs + a are mutually singular for every (possibly zero) constant a.