6198 Townswood ct., Mississauga ON, L5N2L4, Canada

Received June 26, 2021.

Dedicated to the memory of I.V. Ostrovskii

Abstract

Ostrovskii's generalization of the Marcinkiewicz theorem implies that
if an entire characteristic functions of a probability distribution satisfies
$\log\log M(r,f)=o(r)$ and is zero-free then the distribution is normal. We
show that under the same growth condition, absence of zeros in a wide vertical strip implies that the distribution is close to a normal one. This generalizes and simplifies a recent result of Michelen and Sahasrabudhe.

Mathematics Subject Classification 2010: 60E10 Key words: characteristic function, ridge function, normal distribution