Journal of Mathematical Physics, Analysis, Geometry 2021, vol. 17, No 4, pp. 463-467   https://doi.org/10.15407/mag17.04.463     ( to contents , go back )

### Stability in the Marcinkiewicz theorem

Alexandre Eremenko

Purdue University, West Lafayette IN 47907, USA
E-mail: eremenko@purdue.edu

Alexander Fryntov

6198 Townswood ct., Mississauga ON, L5N2L4, Canada

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.