Stability in the Marcinkiewicz theorem

TL;DR

This paper explores conditions under which a probability distribution is close to normal, based on the growth and zero-free properties of its characteristic function, extending previous results with a simpler approach.

Contribution

It generalizes and simplifies a recent result by showing that zero-free characteristic functions with specific growth imply near-normality of the distribution.

Findings

Zero-free characteristic functions with specific growth imply near-normality.

The result generalizes previous theorems and simplifies their proofs.

Distribution is close to normal under the given conditions.

Abstract

Ostrovskii's generalization of the Marcinkiewicz theorem implies that if an entire characteristic functions of a probability distribution satisfies $\log^+\log|f(z)|=o(|z|),\; z\to\infty,$ 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.