Stability in the Marcinkiewicz theorem

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.

6 pages

References in corpus (1)

• Central limit theorems and the geometry of polynomials