  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.
