Stochastic domination for iterated convolutions and catalytic majorization

We study how iterated convolutions of probability measures compare under stochastic domination. We give necessary and sufficient conditions for the existence of an integer $n$ such that ${\mu }^{\ast n}$ is stochastically dominated by ${\nu }^{\ast n}$ for two given probability measures $\mu$ and $\nu$. As a consequence we obtain a similar theorem on the majorization order for vectors in ${\mathbb{R}}^d$. In particular we prove results about catalysis in quantum information theory.