Stochastic domination for iterated convolutions and catalytic majorization

TL;DR

This paper investigates conditions under which iterated convolutions of probability measures can be ordered via stochastic domination and extends these results to majorization in vector spaces, with applications to quantum information theory.

Contribution

It provides necessary and sufficient conditions for stochastic domination of iterated convolutions and extends the results to vector majorization, including applications to catalysis in quantum information.

Findings

Characterization of when $\mu^{*n}$ is stochastically dominated by $ u^{*n}$

Extension of stochastic domination results to vector majorization

Applications to quantum information theory catalysis

Abstract

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 $\mu^{*n}$ is stochastically dominated by $\nu^{*n}$ for two given probability measures $\mu$ and $\nu$. As a consequence we obtain a similar theorem on the majorization order for vectors in $\R^d$. In particular we prove results about catalysis in quantum information theory.