Characterizing the asymptotic and catalytic stochastic orders on topological abelian groups

Understand

We study the usual stochastic order between probability measures on preordered topological abelian groups, focusing on asymptotic and catalytic versions of the order.

• In the asymptotic version, a measure $\mu$ dominates a measure $\nu$ if the i.i.d.~random walk generated by $\mu$ first-order dominates the one generated by $\nu$ at late times.
• In the catalytic version, $\mu$ dominates $\nu$ if there is a third $\tau$ such that the convolution $\mu \ast \tau$ first-order dominates $\nu \ast \tau$.
• Provided that the preorder on $G$ is induced by a suitably large positive cone and that both measures are compactly supported Radon, our main result gives a sufficient condition for asymptotic and catalytic dominance to hold in terms of a family of inequalities closely related to the cumulant-generating functions.