Characterizing the asymptotic and catalytic stochastic orders on topological abelian groups

Abstract

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. While this sufficient condition requires these inequalities to be strict, the non-strict versions of these inequalities are easily seen to be necessary. In this sense, our result gives conditions that are necessary and sufficient in generic cases. This result has been known for $G = \mathbb{R}$, but is new already for $\mathbb{R}^n$ with $n > 1$. It is a direct application of a recently proven theorem of real algebra, namely a \emph{Vergleichsstellensatz} for preordered semirings. We finally use our result to derive a formula for the rate at which the probabilities of a random walk decay \emph{relative} to those of another, now for walks on a preordered topological vector space with compactly supported Radon steps. Taking one of these walks to be deterministic reproduces a version of Cram\'er's large deviation theorem for infinite dimensions.