  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.
