Strassen's Positivstellensatz is a powerful but little known theorem on
preordered commutative semirings satisfying a boundedness condition similar to
Archimedeanicity. It characterizes the relaxed preorder induced by all monotone
homomorphisms to $\mathbb{R}_+$ in terms of a condition involving large powers.
Here, we generalize and strengthen Strassen's result. As a generalization, we
replace the boundedness condition by a polynomial growth condition; as a
strengthening, we prove two further equivalent characterizations of the
homomorphism-induced preorder in our generalized setting.
We then present two applications to large deviation theory, giving results on
the asymptotic comparison of one random walk relative to another. This gives a
probabilistic interpretation for one moment-generating function to dominate
another, in the context of bounded random variables.