A generalization of Strassen's Positivstellensatz and its application to large deviation theory
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Submission date: 04. Oct. 2018
MSC-Numbers: 06F25, 16Y60, 60F10, 12J15, 14P10
Keywords and phrases: ordered semiring, abstract Positivstellensatz, large deviations
Link to arXiv:See the arXiv entry of this preprint.
Strassen’s Positivstellensatz is a powerful but little known result for preordered semirings satisfying a boundedness condition, and characterizes the relaxed preorder induced by all monotone homomorphisms to ℝ+ in terms of a condition involving large powers. Here, we generalize and strengthen Strassen’s theorem: 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 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.