Stone-Weierstrass type theorems for large deviations

TL;DR

This paper extends large deviations theory by providing a general version of Bryc's theorem applicable to any topological space and algebra of functions, relaxing tightness conditions and generalizing classical results.

Contribution

It introduces a broad framework for large deviations without requiring exponential tightness, generalizes Prohorov-type theorems, and offers a variational form of rate functions on general spaces.

Findings

Generalized Bryc's theorem for any topological space

Relaxed conditions for large deviations without exponential tightness

Extended Prohorov-type theorems to arbitrary spaces

Abstract

We give a general version of Bryc's theorem valid on any topological space and with any algebra $\mathcal{A}$ of real-valued continuous functions separating the points, or any well-separating class. In absence of exponential tightness, and when the underlying space is locally compact regular and $\mathcal{A}$ constituted by functions vanishing at infinity, we give a sufficient condition on the functional $\Lambda(\cdot)_{\mid \mathcal{A}}$ to get large deviations with not necessarily tight rate function. We obtain the general variational form of any rate function on a completely regular space; when either exponential tightness holds or the space is locally compact Hausdorff, we get it in terms of any algebra as above. Prohorov-type theorems are generalized to any space, and when it is locally compact regular the exponential tightness can be replaced by a (strictly weaker) condition on $\Lambda(\cdot)_{\mid \mathcal{A}}$.