Exponential Approximations in Completely Regular Topological Spaces and Extensions of Sanov’s Theorem

This paper is devoted to the well known transformations that preserve a large deviation principle (LDP), namely, the contraction principle with approximately continuous maps and the concepts of exponential equivalence and exponential approximations. We generalize these transformations to completely regular topological state spaces, give some examples and, as an illustration, reprove a generalization of Sanov’s theorem, due to de Acosta [1]. Using partition-dependent couplings, we then extend this version of Sanov’s theorem to triangular arrays and prove a LDP for the empirical measures of exchangeable sequences with a general measurable state space.