A proof of Sanov's theorem via discretizations (Q6046202)

The authors present an alternative proof of Sanov's theorem for Polish spaces in the weak topology that follows via discretization arguments. Sanov's theorem is a well-known result in the theory of large deviations principles. It provides the large deviations profile of the empirical measure of a sequence of i.i.d. random variables and characterizes its rate function as the relative entropy. So, the present paper provides an alternative proof of this fact, by exploring the metric structure of the weak topology with the variational formulation of the relative entropy. The proof does not require profound knowledge of large deviations theory or general topology. Instead, the authors combine the simpler version of Sanov's theorem for discrete finite spaces and well-chosen finite discretizations of the Polish space. The main tool of the proof is an explicit control of the rate of convergence for the approximated measures.