A proof of Sanov's Theorem via discretizations

TL;DR

This paper offers a new proof of Sanov's theorem for Polish spaces using discretization techniques, simplifying the proof by leveraging finite space results and controlling convergence rates.

Contribution

It introduces an alternative proof method for Sanov's theorem in Polish spaces through discretization and explicit convergence control.

Findings

Proof of Sanov's theorem via discretization

Explicit convergence rate control for measures

Simplified proof approach for Polish spaces

Abstract

We present an alternative proof of Sanov's theorem for Polish spaces in the weak topology that follows via discretization arguments. We combine the simpler version of Sanov's Theorem for discrete finite spaces and well chosen finite discretizations of the Polish space. The main tool in our proof is an explicit control on the rate of convergence for the approximated measures.