Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 224 / 2 / a / Solution

Let $X_1,X_2,\dots$ be i.i.d. with mass function $Q$ on a finite alphabet, and let ${\hat{P}}_n$ be their empirical distribution. Sanov theorem states that for every set $\Gamma$ of probability mass functions,andwhere interior and closure use the probability-simplex topology. Thus the empirical distributions satisfy a large deviation principle with rate function $D_e(\cdot \Vert Q)$.