Sanov's theorem

In mathematics and information theory, Sanov's theorem gives a bound on the probability of observing an atypical sequence of samples from a given probability distribution. In the language of large deviations theory, Sanov's theorem identifies the rate function for large deviations of the empirical measure of a sequence of i.i.d. random variables.
Let A be a set of probability distributions over an alphabet X, and let q be an arbitrary distribution over X. Suppose we draw n i.i.d. samples from q, represented by the vector. Then, we have the following bound on the probability that the empirical measure of the samples falls within the set A:
where

is the joint probability distribution on, and
is the information projection of q onto A.
, the KL divergence, is given by:

In words, the probability of drawing an atypical distribution is bounded by a function of the KL divergence from the true distribution to the atypical one; in the case that we consider a set of possible atypical distributions, there is a dominant atypical distribution, given by the information projection.
Furthermore, if A is a closed set, then

Technical statement

Define:

is a finite set with size. Understood as “alphabet”.
is the simplex spanned by the alphabet. It is a subset of.
is a random variable taking values in. Take samples from the distribution, then is the frequency probability vector for the sample.
is the space of values that can take. In other words, it is

Then, Sanov's theorem states:

For every measurable subset,
For every open subset,

Here, means the interior, and means the closure.