Large deviation principle

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A large deviation principle (LDP) for a sequence of probability measures $(\mu_n)_{n\ge 1}$ on a topological space $E$ (with its Borel $\sigma$ -algebra) consists of a speed $(a_n)_{n\ge 1}$ with $a_n\to\infty$ and a rate function $I:E\to[0,\infty]$ such that:

for every closed set $F\subseteq E$ , $\limsup_{n\to\infty}\frac{1}{a_n}\log \mu_n(F)\le -\inf_{x\in F} I(x),$

for every open set $G\subseteq E$ , $\liminf_{n\to\infty}\frac{1}{a_n}\log \mu_n(G)\ge -\inf_{x\in G} I(x).$

Typical setting

In many applications, $\mu_n$ is the law of a sequence of random variables defined on a probability space. Additional structure such as a good rate function or exponential tightness often ensures useful compactness properties and helps upgrade “local” bounds to a full LDP.

Examples

If $(X_i)_{i\ge1}$ is an i.i.d. sequence with suitable exponential moments, the empirical mean $\bar X_n=\frac1n\sum_{i=1}^n X_i$ satisfies an LDP at speed $a_n=n$ ; this is the content of Cramér's theorem.
The empirical measure of i.i.d. samples on a finite alphabet satisfies an LDP at speed $n$ with a relative-entropy-type rate; this is Sanov's theorem.