Gärtner–Ellis theorem

A large deviation principle obtained from limits of scaled log moment generating functions.

Gärtner–Ellis theorem

Gärtner–Ellis theorem: Let $(Z_n)$ be $\mathbb{R}^d$ -valued random variables and let $a_n\to\infty$ . Define the scaled log moment generating function

\Lambda_n(\theta)=\frac{1}{a_n}\log \mathbb{E}\bigl[\exp\bigl(a_n\langle \theta, Z_n\rangle\bigr)\bigr],\qquad \theta\in\mathbb{R}^d,

and assume the pointwise limit $\Lambda(\theta)=\lim_{n\to\infty}\Lambda_n(\theta)$ exists in $(-\infty,\infty]$ for all $\theta$ . Suppose that $\Lambda$ is lower semicontinuous, its effective domain $\{\theta:\Lambda(\theta)<\infty\}$ has nonempty interior, and $\Lambda$ is differentiable on the interior of its effective domain and steep (meaning $\|\nabla \Lambda(\theta_k)\|\to\infty$ whenever $(\theta_k)$ approaches the boundary of the effective domain). Then $(Z_n)$ satisfies a large deviation principle on $\mathbb{R}^d$ with speed $a_n$ and rate function

I(x)=\sup_{\theta\in\mathbb{R}^d}\bigl\{\langle \theta,x\rangle-\Lambda(\theta)\bigr\}.

The function $I$ is the Legendre–Fenchel transform of $\Lambda$ . For empirical means of i.i.d. real variables, this recovers Cramér's theorem under standard regularity assumptions.