Laplace principle - Imported Knowlpedia Content

A Laplace principle for a sequence of probability measures $(\mu_n)$ on a space $X$ , with speed $a_n\to\infty$ and rate function $I\colon X\to[0,\infty]$ , is the statement that for every bounded continuous function $\varphi\colon X\to\mathbb{R}$ ,

\lim_{n\to\infty}\frac{1}{a_n}\log \int_X \exp\bigl(-a_n\varphi(x)\bigr)\,\mu_n(dx) = -\inf_{x\in X}\bigl\{\varphi(x)+I(x)\bigr\}.

This is the Laplace-transform formulation of the large deviation principle. Under standard hypotheses (for example, $X$ Polish and $(\mu_n)$ exponentially tight), the Laplace principle with a good rate function is equivalent to an LDP with the same rate function.

Examples

By Cramér's theorem, the empirical mean of an i.i.d. sequence of real-valued random variables satisfies the Laplace principle with speed $n$ and rate given by the Cramér transform.
By Sanov's theorem, the empirical measure of an i.i.d. sample satisfies the Laplace principle with speed $n$ and rate given by relative entropy with respect to the common law.