Varadhan's lemma

Asymptotic evaluation of exponential integrals under a large deviation principle.

Varadhan’s lemma

Varadhan’s lemma: Let $X$ be a Polish space and let $(\mu_n)$ be probability measures on $X$ that satisfy a large deviation principle with speed $a_n\to\infty$ and good rate function $I\colon X\to[0,\infty]$ . If $f\colon X\to\mathbb{R}$ is continuous and bounded above, then

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

Taking $f=-\varphi$ yields the Laplace principle . In many applications, the “goodness” of $I$ is obtained by combining an LDP with exponential tightness .