Definition:Large Deviation Principle

Definition

Let $X$ be a topological space.

Let $\BB$ be a $\sigma$-algebra over $X$.

Let $\sequence {\mu_\epsilon}_{\epsilon \in \R_{>0} }$ be a sequence of probability measures on $\struct {X, \BB}$.

Then $\sequence {\mu_\epsilon}_{\epsilon \in \R_{>0} }$ satisfies the large deviation principle with a rate function $I$ if and only if:

$\ds - \inf_{x \mathop \in \Gamma^\circ} \map I x \le \liminf_{\epsilon \mathop \to 0} \epsilon \log \map {\mu_\epsilon} \Gamma \le \limsup_{\epsilon \mathop\to 0} \epsilon \log \map {\mu_\epsilon} \Gamma \le - \inf_{x \mathop \in \Gamma^-} \map I x$

for all $\Gamma \in \BB$, where:

$\Gamma^\circ$ is the interior of $\Gamma$

$\Gamma^-$ is the closure of $\Gamma$

Sources

• 1998: Amir Dembo and Ofer Zeitouni: Large Deviations Techniques and Applications (2nd ed.): $1.2$ The Large Deviation Principle