Definition:Good Rate Function

Definition

Let $X$ be a topological space.

Let $\overline \R_\ge$ be the positive extended real number line.

Let $I : X \to \overline \R_\ge$ be a rate function.

Then $I$ is good if and only if for all $\alpha \in \R_{\ge 0}$:

$\operatorname {lev} \limits_{\mathop \le \alpha} I$ is compact

where $\operatorname {lev} \limits_{\mathop \le \alpha} I$ denotes the lower level set of $I$ at $\alpha$.

Sources

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