Rate function - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A rate function on a topological space $E$ is a lower semicontinuous function $I:E\to[0,\infty]$ that is not identically $+\infty$ . Equivalently, for every $\alpha\in\mathbb R$ , its sublevel set

\{x\in E:I(x)\le \alpha\}

is closed in $E$ .

Interpretation

In a large deviation principle, $I$ governs exponential decay: values of $I$ closer to zero correspond to less strongly suppressed outcomes. A good rate function additionally has compact sublevel sets.

Examples

On $E=\mathbb R$ , the function $I(x)=x^2/2$ is a rate function.
For a nonempty closed set $C\subseteq E$ , the function $I(x)=\begin{cases} 0,&x\in C,\\ +\infty,&x\notin C \end{cases}$ is a rate function.