Good rate function - Imported Knowlpedia Content

A good rate function on a topological space $E$ is a rate function $I:E\to[0,\infty]$ such that for every $\alpha<\infty$ the sublevel set

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

is compact in $E$ .

Good rate functions are the natural large-deviation analogue of coercive “energy” functionals: they ensure that the variational problems appearing in a large deviation principle are attained on compact sets and interact well with exponential tightness. In metrizable settings (e.g. Polish spaces), goodness is often the key compactness hypothesis used to pass from bounds on nice sets to bounds on all Borel sets.

Examples

On $E=\mathbb R^d$ , any function of the form $I(x)=\|x\|^2$ is good: the sets $\{x:\|x\|^2\le \alpha\}$ are closed and bounded, hence compact in $\mathbb R^d$ .
If $E$ is compact and $I$ is a rate function, then $I$ is automatically good because every closed subset of a compact space is compact.