Good rate function

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

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.