Rate function

A rate function on a topological space $E$ is a function $I:E\to[0,\infty]$ that is lower semicontinuous, meaning that for every $\alpha\in\mathbb R$ the sublevel set

is closed in $E$, and such that $I$ is not identically $+\infty$.

Rate functions quantify the exponential scale of rare-event probabilities in a large deviation principle : heuristically, $\mu_n(A)\approx \exp(-a_n \inf_{x\in A} I(x))$ for large $n$ and speed $a_n$. A particularly well-behaved class is given by good rate functions , whose sublevel sets are compact.

Examples:

• On $E=\mathbb R$, the function $I(x)=\frac{x^2}{2}$ is a rate function (it is continuous, hence lower semicontinuous).

• For a closed set $C\subseteq E$, the indicator-type function

  $$I(x)=\begin{cases} 0,& x\in C,\\ +\infty,& x\notin C, \end{cases}$$

  is a rate function; it forces mass to concentrate on $C$ at the exponential scale.