Rate function

Summary

A rate function is a lower semicontinuous function $I$ from a topological space to $[0, \infty]$. In large deviations theory, it gives the exponential rate at which the probability of a rare event decays: roughly, $P(Z_n \approx x) \approx e^{-n I(x)}$. It is called good when its sublevel sets are compact. Examples are the Legendre transforms in Cramér's theorem and the Kullback–Leibler divergence in Sanov's theorem.