Rate function

Remarks

Example: $I(x) = x^2/2$ on $\R$ is a good rate function. It is continuous, so lower semicontinuous, and $\set{I \le a} = [-\sqrt{2a}, \sqrt{2a}]$ is compact.

Example: on $\R$, let $I(x) = x \log x + (1 - x) \log(1 - x) + \log 2$ for $x \in [0, 1]$, with $0 \log 0 = 0$, and $I(x) = \infty$ otherwise. It is a good rate function, the one for fair coin tossing in Cramér's theorem.

Non-example: on $\R$, let $I(0) = 1$ and $I(x) = 0$ for $x \neq 0$. Then $\set{I \le 0} = \R \setminus \set{0}$ is not closed, so $I$ is not lower semicontinuous.

Non-example of goodness: $I = 0$ on $\R$ is a rate function, but $\set{I \le 0} = \R$ is not compact.

In a large deviation principle, the rate function is unique when $\mathcal{X}$ is a regular topological space, for example a metric space. A good rate function attains its infimum on every non-empty closed set.