Cramér transform - Knowlpedia

A Cramér transform associated with a log moment generating function $\Lambda:\mathbb R^d\to(-\infty,\infty]$ is the function $I:\mathbb R^d\to[0,\infty]$ defined by

I(x)=\sup_{\theta\in\mathbb R^d}\big\{\langle \theta,x\rangle-\Lambda(\theta)\big\}.

This is the Legendre–Fenchel transform of $\Lambda$ (equivalently, the Fenchel conjugate of $\Lambda$ ).

Remarks

In large deviations, when $\Lambda$ is the log moment generating function of a random variable, $I$ is the canonical candidate rate function for the LDP of empirical means; this is made precise by Cramér's theorem and, in broader settings, by the Gärtner–Ellis theorem.

Examples

If $\Lambda(\theta)=\frac{\sigma^2\theta^2}{2}$ on $\mathbb R$ (Gaussian case), then

I(x)=\frac{x^2}{2\sigma^2}.

If $X\sim \mathrm{Bernoulli}(p)$ with log-MGF $\Lambda(\theta)=\log\big((1-p)+p e^{\theta}\big)$ , then the Cramér transform is

I(x)= \begin{cases} x\log\!\left(\frac{x}{p}\right)+(1-x)\log\!\left(\frac{1-x}{1-p}\right), & x\in[0,1],\\ +\infty, & x\notin[0,1]. \end{cases}