Log moment generating function

The logarithm of the moment generating function, viewed as a convex functional of the parameter.

Log moment generating function

A log moment generating function (log-MGF) of an $\mathbb R^d$ -valued random variable $X$ is the function $\Lambda:\mathbb R^d\to(-\infty,\infty]$ defined by

\Lambda(\theta)=\log \mathbb E\big[e^{\langle \theta, X\rangle}\big],

where $\langle\theta,X\rangle$ is the Euclidean inner product and the expectation is taken in the sense of expectation . Equivalently, if $\mu$ is the law of $X$ , then

\Lambda(\theta)=\log\int_{\mathbb R^d} e^{\langle\theta,x\rangle}\,\mu(dx),

where the integral is a special case of the Lebesgue integral .

The log-MGF is a central object in large deviations: it is convex and encodes exponential moment growth, and its convex dual gives the Cramér transform . In particular, for sums of an i.i.d. sequence , the log-MGF is the starting point for Cramér's theorem and the Gärtner–Ellis theorem .

Examples:

If $X\sim \mathcal N(0,\sigma^2)$ on $\mathbb R$ , then $\Lambda(\theta)=\frac{\sigma^2\theta^2}{2}$ for all $\theta\in\mathbb R$ .
If $X\sim \mathrm{Bernoulli}(p)$ on $\{0,1\}$ , then $\Lambda(\theta)=\log\big((1-p)+p e^{\theta}\big)$ for all $\theta\in\mathbb R$ .