 Legendre transform of a cumulant-generating function

The Legendre transform of a cumulant-generating function $K(\theta )=\log \mathbb{E}{e}^{\theta X}$ iswhere parameters with $K(\theta )=+\infty$ contribute nothing to the supremum. More generally this is the Legendre-Fenchel transform. It is convex because it is a supremum of affine functions of $x$, and is nonnegative because $K(0)=0$ allows the choice $\theta =0$.

If $K$ is differentiable and the supremum is attained at an interior parameter, its maximizer solves $K^{\prime }(\theta )=x$. For a Normal distribution with mean $\mu$ and variance ${\sigma }^2>0$, $K(\theta )=\mu \theta +{\sigma }^2{\theta }^2/2$, so $\theta =(x-\mu )/{\sigma }^2$ and $I(x)=(x-\mu {)}^2/(2{\sigma }^2)$. The zero cost occurs at the mean and the cost grows quadratically away from it.

For iid sample means, Cramér's theorem identifies this transform as the large-deviation Rate function when the Moment-generating function is finite on a neighbourhood of zero. The optimization also appears directly in a Chernoff bound: optimize the exponential Markov inequality over nonnegative parameters to bound an upper-tail probability. The transform converts exponential growth of a generating function into exponential decay of a rare-event probability.