Cumulant-generating function (${\psi }_X(\theta )$)

The cumulant-generating function is ${\psi }_X(\theta )=\log \mathbb{E}{e}^{\theta X}$, interpreted as an extended-real convex function where the exponential moment may be infinite.

The Legendre transform is the convex function

For independent identically distributed real random variables, the empirical mean satisfies a large-deviation principle with rate function given by the Legendre transform of a cumulant-generating function.

Exponential tilting by $\theta$ replaces a law $P$ byUnder the tilted law, the mean is ${\psi }^{\prime }(\theta )$ whenever the derivative exists.