{c}
{wiki=Cramér's_theorem_(large_deviations)}

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>.

Cramér theorem

Put $Y=g(X)-\mathbb Eg(X)$ and $H(\lambda)=\log\mathbb Ee^{\lambda Y}$. Apply the assumed Poisson log-Sobolev inequality to $f=e^{\lambda g}$. Since $|Dg|\leq1$ and $|e^a-1|\leq\lambda e^\lambda$ for $|a|\leq\lambda$,
$$
\frac{|D(e^{\lambda g})(x)|^2}{e^{\lambda g(x)}}
=e^{\lambda g(x)}|e^{\lambda Dg(x)}-1|^2
\leq\lambda^2e^{2\lambda}e^{\lambda g(x)}.
$$
After division by $\mathbb Ee^{\lambda g}$, the entropy bound becomes
$$
\lambda H'(\lambda)-H(\lambda)\leq C\lambda^2e^{2\lambda}.
$$
Because $(H(\lambda)/\lambda)'=(\lambda H'-H)/\lambda^2$ and $H'(0)=0$, integration from $0$ to $\lambda$ gives
$$
H(\lambda)\leq\psi(\lambda)
:=\frac{C\lambda}{2}(e^{2\lambda}-1).
$$
The <Chernoff bound> and optimization over $\lambda\geq0$ therefore yield
$$
\mathbb P(Y\geq t)
\leq\exp\left\{-\sup_{\lambda\geq0}
[\lambda t-\psi(\lambda)]\right\}
=e^{-\psi^*(t)},
$$
where $\psi^*$ is the <Legendre transform of a cumulant-generating function>, also called the Chernoff-Cramér transform.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 208 / 3 / d / Solution

For an integrable real <random variable> $X$, its <cumulant-generating function> is the extended-real <convex function>
$$
\psi(\theta)=\log\mathbb E[e^{\theta X}],
\qquad \theta\in\mathbb R,
$$
where $\psi(\theta)=+\infty$ if the <exponential moment> diverges. Its <Legendre transform of a cumulant-generating function> is
$$
\psi^*(x)=\sup_{\theta\in\mathbb R}\{\theta x-\psi(\theta)\}.
$$

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 2 / a / Solution

{title2=$\psi^*$}
{wiki=Legendre_transformation}

The Legendre transform is the convex function
$$
\psi^*(x)=\sup_{\theta\in\mathbb R}\{\theta x-\psi(\theta)\}.
$$


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 Incoming links: Legendre transform of a cumulant-generating function