= Gärtner–Ellis theorem
{c}
{title2=$I=\Lambda^*$}
Let $X_N\in\mathbb R^d$ and $a_N\to\infty$. Define the scaled <cumulant-generating function>
$$
\Lambda_N(\theta)=a_N^{-1}\log\mathbb E e^{a_N\langle\theta,X_N\rangle}.
$$
Suppose its pointwise limit $\Lambda$ exists, is <lower semicontinuous>, has $0$ in the interior of its effective domain, and is essentially smooth: it is differentiable on that interior, which is nonempty, and its gradient norm diverges at every finite boundary point approached from the interior. Then $X_N$ satisfies a <large deviation principle> with <good rate function> given by the <Legendre-Fenchel transform>
$$
\Lambda^*(x)=\sup_\theta\{\langle\theta,x\rangle-\Lambda(\theta)\}.
$$
The <Chernoff bound> supplies the upper estimate. The lower estimate uses <exponential tilting> at a parameter whose gradient selects the required mean; essential smoothness supplies the boundary approximation needed for the full lower bound. A finite boundary slope can leave an affine branch of $\Lambda^*$ outside this argument, so a separate lower-bound proof is then necessary.
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