= Laplace principle for probability measures
{c}
{title2=$\lim_n a_n^{-1}\log\mathbb E e^{a_nf(X_n)}=\sup_x[f(x)-I(x)]$}
A family satisfies this principle with speed $a_n\to\infty$ and <rate function> $I$ when the displayed variational identity holds for every bounded <continuous> real function $f$. A <large deviation principle> implies it by the <Varadhan lemma>. The lower bound restricts the <expectation> to a neighborhood of a chosen point; the upper bound partitions the bounded range of $f$ into small intervals and applies the closed-set probability bounds.
For nonnegative $X_n$, the identity also gives
$$
-\lim_n a_n^{-1}\log\mathbb E e^{-\mu a_nX_n}
=\inf_{s\geq0}[I(s)+\mu s],\qquad \mu\geq0.
$$
For $\mu>0$, truncate $f(s)=-\mu s$ below at $-\mu R$. The resulting bounded-function identity applies, while the difference of the two exponential <expectations> is at most $e^{-\mu Ra_n}$. Choosing $R$ sufficiently large makes this error negligible on the claimed exponential scale. The case $\mu=0$ is immediate. This extension identifies moment-decay rates of a <passive scalar> from fluctuations of its logarithmic stretching.
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