Laplace principle for probability measures (${\lim }_n{a}_n^{-1}\log \mathbb{E}{e}^{a_{n}f(X_n)}={\sup }_x[f(x)-I(x)]$)

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 givesFor $\mu >0$, 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 $e^{-\mu R{a}_n}$. 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.