Varadhan's lemma (${\lim }_n{a}_n^{-1}\log \mathbb{E}{e}^{a_{n}f(X_n)}={\sup }_x(f(x)-I(x))$)

If random elements satisfy a large deviation principle at speed $a_n\to \infty$ with rate function I, then a bounded continuous real function f satisfies the displayed exponential-integral identity. For the upper bound, partition its bounded range into finitely many closed intervals of width epsilon and apply the closed-set upper bound to their preimages. The largest exponential term controls the finite sum. For the lower bound, restrict to an open neighborhood where f is within epsilon of its value at a chosen point and use the open-set lower bound. Let epsilon tend to zero and optimize over the point. Boundedness avoids a separate exponential tail assumption.

For a continuous $F$ and a family satisfying a large deviation principle at speed $L$, the displayed exponential tail condition extends Varadhan's lemma to $F$ unbounded above. Apply the bounded-above result to $\min (F,M)$ and split the full integral into $F\le M$ and its tail; the tail is negligible and the truncated variational suprema increase to ${\sup }_x(F(x)-I(x))$. A sufficient condition is ${\mathrm{limsup}}_L{L}^{-1}\log \mathbb{E}{e}^{\gamma LF(X_L)}<\infty$ for some $\gamma >1$, since the tail is at most $e^{-(\gamma -1)LM}\mathbb{E}{e}^{\gamma LF(X_L)}$.