= Exponential tail extension of Varadhan's lemma
{title2=$\lim_{M\to\infty}\limsup_L L^{-1}\log\mathbb E[e^{LF(X_L)};F(X_L)>M]=-\infty$}
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 $F\leq M$ and its tail; the tail is negligible and the truncated variational suprema increase to $\sup_x(F(x)-I(x))$. A sufficient condition is $\limsup_L L^{-1}\log\mathbb E e^{\gamma LF(X_L)}<\infty$ for some $\gamma>1$, since the tail is at most $e^{-(\gamma-1)LM}\mathbb E e^{\gamma LF(X_L)}$.
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