 Exponential tail extension of Varadhan's lemma

ID: exponential-tail-extension-of-varadhan-s-lemma

For a continuous

F

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

lim sup_{L} L^{- 1} lo g E e^{γ L F (X_{L})} < \infty

for some

γ > 1

, since the tail is at most

e^{- (γ - 1) L M} E e^{γ L F (X_{L})}

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