Varadhan's lemma is a fundamental result in probability theory, particularly in the field of large deviations. It provides a way to evaluate the asymptotic behavior of certain probabilities as a parameter goes to infinity, often in the context of sequences of random variables or stochastic processes.
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If random elements satisfy a large deviation principle at speed 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.