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Asymptotic evaluation of exponential integrals under a large deviation principle.
Varadhan's lemma: Let be a Polish space and let be probability measures on that satisfy a large deviation principle with speed and good rate function . If is continuous and bounded above, then
Taking yields the Laplace principle. In many applications, the “goodness” of is obtained by combining an LDP with exponential tightness.
A probability measure is a function defined on a sigma-algebra of subsets of a set such that , for every pairwise disjoint sequence , and .
A probability measure is a special case of a measure and is the key ingredient in a probability space; it assigns probabilities to events (measurable sets).
A large deviation principle (LDP) for a sequence of probability measures on a topological space (with its Borel -algebra) consists of a speed with and a rate function such that:
A good rate function on a topological space is a rate function such that for every the sublevel set
is compact in .
Good rate functions are the natural large-deviation analogue of coercive “energy” functionals: they ensure that the variational problems appearing in a large deviation principle are attained on compact sets and interact well with exponential tightness. In metrizable settings (e.g. Polish spaces), goodness is often the key compactness hypothesis used to pass from bounds on nice sets to bounds on all Borel sets.
A Laplace principle for a sequence of probability measures on a space , with speed and rate function , is the statement that for every bounded continuous function ,
A sequence of probability measures on a topological space is exponentially tight at speed with if for every there exists a compact set such that
Exponential tightness is a strengthened form of ordinary tightness for probability measures: it not only forces most mass into compacts, but does so with exponentially small tails at the LDP speed. It is frequently paired with a rate function to obtain or upgrade a large deviation principle, and it is a standard hypothesis in results like the Gärtner–Ellis theorem.