Laplace principle (large deviations theory)

In mathematics, Laplace's principle is a basic theorem in large deviations theory, similar to Varadhan's lemma. It gives an asymptotic expression for the Lebesgue integral of exp(−θφ(x)) over a fixed set A as θ becomes large. Such expressions can be used, for example, in statistical mechanics to determining the limiting behaviour of a system as the temperature tends to absolute zero.

Let A be a Lebesgue-measurable subset of d-dimensional Euclidean space R^d and let φ : R^d → R be a measurable function with

Then

where ess inf denotes the essential infimum. Heuristically, this may be read as saying that for large θ,

The Laplace principle can be applied to the family of probability measures P_θ given by

to give an asymptotic expression for the probability of some set/event A as θ becomes large. For example, if X is a standard normally distributed random variable on R, then

for every measurable set A.

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