Laplace principle (large deviations theory)

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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.

Statement of the result[edit]

Let A be a Lebesgue-measurable subset of d-dimensional Euclidean space Rd and let φ : Rd → 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,

Application[edit]

The Laplace principle can be applied to the family of probability measures Pθ 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.

See also[edit]

References[edit]

  • Dembo, Amir; Zeitouni, Ofer (1998). Large deviations techniques and applications. Applications of Mathematics (New York) 38 (Second ed.). New York: Springer-Verlag. pp. xvi+396. ISBN 0-387-98406-2.  MR1619036