Laplace principle (large deviations theory)

In mathematics, Laplace's principle is a basic theorem in large deviations theory which is 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

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

$\underset{A}{\int }{e}^{-\phi (x)}dx<\mathrm{\infty }.$

Then

$\underset{\theta \to \mathrm{\infty }}{\lim }\frac{1}{\theta }\log \underset{A}{\int }{e}^{-\theta \phi (x)}dx=-{\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{f}}_{x\in A}\phi (x),$

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

$\underset{A}{\int }{e}^{-\theta \phi (x)}dx\approx \exp (-\theta {\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{f}}_{x\in A}\phi (x)).$

Application

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

${\mathbf{𝐏}}_{\theta }(A)=\left(\underset{A}{\int }{e}^{-\theta \phi (x)}dx\right)/\left(\underset{{\mathbf{𝐑}}^d}{\int }{e}^{-\theta \phi (y)}dy\right)$

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

$\underset{\epsilon \downarrow 0}{\lim }\epsilon \log \mathbf{𝐏}[\sqrt{\epsilon }X\in A]=-{\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{f}}_{x\in A}\frac{x^2}{2}$

for every measurable set A.

See also

• Laplace's method

References

• 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