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

: $\int_A e^{-\varphi(x)} \,dx < \infty.$

Then

: $\lim_{\theta \to \infty} \frac1{\theta} \log \int_A e^{-\theta \varphi(x)} \, dx = - \mathop{\mathrm{ess \, inf}}_{x \in A} \varphi(x),$

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

: $\int_A e^{-\theta \varphi(x)} \, dx \approx \exp \left(-\theta \mathop{\mathrm{ess \, inf}}_{x \in A} \varphi(x) \right).$

Application

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

: $\mathbf{P}_\theta (A) = \left( \int_A e^{-\theta \varphi(x)} \, dx \right) \bigg/ \left( \int_{\mathbf{R}^{d}} e^{-\theta \varphi(y)} \, dy \right)$

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

: $\lim_{\varepsilon \downarrow 0} \varepsilon \log \mathbf{P} \big[ \sqrt{\varepsilon} X \in A \big] = - \mathop{\mathrm{ess \, inf}}_{x \in A} \frac{x^2}{2}$

for every measurable set A.