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 Rd and let φ : Rd → R be a measurable function with
- [math]\displaystyle{ \int_A e^{-\varphi(x)} \,dx \lt \infty. }[/math]
Then
- [math]\displaystyle{ \lim_{\theta \to \infty} \frac1{\theta} \log \int_A e^{-\theta \varphi(x)} \, dx = - \mathop{\mathrm{ess \, inf}}_{x \in A} \varphi(x), }[/math]
where ess inf denotes the essential infimum. Heuristically, this may be read as saying that for large θ,
- [math]\displaystyle{ \int_A e^{-\theta \varphi(x)} \, dx \approx \exp \left(-\theta \mathop{\mathrm{ess \, inf}}_{x \in A} \varphi(x) \right). }[/math]
Application
The Laplace principle can be applied to the family of probability measures Pθ given by
- [math]\displaystyle{ \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) }[/math]
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
- [math]\displaystyle{ \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} }[/math]
for every measurable set A.
See also
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
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