#jsDisabledContent { display:none; } My Account |  Register |  Help

# Laplace principle (large deviations theory)

Article Id: WHEBN0012795419
Reproduction Date:

 Title: Laplace principle (large deviations theory) Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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.

## Contents

• Statement of the result 1
• Application 2
• References 4

## 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

\int_A e^{- \varphi(x)} \, \mathrm{d} x < + \infty.

Then

\lim_{\theta \to + \infty} \frac1{\theta} \log \int_{A} e^{- \theta \varphi(x)} \, \mathrm{d} x = - \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)} \, \mathrm{d} x \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 Pθ given by

\mathbf{P}_{\theta} (A) = \left( \int_{A} e^{- \theta \varphi(x)} \, \mathrm{d} x \right) \Big/ \left( \int_{\mathbf{R}^{d}} e^{- \theta \varphi(y)} \, \mathrm{d} y \right)

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

\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.

## References

• Dembo, Amir; Zeitouni, Ofer (1998). Large deviations techniques and applications. Applications of Mathematics (New York) 38 (Second edition ed.). New York: Springer-Verlag. pp. xvi+396. MR 1619036
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.