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.

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)}}\,{\mathrm {d}}x<+\infty .

Then

\lim _{{\theta \to +\infty }}{\frac 1{\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 P_θ 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 ed.). New York: Springer-Verlag. pp. xvi+396. ISBN 0-387-98406-2. MR 1619036

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Laplace principle (large deviations theory)

Statement of the result

Application

See also

References