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

${\displaystyle \int _{A}e^{-\varphi (x)}\,dx<\infty .}$

Then

${\displaystyle \lim _{\theta \to \infty }{\frac {1}{\theta }}\log \int _{A}e^{-\theta \varphi (x)}\,dx=-\mathop {\mathrm {ess\,inf} } _{x\in A}\varphi (x),}$

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

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

Application

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

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

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

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

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