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.

tatement 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

:

Then

:$lim\_\{\; heta\; o\; +\; infty\}\; frac1\{\; heta\}\; log\; int\_\{A\}\; e^\{-\; heta\; varphi(x)\}\; ,\; mathrm\{d\}\; x\; =\; -\; mathop\{mathrm\{ess\; ,\; inf\_\{x\; in\; A\}\; varphi(x),$

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

:$int\_\{A\}\; e^\{-\; heta\; varphi(x)\}\; ,\; mathrm\{d\}\; x\; approx\; exp\; left(\; -\; heta\; mathop\{mathrm\{ess\; ,\; inf\_\{x\; in\; A\}\; varphi(x)\; ight).$

Application

The Laplace principle can be applied to the family of probability measures P_"θ" given by

:$mathbf\{P\}\_\{\; heta\}\; (A)\; =\; left(\; int\_\{A\}\; e^\{-\; heta\; varphi(x)\}\; ,\; mathrm\{d\}\; x\; ight)\; Big/\; left(\; int\_\{mathbf\{R\}^\{d\; e^\{-\; heta\; varphi(y)\}\; ,\; mathrm\{d\}\; y\; ight)$

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

:

for every measurable set "A".

References

* cite book
last= Dembo
first = Amir
coauthors = Zeitouni, Ofer
title = Large deviations techniques and applications
series = Applications of Mathematics (New York) 38
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 1998
pages = xvi+396
isbn = 0-387-98406-2 MathSciNet|id=1619036