A Course on Large Deviations with an Introduction to Gibbs Measures.

Yet some more
generalities
4.1. Contraction principle
Chapter 4
Sometimes the problem at hand can be formulated as a mapping of another
problem on a different space. It is reasonable that an LDP for the latter
transfers to one for the former with the rate function at point y being the
smallest value of the original rate function at preimages of y. This is what
the contraction principle (or push-forward principle) is about.
Contraction principle. Suppose f is a continuous function from X to Y,
two Hausdorff topological spaces, and LDP(µn, rn, I) holds on X . Define
νn = µn ◦ f −1 ∈ M1(Y); i.e. νn(A) = µn(f −1 (A)). Define also
J(y) = inf
f(x)=y I(x),
with the convention that the inf over an empty set is infinite. Let J be the
lower semicontinuous regularization of J. That is,
J(y) = sup inf J : y ∈ G, G is open .
G
(a) LDP(νn, rn, J) holds on Y.
(b) If I is tight, then J ≡ J is tight as well.
Proof. By Lemma 2.13 it suffices to prove that J satisfies the large deviation
bounds (2.3) and (2.4). Take a closed set F ⊂ Y. Then
lim
n→∞
1
rn
log µn(f −1 (F )) ≤ − inf
x∈f −1 I(x) = − inf
(F ) y∈F inf I(x) = − inf
f(x)=y y∈F
J(y).
29