Contraction principle (large deviations theory)

In mathematics &mdash; specifically, in large deviations theory &mdash; the contraction principle is a theorem that states how a large deviation principle on one space "pushes forward" (via the pushforward of a probability measure) to a large deviation principle on another space via a continuous function.

Statement
Let X and Y be Hausdorff topological spaces and let (&mu;&epsilon;)&epsilon;>0 be a family of probability measures on X that satisfies the large deviation principle with rate function I : X → [0, +∞]. Let T : X → Y be a continuous function, and let &nu;&epsilon; = T∗(&mu;&epsilon;) be the push-forward measure of &mu;&epsilon; by T, i.e., for each measurable set/event E ⊆ Y, &nu;&epsilon;(E) = &mu;&epsilon;(T&minus;1(E)). Let


 * $$J(y) := \inf \{ I(x) \mid x \in X \text{ and } T(x) = y \},$$

with the convention that the infimum of I over the empty set ∅ is +∞. Then:
 * J : Y → [0, +∞] is a rate function on Y,
 * J is a good rate function on Y if I is a good rate function on X, and
 * (&nu;&epsilon;)&epsilon;&gt;0 satisfies the large deviation principle on Y with rate function J.