Contraction principle (large deviations theory)
In mathematics — specifically, in large deviations theory — the contraction principle is a theorem that states how a large deviation principle on one space "pushes forward" to a large deviation principle on another space via a continuous function.Statement
Let X and Y be Hausdorff topological spaces and let ε>0 be a family of probability measures on X that satisfies the large deviation principle with rate function I : X → . Let T : X → Y be a continuous function, and let νε = T∗ be the push-forward measure of με by T, i.e., for each measurable set/event E ⊆ Y, νε = με. Let
with the convention that the infimum of I over the empty set ∅ is +∞. Then:
- J : Y → is a rate function on Y,
- J is a good rate function on Y if I is a good rate function on X, and
- ε>0 satisfies the large deviation principle on Y with rate function J.
