Contraction principle (large deviations theory) explained

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" (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 (μ_ε)_ε>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 ν_ε = T_∗(μ_ε) be the push-forward measure of μ_ε by T, i.e., for each measurable set/event E ⊆ Y, ν_ε(E) = μ_ε(T^-1(E)). Let

J(y) : = inf \{ I(x) \mid x \in X 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
• (ν_ε)_ε>0 satisfies the large deviation principle on Y with rate function J.

References

• Book: Dembo , Amir . Zeitouni, Ofer. Large deviations techniques and applications. Applications of Mathematics (New York) 38. Second. Springer-Verlag. New York. 1998. xvi+396. 0-387-98406-2. 1619036. (See chapter 4.2.1)
• Book: den Hollander , Frank . Large deviations. Fields Institute Monographs 14. American Mathematical Society. Providence, RI. 2000. x+143. 0-8218-1989-5. 1739680.