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 (με)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 νε = T∗(με) be the push-forward measure of με by T, i.e., for each measurable set/event E ⊆ Y, νε(E) = με(T−1(E)). Let
J ( y ) := inf { I ( x ) ∣ x ∈ X and T ( x ) = y } , {\displaystyle 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 (νε)ε>0 satisfies the large deviation principle on Y with rate function J.
References Dembo, Amir; Zeitouni, Ofer (1998). Large deviations techniques and applications. Applications of Mathematics (New York) 38 (Second ed.). New York: Springer-Verlag. pp. xvi+396. ISBN 0-387-98406-2. MR 1619036. (See chapter 4.2.1) den Hollander, Frank (2000). Large deviations. Fields Institute Monographs 14. Providence, RI: American Mathematical Society. pp. x+143. ISBN 0-8218-1989-5. MR 1739680.