Contraction principle (large deviations theory)

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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.

[edit] Statement of the theorem

Let X be a Polish space (i.e., a separable, completely metrizable topological space), 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 Y be another Polish space, 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,

\nu_{\varepsilon} (E) := \mu_{\varepsilon} \big( T^{-1} (E) \big).

Then (νε)ε>0 satisfies the large deviation principle on Y with rate function J : Y → [0, +∞] given by

J(y) := \inf \big\{ I(x) \big| x \in X \mbox{ and } T(x) = y \big\},

with the convention that the infimum of I over the empty set ∅ is +∞.

[edit] References