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 of the theorem

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

$\nu _{\varepsilon }(E):=\mu _{\varepsilon }{\big (}T^{-1}(E){\big )}.$

Then (ν_ε)_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 +∞.

References

• den Hollander, Frank (2000). Large deviations. Fields Institute Monographs 14. Providence, RI: American Mathematical Society. pp. pp. x+143. ISBN 0-8218-1989-5. {{cite book}}: |pages= has extra text (help) MR1739680