Contraction principle (large deviations theory) - Alchetron, the free social encyclopedia

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 of the theorem

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 ) ∣ x ∈ 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

Contraction principle (large deviations theory) Wikipedia

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