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.

Statement of the theorem

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 \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 +∞. 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

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