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" (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 (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(T1(E)). Let

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

References

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