Contraction principle (large deviations theory)
# 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

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

    $J(y):=\inf\{I(x)\mid x\in X{\text{ and }}T(x)=y\},$

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.
