title: Contraction principle (large deviations theory)
text: # 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.
  