Tilted large deviation principle

In mathematics — specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new large deviation principle from an old one by exponential tilting, i.e. integration against an exponential functional.

It can be seen as an alternative formulation of Varadhan's lemma.

Let X be a Polish space (i.e., a separable, completely metrizable topological space), 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 F : X → R be a continuous function that is bounded from above.

For each Borel set S ⊆ X, let and define a new family of probability measures (νε)ε>0 on X by Then (νε)ε>0 satisfies the large deviation principle on X with rate function IF : X → [0, +∞] given by