Tilted large deviation principle
# 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 "tilting", i.e. integration against an exponential functional. It can be seen as an alternative formulation of Varadhan's lemma.

## Statement of the theorem

Let X be a Polish space (i.e., a separable, completely metrizable topological space), 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 F : X → R be a continuous function that is bounded from above. For each Borel set S ⊆ X, let

    $J_{\varepsilon }(S)=\int _{S}e^{F(x)/\varepsilon }\,\mathrm {d} \mu _{\varepsilon }(x)$

and define a new family of probability measures $(ν_ε)_{ε>0}$ on X by

    $\nu _{\varepsilon }(S)={\frac {J_{\varepsilon }(S)}{J_{\varepsilon }(X)}}.$

Then $(ν_ε)_{ε>0}$ satisfies the large deviation principle on X with rate function $I^F$ : X → [0, +∞] given by

    $I^{F}(x)=\sup _{y\in X}{\bigF(y)-I(y){\big ]}-{\bigF(x)-I(x){\big ]}.$
