Tilted large deviation principle

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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.

[edit] Statement of the theorem

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

J_{\varepsilon} (S) = \int_{S} e^{- F(x) / \varepsilon} \, \mathrm{d} \mu_{\varepsilon} (x)

and define a new family of probability measures (νε)ε>0 on X by

\nu_{\varepsilon} (S) = \frac{J_{\varepsilon} (S)}{J_{\varepsilon} (X)}.

Then (νε)ε>0 satisfies the large deviation principle on X with rate function IF : X → [0, +∞] given by

I^{F} (x) = \sup_{y \in X} \big[ F(y) - I(y) \big] - \big[ F(x) - I(x) \big].

[edit] References