Dawson–Gärtner theorem

In mathematics, the Dawson–Gärtner theorem is a result in large deviations theory. Heuristically speaking, the Dawson–Gärtner theorem allows one to transport a large deviation principle on a “smaller” topological space to a “larger” one.

Statement of the theorem
Let (Yj)j∈J be a projective system of Hausdorff topological spaces with maps pij : Yj → Yi. Let X be the projective limit (also known as the inverse limit) of the system (Yj, pij)i,j∈J, i.e.


 * $$X = \varprojlim_{j \in J} Y_{j} = \left\{ \left. y = (y_{j})_{j \in J} \in Y = \prod_{j \in J} Y_{j} \right| i < j \implies y_{i} = p_{ij} (y_{j}) \right\}.$$

Let (&mu;&epsilon;)&epsilon;&gt;0 be a family of probability measures on X. Assume that, for each j ∈ J, the push-forward measures (pj∗&mu;&epsilon;)&epsilon;&gt;0 on Yj satisfy the large deviation principle with good rate function Ij : Yj → R ∪ {+∞}. Then the family (&mu;&epsilon;)&epsilon;&gt;0 satisfies the large deviation principle on X with good rate function I : X → R ∪ {+∞} given by


 * $$I(x) = \sup_{j \in J} I_{j}(p_{j}(x)).$$