Dawson–Gärtner theorem explained

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 (Y_j)_j∈J be a projective system of Hausdorff topological spaces with maps p_ij : Y_j → Y_i. Let X be the projective limit (also known as the inverse limit) of the system (Y_j, p_ij)_i,j∈J, i.e.

X = \varprojlim _{j} Y _{j} = \left\{ \left. y = (y _{j} ) _{j} \in Y = \prod _{j} Y _{j} \right| i < j \implies y _{i} = p _{ij} (y _{j} ) \right\}.

Let (μ_&epsilon;)_&epsilon;>0 be a family of probability measures on X. Assume that, for each j ∈ J, the push-forward measures (p_j∗μ_&epsilon;)_&epsilon;>0 on Y_j satisfy the large deviation principle with good rate function I_j : Y_j → R ∪ . Then the family (μ_&epsilon;)_&epsilon;>0 satisfies the large deviation principle on X with good rate function I : X → R ∪  given by

I(x) = \sup _{j} I _{j} (p _{j} (x)).

References

• Book: Dembo , Amir . Zeitouni, Ofer. Large deviations techniques and applications. Applications of Mathematics (New York) 38. Second. Springer-Verlag. New York. 1998. xvi+396. 0-387-98406-2. 1619036. (See theorem 4.6.1)