Sammon mapping

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.

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

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

$I(x)=\sup _{j\in J}I_{j}(p_{j}(x)).$

References

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