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Dawson–Gärtner theorem

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(Redirected from Dawson-Gärtner theorem) Mathematical result in large deviations theory

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 = lim ← j ∈ J ⁡ Y j = { y = ( y j ) j ∈ J ∈ Y = ∏ j ∈ J Y j | i < j ⟹ y i = p i j ( y j ) } . {\displaystyle 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 (με)ε>0 be a family of probability measures on X. Assume that, for each j ∈ J, the push-forward measures (pj∗με)ε>0 on Yj satisfy the large deviation principle with good rate function Ij : Yj → R ∪ {+∞}. Then the family (με)ε>0 satisfies the large deviation principle on X with good rate function I : X → R ∪ {+∞} given by

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

References

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