Dawson–Gärtner theorem

---

title: "Dawson–Gärtner theorem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["asymptotic-analysis", "large-deviations-theory", "theorems-in-probability-theory"] description: "Mathematical result in large deviations theory" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Dawson–Gärtner_theorem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary 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 (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

Let (μ**ε)ε0 be a family of probability measures on X. Assume that, for each j ∈ J, the push-forward measures (p**j∗μ**ε)ε0 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

• {{cite book | last= Dembo | first = Amir |author2=Zeitouni, Ofer | title = Large deviations techniques and applications | series = Applications of Mathematics (New York) 38 | edition = Second | publisher = Springer-Verlag | location = New York | year = 1998 | pages = xvi+396 | isbn = 0-387-98406-2 | mr = 1619036

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::