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.

tatement 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 of the system ("Y"_"j", "p"_"ij")_{"i","j"∈"J"}, i.e.

:

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

* cite book
last= Dembo
first = Amir
coauthors = Zeitouni, Ofer
title = Large deviations techniques and applications
series = Applications of Mathematics (New York) 38
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 1998
pages = xvi+396
isbn = 0-387-98406-2 MathSciNet|id=1619036 (See theorem 4.6.1)