mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 In mathematics — specifically, in large deviations theory — a rate function is a function used to quantify the probabilities of rare events. It is required to have several properties which assist in the formulation of the large deviat ...

on a “smaller” topological space to a “larger” one.

Statement of the theorem

Let (''Y''_''j'')_{''j''∈''J''} be a

projective system In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ca ...

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_ Y_ = \left\.

Let (''μ''_''ε'')_''ε''>0 be a family of

probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...

s 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_ I_(p_(x)).

References

* (See theorem 4.6.1) {{DEFAULTSORT:Dawson-Gartner theorem Asymptotic analysis Large deviations theory Probability theorems