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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 space|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.

:<math>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\}.</math>

Let (''μ''''ε'')''ε''>0 be a family of [[probability measure]]s on ''X''. Assume that, for each ''j'' ∈ ''J'', the [[push-forward measure]]s (''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

:<math>I(x) = \sup_{j \in J} I_{j}(p_{j}(x)).</math>

==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
| mr = 1619036
}} (See theorem 4.6.1)

{{DEFAULTSORT:Dawson-Gartner theorem}}
[[Category:Asymptotic analysis]]
[[Category:Large deviations theory]]
[[Category:Probability theorems]]

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en>Yaris678: Wikilink