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| In [[mathematics]] — specifically, in [[large deviations theory]] — the '''contraction principle''' is a [[theorem]] that states how a large deviation principle on one space "[[push forward|pushes forward]]" to a large deviation principle on another space ''via'' a [[continuous function]].
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| ==Statement of the theorem==
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| Let ''X'' and ''Y'' be [[Hausdorff space|Hausdorff]] [[topological space]]s and let (''μ''<sub>''ε''</sub>)<sub>''ε''>0</sub> be a family of [[probability measure]]s on ''X'' that satisfies the large deviation principle with [[rate function]] ''I'' : ''X'' → [0, +∞]. Let ''T'' : ''X'' → ''Y'' be a continuous function, and let ''ν''<sub>''ε''</sub> = ''T''<sub>∗</sub>(''μ''<sub>''ε''</sub>) be the [[Pushforward measure|push-forward measure]] of ''μ''<sub>''ε''</sub> by ''T'', i.e., for each [[measurable set]]/event ''E'' ⊆ ''Y'', ''ν''<sub>''ε''</sub>(''E'') = ''μ''<sub>''ε''</sub>(''T''<sup>−1</sup>(''E'')). Let
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| :<math>J(y) := \inf \big\{ I(x) \big| x \in X \mbox{ and } T(x) = y \big\},</math> | |
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| with the convention that the [[infimum]] of ''I'' over the [[empty set]] ∅ is +∞. Then:
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| * ''J'' : ''Y'' → [0, +∞] is a rate function on ''Y'',
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| * ''J'' is a good rate function on ''Y'' if ''I'' is a good rate function on ''X'', and
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| * (''ν''<sub>''ε''</sub>)<sub>''ε''>0</sub> satisfies the large deviation principle on ''Y'' with rate function ''J''.
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| ==References==
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| * {{cite book
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| | last= Dembo
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| | first = Amir
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| | coauthors = Zeitouni, Ofer
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| | title = Large deviations techniques and applications
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| | series = Applications of Mathematics (New York) 38
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| | edition = Second edition
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| | publisher = Springer-Verlag
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| | location = New York
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| | year = 1998
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| | pages = xvi+396
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| | isbn = 0-387-98406-2
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| | mr = 1619036
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| }} (See chapter 4.2.1)
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| * {{cite book
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| | last = den Hollander
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| | first = Frank
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| | title = Large deviations
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| | series = [[Fields Institute]] Monographs 14
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| | publisher = [[American Mathematical Society]]
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| | location = Providence, RI
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| | year = 2000
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| | pages = x+143
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| | isbn = 0-8218-1989-5
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| | mr = 1739680
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| }}
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| [[Category:Asymptotic analysis]]
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| [[Category:Large deviations theory]]
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| [[Category:Mathematical principles]]
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| [[Category:Probability theorems]]
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