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| In [[mathematics]] — specifically, in [[large deviations theory]] — the '''tilted large deviation principle''' is a result that allows one to generate a new [[Rate function|large deviation principle]] from an old one by "tilting", i.e. [[Integral|integration]] against an [[Exponential function|exponential]] [[Functional (mathematics)|functional]]. It can be seen as an alternative formulation of [[Varadhan's lemma]].
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| ==Statement of the theorem==
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| Let ''X'' be a [[Polish space]] (i.e., a [[separable space|separable]], [[Complete metric space|completely metrizable]] [[topological space]]), and let (''μ''<sub>''ε''</sub>)<sub>''ε''>0</sub> be a family of [[Probability space|probability measures]] on ''X'' that satisfies the large deviation principle with [[rate function]] ''I'' : ''X'' → [0, +∞]. Let ''F'' : ''X'' → '''R''' be a [[continuous function]] that is [[bounded function|bounded]] from above. For each Borel set ''S'' ⊆ ''X'', let
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| :<math>J_{\varepsilon} (S) = \int_{S} e^{- F(x) / \varepsilon} \, \mathrm{d} \mu_{\varepsilon} (x)</math> | |
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| and define a new family of probability measures (''ν''<sub>''ε''</sub>)<sub>''ε''>0</sub> on ''X'' by
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| :<math>\nu_{\varepsilon} (S) = \frac{J_{\varepsilon} (S)}{J_{\varepsilon} (X)}.</math>
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| Then (''ν''<sub>''ε''</sub>)<sub>''ε''>0</sub> satisfies the large deviation principle on ''X'' with rate function ''I''<sup>''F''</sup> : ''X'' → [0, +∞] given by
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| :<math>I^{F} (x) = \sup_{y \in X} \big[ F(y) - I(y) \big] - \big[ F(x) - I(x) \big].</math>
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| ==References==
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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 = pp. x+143
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| | isbn = 0-8218-1989-5
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| }} {{MathSciNet|id=1739680}}
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| [[Category:Asymptotic analysis]]
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| [[Category:Mathematical principles]]
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| [[Category:Probability theorems]]
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| [[Category:Large deviations theory]]
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The author's name is Andera and she thinks it seems fairly good. The preferred pastime for him and his kids is to play lacross and he would by no means give it up. Mississippi is where her home is but her husband wants them to transfer. Invoicing is what I do.
Feel free to surf to my website love psychics; see this here,