|
|
| Line 1: |
Line 1: |
| 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]].
| | The name of the writer is Jayson. My spouse and I reside in Kentucky. My day job is an information officer but I've currently applied for an additional one. The preferred pastime for him and his kids is to play lacross and he would by no means give it up.<br><br>My blog - free tarot readings [[http://fashionlinked.com/index.php?do=/profile-13453/info/ fashionlinked.com]] |
| | |
| ==Statement of the theorem==
| |
| | |
| 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
| |
| | |
| :<math>J_{\varepsilon} (S) = \int_{S} e^{- F(x) / \varepsilon} \, \mathrm{d} \mu_{\varepsilon} (x)</math>
| |
| | |
| and define a new family of probability measures (''ν''<sub>''ε''</sub>)<sub>''ε''>0</sub> on ''X'' by
| |
| | |
| :<math>\nu_{\varepsilon} (S) = \frac{J_{\varepsilon} (S)}{J_{\varepsilon} (X)}.</math>
| |
| | |
| 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
| |
| | |
| :<math>I^{F} (x) = \sup_{y \in X} \big[ F(y) - I(y) \big] - \big[ F(x) - I(x) \big].</math>
| |
| | |
| ==References==
| |
| | |
| * {{cite book
| |
| | last = den Hollander
| |
| | first = Frank
| |
| | title = Large deviations
| |
| | series = [[Fields Institute]] Monographs 14
| |
| | publisher = [[American Mathematical Society]]
| |
| | location = Providence, RI
| |
| | year = 2000
| |
| | pages = pp. x+143
| |
| | isbn = 0-8218-1989-5
| |
| }} {{MathSciNet|id=1739680}}
| |
| | |
| [[Category:Asymptotic analysis]]
| |
| [[Category:Mathematical principles]]
| |
| [[Category:Probability theorems]]
| |
| [[Category:Large deviations theory]]
| |
The name of the writer is Jayson. My spouse and I reside in Kentucky. My day job is an information officer but I've currently applied for an additional one. The preferred pastime for him and his kids is to play lacross and he would by no means give it up.
My blog - free tarot readings [fashionlinked.com]