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| In [[mathematics]] — specifically, in [[measure theory]] — '''Fernique's theorem''' is a result about [[Gaussian measure]]s on [[Banach space]]s. It extends the finite-dimensional result that a Gaussian [[random variable]] has [[Exponential function|exponential]] tails. The result was proved in 1970 by the [[mathematician]] [[Xavier Fernique]].
| | 25 yr old Aircraft Maintenance Manufacture (Avionics) Augustine from Thorold, usually spends time with interests for example lasers, [http://gamsolutions.edublogs.org/2014/10/15/134/ Dungeon Hunter 4 Hack] and windsurfing. Finds inspiration through travel and just spent 4 weeks at Simien National Park. |
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| ==Statement of the theorem==
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| Let (''X'', || ||) be a [[separable space|separable]] Banach space. Let ''μ'' be a centered Gaussian measure on ''X'', i.e. a [[probability measure]] defined on the [[Borel set]]s of ''X'' such that, for every [[bounded linear functional]] ''ℓ'' : ''X'' → '''R''', the [[push-forward measure]] ''ℓ''<sub>∗</sub>''μ'' defined on the [[Borel sets]] of '''R''' by
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| :<math>( \ell_{\ast} \mu ) (A) = \mu ( \ell^{-1} (A) ), </math>
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| is a Gaussian measure (a [[normal distribution]]) with zero [[expected value|mean]]. Then there exists ''α'' > 0 such that
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| :<math>\int_{X} \exp ( \alpha \| x \|^{2} ) \, \mathrm{d} \mu (x) < + \infty.</math>
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| ''[[A fortiori]]'', ''μ'' (equivalently, any ''X''-valued random variable ''G'' whose [[probability distribution|law]] is ''μ'') has [[moment (mathematics)|moments]] of all orders: for all ''k'' ≥ 0,
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| :<math>\mathbb{E} [ \| G \|^{k} ] = \int_{X} \| x \|^{k} \, \mathrm{d} \mu (x) < + \infty.</math>
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| ==References==
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| * {{cite journal
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| | last = Fernique
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| | first = Xavier
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| | title = Intégrabilité des vecteurs gaussiens
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| | journal = C. R. Acad. Sci. Paris Sér. A-B
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| | volume = 270
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| | year = 1970
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| | pages = A1698–A1699
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| }} {{MathSciNet|id=0266263}}
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| * [[Giuseppe Da Prato]] and [[Jerzy Zabczyk]], Stochastic equations in infinite dimension, Cambridge University Press, 1992. Theorem 2.6
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| [[Category:Probability theorems]]
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| [[Category:Theorems in measure theory]]
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| {{mathanalysis-stub}}
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25 yr old Aircraft Maintenance Manufacture (Avionics) Augustine from Thorold, usually spends time with interests for example lasers, Dungeon Hunter 4 Hack and windsurfing. Finds inspiration through travel and just spent 4 weeks at Simien National Park.