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| {{Unreferenced|date=December 2009}}
| | The person who wrote the article is known as Jayson Hirano and he completely digs that name. The favorite pastime for him and his kids is to play lacross and he would by no means give it up. Some time ago he chose to reside in North Carolina and he doesn't strategy on clairvoyants ([http://ustanford.com/index.php?do=/profile-38218/info/ try this out]) altering it. My working day job is an invoicing officer but I've already utilized for another 1.<br><br>Also visit my site psychics online ([http://medialab.zendesk.com/entries/54181460-Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- medialab.zendesk.com]) - [http://jplusfn.gaplus.kr/xe/qna/78647 clairvoyant psychic] |
| In [[mathematics]], '''Gaussian measure''' is a [[Borel measure]] on finite-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup>, closely related to the [[normal distribution]] in [[statistics]]. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the [[Germany|German]] [[mathematician]] [[Carl Friedrich Gauss]]. One reason why Gaussian measures are so ubiquitous in probability theory is the [[Central Limit Theorem]]. Loosely speaking, it states that if a random variable
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| ''X'' is obtained by summing a large number ''N'' of independent random variables of order 1, then ''X'' is of order <math>\sqrt{N}</math> and its law is
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| approximately Gaussian.
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| ==Definitions==
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| Let ''n'' ∈ '''N''' and let ''B''<sub>0</sub>('''R'''<sup>''n''</sup>) denote the [[complete measure|completion]] of the [[Borel sigma algebra|Borel ''σ''-algebra]] on '''R'''<sup>''n''</sup>. Let ''λ''<sup>''n''</sup> : ''B''<sub>0</sub>('''R'''<sup>''n''</sup>) → [0, +∞] denote the usual ''n''-dimensional [[Lebesgue measure]]. Then the '''standard Gaussian measure''' ''γ''<sup>''n''</sup> : ''B''<sub>0</sub>('''R'''<sup>''n''</sup>) → [0, 1] is defined by
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| :<math>\gamma^{n} (A) = \frac{1}{\sqrt{2 \pi}^{n}} \int_{A} \exp \left( - \frac{1}{2} \| x \|_{\mathbb{R}^{n}}^{2} \right) \, \mathrm{d} \lambda^{n} (x)</math>
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| for any measurable set ''A'' ∈ ''B''<sub>0</sub>('''R'''<sup>''n''</sup>). In terms of the [[Radon–Nikodym derivative]],
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| :<math>\frac{\mathrm{d} \gamma^{n}}{\mathrm{d} \lambda^{n}} (x) = \frac{1}{\sqrt{2 \pi}^{n}} \exp \left( - \frac{1}{2} \| x \|_{\mathbb{R}^{n}}^{2} \right).</math>
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| More generally, the Gaussian measure with [[mean]] ''μ'' ∈ '''R'''<sup>''n''</sup> and [[variance]] ''σ''<sup>2</sup> > 0 is given by
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| :<math>\gamma_{\mu, \sigma^{2}}^{n} (A) := \frac{1}{\sqrt{2 \pi \sigma^{2}}^{n}} \int_{A} \exp \left( - \frac{1}{2 \sigma^{2}} \| x - \mu \|_{\mathbb{R}^{n}}^{2} \right) \, \mathrm{d} \lambda^{n} (x).</math>
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| Gaussian measures with mean ''μ'' = 0 are known as '''centred Gaussian measures'''.
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| The [[Dirac measure]] ''δ''<sub>''μ''</sub> is the [[weak convergence of measures|weak limit]] of <math>\gamma_{\mu, \sigma^{2}}^{n}</math> as ''σ'' → 0, and is considered to be a '''degenerate Gaussian measure'''; in contrast, Gaussian measures with finite, non-zero variance are called '''non-degenerate Gaussian measures'''.
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| ==Properties of Gaussian measure==
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| The standard Gaussian measure ''γ''<sup>''n''</sup> on '''R'''<sup>''n''</sup>
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| * is a [[Borel measure]] (in fact, as remarked above, it is defined on the completion of the Borel sigma algebra, which is a finer structure);
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| * is [[Equivalence (measure theory)|equivalent]] to Lebesgue measure: <math>\lambda^{n} \ll \gamma^{n} \ll \lambda^{n}</math>, where <math>\ll</math> stands for [[absolute continuity]] of measures;
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| * is [[Support (measure theory)|supported]] on all of Euclidean space: supp(''γ''<sup>''n''</sup>) = '''R'''<sup>''n''</sup>;
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| * is a [[probability measure]] (''γ''<sup>''n''</sup>('''R'''<sup>''n''</sup>) = 1), and so it is [[Locally finite measure|locally finite]];
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| * is [[Strictly positive measure|strictly positive]]: every non-empty [[open set]] has positive measure;
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| * is [[Inner regular measure|inner regular]]: for all Borel sets ''A'',
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| :: <math>\gamma^{n} (A) = \sup \{ \gamma^{n} (K) | K \subseteq A, K \mbox{ is compact} \},</math>
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| so Gaussian measure is a [[Radon measure]];
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| * is not [[Translation (geometry)|translation]]-[[Invariant (mathematics)|invariant]], but does satisfy the relation
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| :: <math>\frac{\mathrm{d} (T_{h})_{*} (\gamma^{n})}{\mathrm{d} \gamma^{n}} (x) = \exp \left( \langle h, x \rangle_{\mathbb{R}^{n}} - \frac{1}{2} \| h \|_{\mathbb{R}^{n}}^{2} \right),</math>
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| :where the [[derivative]] on the left-hand side is the [[Radon–Nikodym derivative]], and (''T''<sub>''h''</sub>)<sub>∗</sub>(''γ''<sup>''n''</sup>) is the [[pushforward measure|push forward]] of standard Gaussian measure by the translation map ''T''<sub>''h''</sub> : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>, ''T''<sub>''h''</sub>(''x'') = ''x'' + ''h'';
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| * is the probability measure associated to a [[normal distribution|normal]] [[probability distribution]]:
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| :: <math>Z \sim \mathrm{Normal} (\mu, \sigma^{2}) \implies \mathbb{P} (Z \in A) = \gamma_{\mu, \sigma^{2}}^{n} (A).</math>
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| ==Gaussian measures on infinite-dimensional spaces==
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| It can be shown that [[There is no infinite-dimensional Lebesgue measure|there is no analogue of Lebesgue measure]] on an infinite-dimensional [[vector space]]. Even so, it is possible to define Gaussian measures on infinite-dimensional spaces, the main example being the [[abstract Wiener space]] construction. A Borel measure ''γ'' on a [[separable space|separable]] [[Banach space]] ''E'' is said to be a '''non-degenerate (centered) Gaussian measure''' if, for every [[linear functional]] ''L'' ∈ ''E''<sup>∗</sup> except ''L'' = 0, the [[push-forward measure]] ''L''<sub>∗</sub>(''γ'') is a non-degenerate (centered) Gaussian measure on '''R''' in the sense defined above.
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| For example, [[Classical Wiener space|classical Wiener measure]] on the space of [[continuous function|continuous]] [[path (topology)|paths]] is a Gaussian measure.
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| ==See also==
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| * [[Cameron–Martin theorem]]
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| {{DEFAULTSORT:Gaussian Measure}}
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| [[Category:Measures (measure theory)]]
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| [[Category:Stochastic processes]]
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