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| In [[mathematics]], the '''Khintchine inequality''', named after [[Aleksandr Khinchin]] and spelled in multiple ways in the Roman alphabet, is a theorem from [[probability]], and is also frequently used in [[mathematical analysis|analysis]]. Heuristically, it says that if we pick <math> N </math> [[complex numbers]] <math> x_1,\dots,x_N \in\mathbb{C}</math>, and add them together each multiplied by a random sign <math>\pm 1 </math>, then the [[expected value]] of its [[absolute value|modulus]], or the modulus it will be closest to on average, will be not too far off from <math> \sqrt{|x_1|^{2}+\cdots + |x_N|^{2}}</math>.
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| ==Statement of theorem==
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| Let <math> \{\epsilon_{n}\}_{n=1}^{N} </math> be [[i.i.d.]] [[random variables]]
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| with <math>P(\epsilon_n=\pm1)=\frac12</math> for every <math>n=1\ldots N</math>,
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| i.e., a sequence with [[Rademacher distribution]].
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| Let <math> 0<p<\infty</math> and let <math> x_1,...,x_N\in \mathbb{C}</math>.
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| Then
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| :<math> A_p \left( \sum_{n=1}^{N}|x_{n}|^{2} \right)^{\frac{1}{2}} \leq \left(\mathbb{E}\Big|\sum_{n=1}^{N}\epsilon_{n}x_{n}\Big|^{p} \right)^{1/p} \leq B_p \left(\sum_{n=1}^{N}|x_{n}|^{2}\right)^{\frac{1}{2}} </math>
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| for some constants <math> A_p,B_p>0 </math> depending only on <math>p</math> (see [[Expected value]] for notation). The sharp values of the constants <math>A_p,B_p</math> were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof).
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| ==Uses in analysis==
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| The uses of this inequality are not limited to applications in [[probability theory]]. One example of its use in [[Mathematical Analysis|analysis]] is the following: if we let <math>T</math> be a [[linear operator]] between two [[Lp space|L<sup>''p''</sup> spaces]] <math> L^p(X,\mu)</math> and <math> L^p(Y,\nu) </math>, <math>1\leq p<\infty</math>, with bounded [[operator norm|norm]] <math> \|T\|<\infty </math>, then one can use Khintchine's inequality to show that
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| :<math> \left\|\left(\sum_{n=1}^{N}|Tf_n|^{2} \right)^{\frac{1}{2}}\right\|_{L^p(Y,\nu)}\leq C_p\left\|\left(\sum_{n=1}^{N}|f_{n}|^{2}\right)^{\frac{1}{2}}\right\|_{L^p(X,\mu)} </math>
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| for some constant <math>C_p>0</math> depending only on <math>p</math> and <math>\|T\|</math>. | |
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| == See also ==
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| * [[Marcinkiewicz–Zygmund inequality]]
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| ==References==
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| #[[Thomas Wolff|Thomas H. Wolff]], "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003. ISBN 0-8218-3449-5
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| #Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math. 70 (1981), no. 3, 231–283 (1982).
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| #[[Fedor Nazarov]] and Anatoliy Podkorytov, "Ball, Haagerup, and distribution functions", Complex analysis, operators, and related topics, 247–267, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000.
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| [[Category:Mathematical analysis]]
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| [[Category:Probabilistic inequalities]]
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Greetings! I am Marvella and I really feel comfortable when people use the full name. For many years I've been operating as a payroll clerk. California is exactly where her home std test is but she requirements to move simply because of her family members. He is truly fond of performing ceramics but he is struggling to find time for it.