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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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