Khintchine inequality

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In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick complex numbers , and add them together each multiplied by a random sign , then the expected value of the sum's modulus, or the modulus it will be closest to on average, will be not too far off from .

Statement

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Let   be i.i.d. random variables with   for  , i.e., a sequence with Rademacher distribution. Let   and let  . Then

 

for some constants   depending only on   (see Expected value for notation). The sharp values of the constants   were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof). It is a simple matter to see that   when  , and   when  .

Haagerup found that

 

where   and   is the Gamma function. One may note in particular that   matches exactly the moments of a normal distribution.

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 analysis is the following: if we let   be a linear operator between two Lp spaces   and  ,  , with bounded norm  , then one can use Khintchine's inequality to show that

 

for some constant   depending only on   and  .[citation needed]

Generalizations

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For the case of Rademacher random variables, Pawel Hitczenko showed[1] that the sharpest version is:

 

where  , and   and   are universal constants independent of  .

Here we assume that the   are non-negative and non-increasing.

See also

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References

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  1. ^ Pawel Hitczenko, "On the Rademacher Series". Probability in Banach Spaces, 9 pp 31-36. ISBN 978-1-4612-0253-0
  1. Thomas H. Wolff, "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003. ISBN 0-8218-3449-5
  2. Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math. 70 (1981), no. 3, 231–283 (1982).
  3. 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.