Khintchine inequality
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

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

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 . Generalizations

For the case of Rademacher random variables, Pawel Hitczenko showed 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 
 Marcinkiewicz–Zygmund inequality
 Burkholder-Davis-Gundy inequality

References

Thomas H. Wolff, "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003. Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math. 70 (1981), no. 3, 231–283 (1982). 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. Theorems in analysis
Probabilistic inequalities