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 . The sharp values of the constants were found by Haagerup. 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 are universal constants independent of.
Here we assume that the are non-negative and non-increasing.