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.

complex numbers

, which can be pictured as vectors in a plane.

random signs

ϵ

, with equal independent probability.

The inequality intuitively states that

ϵ

ε

random variables with

ε

, i.e., a sequence with Rademacher distribution.

(see Expected value for notation).

More succinctly,

for any sequence

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

Haagerup found that where

is the Gamma function.

matches exactly the moments of a normal distribution.

The uses of this inequality are not limited to applications in probability theory.

be a linear operator between two Lp spaces

, with bounded norm

, then one can use Khintchine's inequality to show that for some constant

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

are universal constants independent of

are non-negative and non-increasing.