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 N complex numbers, and add them together each multiplied by a random sign \pm 1, 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 0<p<\infty and let. Then for some constants A_p,B_p>0 depending only on p (see Expected value for notation). The sharp values of the constants A_p,B_p were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof). It is a simple matter to see that A_p = 1 when p \ge 2, and B_p = 1 when 0 < p \le 2. Haagerup found that where and \Gamma is the Gamma function. One may note in particular that B_p 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 T be a linear operator between two Lp spaces L^p(X,\mu) and L^p(Y,\nu),, with bounded norm , then one can use Khintchine's inequality to show that for some constant C_p>0 depending only on p and |T|.

Generalizations

For the case of Rademacher random variables, Pawel Hitczenko showed that the sharpest version is: where, and A and B are universal constants independent of p. Here we assume that the x_i are non-negative and non-increasing.