In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Roman alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick N complex numbers x 1 , … , x N ∈ C , and add them together each multiplied by a random sign ± 1 , then the expected value of its modulus, or the modulus it will be closest to on average, will be not too far off from | x 1 | 2 + ⋯ + | x N | 2 .
Let { ε n } n = 1 N be i.i.d. random variables with P ( ε n = ± 1 ) = 1 2 for n = 1 , … , N , i.e., a sequence with Rademacher distribution. Let 0 < p < ∞ and let x 1 , … , x N ∈ C . Then
A p ( ∑ n = 1 N | x n | 2 ) 1 / 2 ≤ ( E | ∑ n = 1 N ε n x n | p ) 1 / p ≤ B p ( ∑ n = 1 N | x n | 2 ) 1 / 2 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 ≥ 2 , and B p = 1 when 0 < p ≤ 2 .
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 , μ ) and L p ( Y , ν ) , 1 ≤ p < ∞ , with bounded norm ∥ T ∥ < ∞ , then one can use Khintchine's inequality to show that
∥ ( ∑ n = 1 N | T f n | 2 ) 1 / 2 ∥ L p ( Y , ν ) ≤ C p ∥ ( ∑ n = 1 N | f n | 2 ) 1 / 2 ∥ L p ( X , μ ) for some constant C p > 0 depending only on p and ∥ T ∥ .
