Khintchine inequality
# 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 $N$ complex numbers $x_{1},\dots ,x_{N}\in \mathbb {C}$, 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 ${\sqrt {|x_{1}|^{2}+\cdots +|x_{N}|^{2}}}$.

## Statement

Let $\{\varepsilon _{n}\}_{n=1}^{N}$ be i.i.d. random variables with $P(\varepsilon _{n}=\pm 1)={\frac {1}{2}}$ for $n=1,\ldots ,N$, i.e., a sequence with Rademacher distribution. Let $0<p<\infty$ and let $x_{1},\ldots ,x_{N}\in \mathbb {C}$. Then

    $A_{p}\left(\sum _{n=1}^{N}|x_{n}|^{2}\right)^{1/2}\leq \left(\operatorname {E} \left|\sum _{n=1}^{N}\varepsilon _{n}x_{n}\right|^{p}\right)^{1/p}\leq B_{p}\left(\sum _{n=1}^{N}|x_{n}|^{2}\right)^{1/2}$

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 $p\geq 2$, and $B_{p}=1$ when $0<p\leq 2$.

Haagerup found that

    ${\begin{aligned}A_{p}&={\begin{cases}2^{1/2-1/p}&0<p\leq p_{0},\\2^{1/2}(\Gamma ((p+1)/2)/{\sqrt {\pi }})^{1/p}&p_{0}<p<2\\1&2\leq p<\infty \end{cases}}\\&{\text{and}}\\B_{p}&={\begin{cases}1&0<p\leq 2\\2^{1/2}(\Gamma ((p+1)/2)/{\sqrt {\pi }})^{1/p}&2<p<\infty \end{cases}},\end{aligned}}$

where $p_{0}\approx 1.847$ 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 $L^p$ spaces $L^{p}(X,\mu )$ and $L^{p}(Y,\nu )$, $1<p<\infty$, with bounded norm $\|T\|<\infty$, then one can use Khintchine's inequality to show that

    $\left\|\left(\sum _{n=1}^{N}|Tf_{n}|^{2}\right)^{1/2}\right\|_{L^{p}(Y,\nu )}\leq C_{p}\left\|\left(\sum _{n=1}^{N}|f_{n}|^{2}\right)^{1/2}\right\|_{L^{p}(X,\mu )}$

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:

    $A\left({\sqrt {p}}\left(\sum _{n=b+1}^{N}x_{n}^{2}\right)^{1/2}+\sum _{n=1}^{b}x_{n}\right)\leq \left(\operatorname {E} \left|\sum _{n=1}^{N}\varepsilon _{n}x_{n}\right|^{p}\right)^{1/p}\leq B\left({\sqrt {p}}\left(\sum _{n=b+1}^{N}x_{n}^{2}\right)^{1/2}+\sum _{n=1}^{b}x_{n}\right)$

where $b=\lfloor p\rfloor$, and $A$ and $B$ are universal constants independent of $p$.

Here we assume that the $x_{i}$ are non-negative and non-increasing.

