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 ${\displaystyle N}$ complex numbers ${\displaystyle x_{1},\dots ,x_{N}\in \mathbb {C} }$, and add them together each multiplied by a random sign ${\displaystyle \pm 1}$, 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 ${\displaystyle {\sqrt {|x_{1}|^{2}+\cdots +|x_{N}|^{2}}}}$.

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

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

for some constants ${\displaystyle A_{p},B_{p}>0}$ depending only on ${\displaystyle p}$ (see Expected value for notation). The sharp values of the constants ${\displaystyle A_{p},B_{p}}$ were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof). It is a simple matter to see that ${\displaystyle A_{p}=1}$ when ${\displaystyle p\geq 2}$, and ${\displaystyle B_{p}=1}$ when ${\displaystyle 0<p\leq 2}$.

Haagerup found that

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

where ${\displaystyle p_{0}\approx 1.847}$ and ${\displaystyle \Gamma }$ is the Gamma function. One may note in particular that ${\displaystyle B_{p}}$ matches exactly the moments of a normal distribution.

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 ${\displaystyle T}$ be a linear operator between two L^p spaces ${\displaystyle L^{p}(X,\mu )}$ and ${\displaystyle L^{p}(Y,\nu )}$, ${\displaystyle 1<p<\infty }$, with bounded norm ${\displaystyle \|T\|<\infty }$, then one can use Khintchine's inequality to show that

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

for some constant ${\displaystyle C_{p}>0}$ depending only on ${\displaystyle p}$ and ${\displaystyle \|T\|}$.^{[citation needed]}

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

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

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

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

• Marcinkiewicz–Zygmund inequality
• Burkholder-Davis-Gundy inequality