Khintchine inequality

The Khintchine inequality, is a result in probability also frequently used in analysis bounding the expectation a weighted sum of Rademacher random variables with square-summable weights. It is named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet.

It states that for each ${\displaystyle p\in (0,\infty )}$ there exist constants ${\displaystyle A_{p},B_{p}>0}$ depending only on ${\displaystyle p}$ such that for every sequence ${\displaystyle x=(x_{1},x_{2},\dots )\in \ell ^{2}}$, and i.i.d. Rademacher random variables ${\displaystyle \epsilon _{1},\epsilon _{2},\dots }$,

${\displaystyle A_{p}\leq {\frac {\mathbb {E} \left[\left|\sum _{n=1}^{\infty }\epsilon _{n}x_{n}\right|^{p}\right]^{1/p}}{\|x\|_{2}}}\leq B_{p}.}$

As a particular case, consider ${\displaystyle N}$ complex numbers ${\displaystyle x_{1},\dots ,x_{N}\in \mathbb {C} }$, which can be pictured as vectors in a plane. Now sample ${\displaystyle N}$ random signs ${\displaystyle \epsilon _{1},\dots ,\epsilon _{N}\in \{-1,+1\}}$, with equal independent probability. The inequality states that $${\displaystyle {\Big |}\sum _{i}\epsilon _{i}x_{i}{\Big |}\approx {\sqrt {|x_{1}|^{2}+\cdots +|x_{N}|^{2}}}}$$ with a bounded error.

Statement

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). More succinctly, $${\displaystyle \left(\operatorname {E} \left|\sum _{n=1}^{N}\varepsilon _{n}x_{n}\right|^{p}\right)^{1/p}\in [A_{p},B_{p}]}$$for any sequence ${\displaystyle x}$ with unit ${\displaystyle \ell ^{2}}$ norm.

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.

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 ${\displaystyle T}$ 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 ${\displaystyle \|T\|}$. ^[1]

Generalizations

For the case of Rademacher random variables, Pawel Hitczenko showed ^[2] 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.

See also

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

References

1. Tao, Terence. "Amplification, arbitrage, and the tensor power trick" . Retrieved 13 April 2025.
2. Pawel Hitczenko, "On the Rademacher Series". Probability in Banach Spaces, 9 pp 31-36. ISBN   978-1-4612-0253-0

1. Thomas H. Wolff, "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003. ISBN   0-8218-3449-5
2. Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math. 70 (1981), no. 3, 231–283 (1982).
3. Fedor Nazarov and Anatoliy Podkorytov, "Ball, Haagerup, and distribution functions", Complex analysis, operators, and related topics, 247–267, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000.