Khintchine inequality

Short description: Theorem in probability

thumb 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 $p\in (0,\mathrm{\infty })$ there exist constants $A_p,B_p>0$ depending only on $p$ such that for every sequence $x=(x_1,x_2,\dots )\in {\ell }^2$, and i.i.d. Rademacher random variables ${ϵ}_1,{ϵ}_2,\dots$,

$A_p\le \frac{\mathbb{𝔼}{\left[{\left|\sum _{n=1}^{\mathrm{\infty }}{ϵ}_n{x}_n\right|}^p\right]}^{1/p}}{\Vert x{\Vert }_2}\le B_p.$

As a particular case, consider $N$ complex numbers $x_1,\dots ,x_N\in \mathbb{ℂ}$, which can be pictured as vectors in a plane. Now sample $N$ random signs ${ϵ}_1,\dots ,{ϵ}_N\in \{-1,+1\}$, with equal independent probability. The inequality states that $$\left|\sum _i{ϵ}_i{x}_i\right|\approx \sqrt{|x_1{|}^2+\cdots +|x_N{|}^2}$$ with a bounded error.

Statement

Let $\{{\epsilon }_n{\}}_{n=1}^N$ be i.i.d. random variables with $P({\epsilon }_n=\pm 1)=\frac{1}{2}$ for $n=1,\dots ,N$, i.e., a sequence with Rademacher distribution. Let $0<p<\mathrm{\infty }$ and let $x_1,\dots ,x_N\in \mathbb{ℂ}$. Then

$A_p{(\sum _{n=1}^N|x_n{|}^2)}^{1/2}\le {(\mathrm{E}{\left|\sum _{n=1}^N{\epsilon }_n{x}_n\right|}^p)}^{1/p}\le B_p{(\sum _{n=1}^N|x_n{|}^2)}^{1/2}$

for some constants $A_p,B_p>0$ depending only on $p$ (see Expected value for notation). More succinctly, $${(\mathrm{E}{\left|\sum _{n=1}^N{\epsilon }_n{x}_n\right|}^p)}^{1/p}\in [A_p,B_p]$$for any sequence $x$ with unit ${\ell }^2$ norm.

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

$\begin{array}{rl}{A}_p& =\{\begin{array}{ll}{2}^{1/2-1/p}& 0<p\le p_0,\\ 2^{1/2}(\mathrm{\Gamma }((p+1)/2)/\sqrt{\pi }{)}^{1/p}& p_0<p<2\\ 1& 2\le p<\mathrm{\infty }\end{array}\\ & \text{and}\\ B_p& =\{\begin{array}{ll}1& 0<p\le 2\\ 2^{1/2}(\mathrm{\Gamma }((p+1)/2)/\sqrt{\pi }{)}^{1/p}& 2<p<\mathrm{\infty }\end{array},\end{array}$

where $p_0\approx 1.847$ and $\mathrm{\Gamma }$ 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 $L^p(Y,\nu )$, $1<p<\mathrm{\infty }$, with bounded norm $\Vert T\Vert <\mathrm{\infty }$, then one can use Khintchine's inequality to show that

${\Vert {(\sum _{n=1}^N|T{f}_n{|}^2)}^{1/2}\Vert }_{L^p(Y,\nu )}\le C_p{\Vert {(\sum _{n=1}^N|f_n{|}^2)}^{1/2}\Vert }_{L^p(X,\mu )}$

for some constant $C_p>0$ depending only on $p$ and $\Vert T\Vert$.

Generalizations

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

$A(\sqrt{p}{\left(\sum _{n=b+1}^N{x}_n^2\right)}^{1/2}+\sum _{n=1}^b{x}_n)\le {(\mathrm{E}{\left|\sum _{n=1}^N{\epsilon }_n{x}_n\right|}^p)}^{1/p}\le B(\sqrt{p}{\left(\sum _{n=b+1}^N{x}_n^2\right)}^{1/2}+\sum _{n=1}^b{x}_n)$

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.

See also

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

References

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.