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 $x_1,\dots,x_N \in\mathbb$, 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 $\sqrt$.

Statement

Let $\_^N$ be i.i.d. random variables
with $P(\varepsilon_n=\pm1)=\frac12$ for $n=1,\ldots, N$,
i.e., a sequence with Rademacher distribution. Let 0 and let $x_1,\ldots,x_N\in \mathbb$. Then

:$A_p \left( \sum_^N , x_n, ^2 \right)^ \leq \left(\operatorname \left, \sum_^N \varepsilon_n x_n\^p \right)^ \leq B_p \left(\sum_^N , x_n, ^2\right)^$

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 $p \ge 2$, and $B_p = 1$ when $0 < p \le 2$.

Haagerup found that
:
\begin
A_p &= \begin
2^ & 0
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

:$\left\, \left(\sum_^N , Tf_n, ^2 \right)^ \right\, _\leq C_p \left\, \left(\sum_^N , f_n, ^2\right)^ \right\, _$

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

Generalizations

For the case of Rademacher random variables, Pawel Hitczenko showed Pawel Hitczenko, "On the Rademacher Series". Probability in Banach Spaces, 9 pp 31-36. that the sharpest version is:

:$A \left(\sqrt\left(\sum_^N x_n^2\right)^ + \sum_^b x_n\right) \leq \left(\operatorname \left, \sum_^N \varepsilon_n x_n\^p \right)^ \leq B \left(\sqrt\left(\sum_^N x_n^2\right)^ + \sum_^b x_n\right)$

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.

See also

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

References

# Thomas H. Wolff, "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003. {{ISBN, 0-8218-3449-5
#Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math. 70 (1981), no. 3, 231–283 (1982).
# 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.

Theorems in analysis
Probabilistic inequalities