Khintchine inequality

Short description: Theorem in probability

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 [math]\displaystyle{ N }[/math] complex numbers [math]\displaystyle{ x_1,\dots,x_N \in\mathbb{C} }[/math], and add them together each multiplied by a random sign [math]\displaystyle{ \pm 1 }[/math], 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 [math]\displaystyle{ \sqrt{|x_1|^{2}+\cdots + |x_N|^{2}} }[/math].

Statement

Let [math]\displaystyle{ \{\varepsilon_n\}_{n=1}^N }[/math] be i.i.d. random variables with [math]\displaystyle{ P(\varepsilon_n=\pm1)=\frac12 }[/math] for [math]\displaystyle{ n=1,\ldots, N }[/math], i.e., a sequence with Rademacher distribution. Let [math]\displaystyle{ 0\lt p\lt \infty }[/math] and let [math]\displaystyle{ x_1,\ldots,x_N\in \mathbb{C} }[/math]. Then

[math]\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} }[/math]

for some constants [math]\displaystyle{ A_p,B_p\gt 0 }[/math] depending only on [math]\displaystyle{ p }[/math] (see Expected value for notation). The sharp values of the constants [math]\displaystyle{ A_p,B_p }[/math] were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof). It is a simple matter to see that [math]\displaystyle{ A_p = 1 }[/math] when [math]\displaystyle{ p \ge 2 }[/math], and [math]\displaystyle{ B_p = 1 }[/math] when [math]\displaystyle{ 0 \lt p \le 2 }[/math].

Haagerup found that

[math]\displaystyle{ \begin{align} A_p &= \begin{cases} 2^{1/2-1/p} & 0\lt p\le p_0, \\ 2^{1/2}(\Gamma((p+1)/2)/\sqrt{\pi})^{1/p} & p_0 \lt p \lt 2\\ 1 & 2 \le p \lt \infty \end{cases} \\ &\text{and} \\ B_p &= \begin{cases} 1 & 0 \lt p \le 2 \\ 2^{1/2}(\Gamma((p+1)/2)/\sqrt\pi)^{1/p} & 2 \lt p \lt \infty \end{cases}, \end{align} }[/math]

where [math]\displaystyle{ p_0\approx 1.847 }[/math] and [math]\displaystyle{ \Gamma }[/math] is the Gamma function. One may note in particular that [math]\displaystyle{ B_p }[/math] 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 [math]\displaystyle{ T }[/math] be a linear operator between two L^p spaces [math]\displaystyle{ L^p(X,\mu) }[/math] and [math]\displaystyle{ L^p(Y,\nu) }[/math], [math]\displaystyle{ 1 \lt p \lt \infty }[/math], with bounded norm [math]\displaystyle{ \|T\|\lt \infty }[/math], then one can use Khintchine's inequality to show that

[math]\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)} }[/math]

for some constant [math]\displaystyle{ C_p\gt 0 }[/math] depending only on [math]\displaystyle{ p }[/math] and [math]\displaystyle{ \|T\| }[/math].

Generalizations

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

[math]\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) }[/math]

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

Here we assume that the [math]\displaystyle{ x_i }[/math] 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.