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 $$ x_1,\dots,x_N \in\mathbb{C}$$, 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 $$ \sqrt{|x_1|^{2}+\cdots + |x_N|^{2}}$$.

Statement
Let $$ \{\varepsilon_n\}_{n=1}^N $$ 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 $$ 00 $$ 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{align} A_p &= \begin{cases} 2^{1/2-1/p} & 0<p\le p_0, \\ 2^{1/2}(\Gamma((p+1)/2)/\sqrt{\pi})^{1/p} & p_0 < p < 2\\ 1 & 2 \le p < \infty \end{cases} \\ &\text{and} \\ B_p &= \begin{cases} 1 & 0 < p \le 2 \\ 2^{1/2}(\Gamma((p+1)/2)/\sqrt\pi)^{1/p} & 2 < p < \infty \end{cases}, \end{align} $$ where $$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 Lp 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


 * $$ \left\|\left(\sum_{n=1}^N |Tf_n|^2 \right)^{1/2} \right\|_{L^p(Y,\nu)}\leq C_p \left\|\left(\sum_{n=1}^N |f_n|^2\right)^{1/2} \right\|_{L^p(X,\mu)} $$

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

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



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

Here we assume that the $$x_i$$ are non-negative and non-increasing.