Restricted Khinchine inequality (Q2788806)

Let \(\epsilon_1,\dots ,\epsilon_n\) be random variables taking the values \(1\) and \(-1\) with probability one half. The well-known Khinchine inequality states that, if the \(\epsilon_i\)'s are independent, then for every \(a\in \mathbb R^n\) and every \(p\geq 2\) one has NEWLINE\[NEWLINE \left(\mathbb{E} \left|\sum_{i=1}^n a_i \epsilon_i \right| ^p\right)^{1/p} \leq \sqrt{p}\, \|a\|_2 = \sqrt{p}\, \left(\mathbb{E} \left|\sum_{i=1}^n a_i \epsilon_i \right| ^2\right)^{1/2} .NEWLINE\]NEWLINE \textit{S. O'Rourke} [Electron. Commun. Probab. 17, No. 28, 13 p. (2012; Zbl 1251.60006)] posed the question whether a Khinchine-type inequality holds under the additional assumptions that \(n\) is even and \(\sum _{i=1}^n \epsilon _i=0\), proving it with the factor \(\sqrt{n} \, p/\log n\) instead of \(\sqrt{p}\). The purpose of the paper under review is to prove that for every \(a\in \mathbb R^n\) and every \(p\geq 2\) one has NEWLINE\[NEWLINE\left(\mathbb{E}_{\Omega} \left|\sum_{i=1}^n a_i \epsilon_i \right| ^p \right)^{1/p} \leq \sqrt{2p}\, \left(\|a\|_2^2 - n b^2\right) \leq \sqrt{2 p}\, \left(\mathbb{E}_{\Omega} \left|\sum_{i=1}^n a_i \epsilon_i \right| ^2\right)^{1/2} ,NEWLINE\]NEWLINE where \(b=\frac{1}{n} \sum _{i=1}^n a_i\) and \(\mathbb{E}_{\Omega}\) denotes the expectation taken with respect to the space NEWLINE\[NEWLINE \Omega =\left\{ \epsilon = (\epsilon _i) _{i=1}^n\, :\, \sum _{i=1}^n \epsilon _i=0\right\}NEWLINE\]NEWLINE endowed with the uniform probability measure.NEWLINENEWLINEThe author also notices that the case when \(a\) has \(0/1\) coordinates corresponds to the \((n, n/2, \ell)\) hypergeometric random variable \(\xi\), where \(\ell=\sum_{i=1}^n a _i\). More precisely, if \(X=\sum _{i=1}^n a_i \epsilon _i/2\), then \(X\) has the same distribution as \(\xi - \mathbb{E}_{\Omega} \xi\). This implies that NEWLINE\[NEWLINE \left(\mathbb{E}_{\Omega} \left| \xi - \mathbb{E}_{\Omega} xi \right| ^p\right)^{1/p} \leq \sqrt{p\ell},NEWLINE\]NEWLINE improving previously known bounds.