On the best constants in the Khintchine inequality for Steinhaus variables
The classical Khintchine inequality says that there exist constants \(A_p\) and \(B_p\), \(0 < p < \infty\), such that for all \(x = (x_j)_{j=1}^n \in \mathbb{R}^n\) \[ A_p \|x\|_2 \leq \left( \mathbb{E} \left| \sum_{j=1}^n x_j r_j \right|^p \right)^{1/p} \leq B_p \|x\|_2, \] where \((r_j)_{j=1}^n\), \(r_j: [0,1] \to \{+1,-1\} \subset \mathbb{R}\), are the Rademacher functions. For the best possible constants \(A_p\) and \(B_p\) see [\textit{U. Haagerup}, Stud. Math. 70, 231--283 (1981; Zbl 0501.46015)]. The Rademacher functions constitute an i.i.d.\ sequence of Bernoulli random variables. The Steinhaus variables \(S_j\) are the complex counterpart of the Rademacher functions. They are an i.i.d.\ sequence of random variables \(S_j : (\Omega,P) \to S^1 \subset \mathbb{C}\) on a probability space \((\Omega,P)\) which are uniformly distributed on the circle \(S^1\). For \(0 < p < \infty\) define \[ a_p := \min \left( \Gamma \left(\frac{p}{2}+1\right)^{1/p}, \sqrt{2} \left( \frac{\Gamma(\frac{p+1}{2})}{\Gamma(\frac{p}{2}+1)\sqrt{\pi}} \right)^{1/p},1 \right) \] and \[ b_p := \max \left( \Gamma\left(\frac{p}{2}+1\right)^{1/p},1 \right). \] Let \((S_j)\) be an i.i.d.\ sequence of Steinhaus random variables. Then for any \(n \in \mathbb{N}\) and any sequence \((x_j)_{j=1}^n \in \mathbb{C}^n\), \[ a_p \|x\|_2 \leq \left( \mathbb{E} \left| \sum_{j=1}^n x_j S_j \right|^p \right)^{1/p} \leq b_p \|x\|_2. \] The constants \(a_p\) and \(b_p\) are the best possible. This result was known for \(1 \leq p < \infty\) and the author proves the remaining cases \(0 < p < 1\) (verifying a conjecture of U.\,Haagerup). The proof relies on real analysis techniques only, in particular estimates involving the Bessel functions \(J_0\) and \(J_1\). The above result implies that, for \(x \in \mathbb{C}^n\) with \(\|x\|_2 = 1\), \[ \mathbb{E} \ln \left| \frac{S_1+S_2}{\sqrt{2}}\right| \leq \mathbb{E} \ln \left| \sum_{j=1}^n x_j S_j \right|, \] answering a question of Baernstein and Culverhouse (see Equation (2.2) in [\textit{A. Baernstein II} and \textit{R. C. Culverhouse}, Stud. Math. 152, No. 3, 231--248 (2002; Zbl 0997.60009)]).
scientific article; zbMATH DE number 6375675
scientific article