On the best constants in Khintchine type inequalities for martingales

Abstract

For discrete martingale-difference sequences $d=\{d_1,\ldots,d_n\}$ we consider Khintchine type inequalities, involving certain square function $\mathfrak S (d)$ considered by Chang-Wilson-Wolff in 1982. In particular, we prove \begin{equation} \left\|\sum_{k=1}^nd_k\right\|_p\le 2^{1/2}\big(\Gamma((p+1)/2))/\sqrt{\pi}\big)^{1/p}\|\mathfrak S(d)\|_\infty,\quad p\ge 3, \end{equation} where the constant on the right hand side is the best possible and the same as known for the Rademacher sums $\sum_{k=1}^na_kr_k$. Moreover, for a fixed $n$ the constant in the inequality can be replaced by $\sum_{k=1}^nr_k/\sqrt{n}$. We apply a technique, reducing the general case to the case of Haar and Rademacher sums, that allows also establish a sub-Gaussian estimate \begin{equation} {\bf E}\left[\exp\left(\lambda\cdot \left(\frac{\sum_{k=1}^nd_k}{\|\mathfrak S(d)\|_\infty}\right)^2\right)\right]\le \frac{1}{\sqrt{1-2\lambda}},\quad 0<\lambda<1/2, \end{equation} where the constant on the right hand side is the best possible.