A representation of martingale differences and orthonormal systems of unconditional convergence almost everywhere (Q1852370)

A special representation of uniformly bounded martingale differences is established, which has interesting applications. We present two of them. Theorem 2. If \((\phi_k: k= 1,2,\dots)\) is a sequence of martingale differences with \(|\phi_k|\leq M_k\) and \(M^2:= \sum^\infty_{k=1} M^2_k< \infty\), then there exist absolute constants \(C\) and \(\gamma= \gamma(M)> 0\) such that for every permutation \(\sigma\) of the natural numbers and every number \(\lambda> 0\), the majorant \[ s^*(\sigma):= \sup_{1\leq n<\infty}\Biggl|\sum^n_{k=1} \phi_{\sigma(k)}\Biggr| \] satisfies the inequality \[ P[s^*(\sigma)> \lambda]\leq Ce^{-\gamma\lambda^2}.\tag{\(*\)} \] Proposition 3. Every uniformly bounded, orthonormal system \(\{f_k:k= 1,2,\dots\}\) contains a subsystem \((f_{k_n}: n= 1,2,\dots; k_n\leq 2^n)\) of unconditional convergence with an estimate of type \((*)\) for the majortant.