A Representation of Martingale Differences and Orthogonormal Systems of Unconditional Convergence Almost Everywhere

Abstract

We establish a special representation of uniformly bounded martingale differences and give some examples of its application. In particular, we prove that any uniformly bounded ONS $\left\{ f_{k}\right\}$ contains a subsystem $\left\{ f_{k_{n}}\right\} $ ($k_{n}\leq 2^{n}$) of\ \ unconditional convergence with exponential estimation of majorants.