On a complete characterization of a.s. convergence of multiple orthogonal series

We present a relation between convergence of multiple and single orthogonal series. This relation implies a complete characterization of all multiple sequences $(a_{n_1...n_d})_{n_1,...,n_d\in\bb N}$ such that for all orthonormal $(\Phi_{n_1...n_d})$ multiple orthogonal series $\sum_{n_1,...,n_d\in\bb N}a_{n_1...n_d}\Phi_{n_1...n_d}$ are a.s.\ convergent.