  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.
