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

TL;DR

This paper provides a comprehensive characterization of when multiple orthogonal series converge almost surely, linking their convergence to that of single orthogonal series, thus advancing understanding of their probabilistic behavior.

Contribution

It offers a complete criterion for the a.s. convergence of multiple orthogonal series based on their relation to single series, filling a key gap in the theory.

Findings

Characterization of all multiple sequences ensuring a.s. convergence

Relation established between multiple and single orthogonal series convergence

Complete criterion for a.s. convergence of multiple orthogonal series

Abstract

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.