A Tauberian theorem for Vilenkin series

There are a number of papers in the literature which contain Cesàro analogues of results already known for martingale sums of Vilenkin-Fourier series. We show that for Vilenkin systems of bounded type, these are not merely analogues but actually generalizations. Indeed, we prove that convergence of the Cesàro means of a Vilenkin series $S$ implies convergence of martingale partial sums of $S$ itself.