  Martingale methods are used to study the almost everywhere convergence of
general function series. Applications are given to ergodic series, which
improves recent results of Fan \cite{FanETDS}, and to dilated series, including
Davenport series, which completes results of Gaposhkin \cite{Gaposhkin67} (see
also \cite{Gaposhkin68}). Application is also given to the almost everywhere
convergence with respect to Riesz products of lacunary series.

Probability