  Given a sequence $(X_n)$ of real or complex random variables and a sequence
of numbers $(a_n)$, an interesting problem is to determine the conditions under
which the series $\sum_{n=1}^\infty a_n X_n$ is almost surely convergent. This
paper extends the classical Menshov--Rademacher theorem on the convergence of
orthogonal series to general series of dependent random variables and derives
interesting sufficient conditions for the almost everywhere convergence of
trigonometric series with respect to singular measures whose Fourier transform
decays to 0 at infinity with positive rate.
