  Let $(X,\mathcal{B},m,\tau)$ be a dynamical system with $\ds
(X,\mathcal{B},m)$ a probability space and $\ds \tau$ an invertible, measure
preserving transformation. The present paper deals with the almost everywhere
convergence in $\ds{L}^1(X)$ of a sequence of operators of weighted averages.
Almost everywhere convergence follows once we obtain an appropriate maximal
estimate and once we provide a dense class where convergence holds almost
everywhere. The weights are given by convolution products of members of a
sequence of probability measures $\ds\{\nu_i\}$ defined on $\ds\mathbb{Z}$. We
then exhibit cases of such averages, where convergence fails.
