  We offer a generalization of the recent result of Tao (building on earlier
results of Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra,
Frantzikinakis and Kra and Ziegler) that the nonconventional ergodic averages
associated to an arbitrary number of commuting probability-preserving
transformations always converge to some limit in L^2. We prove the
corresponding result for a collection of commuting actions of a larger discrete
Abelian group, and gives convergence that is uniform in the start-point of the
averages. While Tao's proof rests on a conversion to a finitary problem, we
invoke only techniques from classical ergodic theory, so giving a new proof of
his result.
