Let \{a_k\}_{k\geq0} be a sequence of complex numbers. We obtain the necessary and sufficient conditions for the convergence of n^{-1}\sum_{k=0}^na_kT^kx for every contraction T on a Hilbert space H and every x \in H. It is shown that a natural strengthening of the conditions does not yield convergence for all weakly almost periodic operators in Banach spaces, and the relations between the conditions are exhibited. For a strictly increasing sequence of positive integers \{k_j\}, we study the problem of when n^{-1}\sum_{j=1}^nT^{k_j}x converges to a T-fixed point for every weakly almost periodic T or for every contraction in a Hilbert space and not for every weakly almost periodic operator.