Let (X, A, µ) be a probability space, let T be a contraction on L p (µ) and let f in L p (µ), (p > 1). We provide suitable conditions over sequences (w k), (u k) and (A k) in such a way that the weighted ergodic limit lim N →∞ 1 A N N −1 k=0 w k T u k (f) = 0 µ-a.e.. As a consequence of our main theorems, we also deal with the so-called one-sided weighted ergodic Hilbert transforms.