On the rate of convergence of weighted oscillating ergodic averages

Let (X, A, mu) be a probability space, let T be a contraction on L-p(mu) and let f in L-p(mu), (p > 1). In this paper, we provide suitable conditions over sequences (w(k)), (u(k)) and (A(k)) in such a way that the limit of the weighted ergodic average is lim(N ->infinity) 1PN-1 k=0 w(k)T(uk) (f) = 0 mu-a.e. We also give applications which concretely 1/A(N) prove the effectiveness of the obtained theorems. More precisely, we construct sequences (w(k)) and (u(k)) that were difficult to deal with previously, which satisfy the conditions of our theorems.