On the rate of convergence of weighted oscillating ergodic averages

Abstract

Let [Formula: see text] be a probability space, let [Formula: see text] be a contraction on [Formula: see text] and let [Formula: see text] in [Formula: see text], ([Formula: see text]). In this paper, we provide suitable conditions over sequences [Formula: see text], [Formula: see text] and [Formula: see text] in such a way that the limit of the weighted ergodic average is [Formula: see text] [Formula: see text]-a.e. We also give applications which concretely prove the effectiveness of the obtained theorems. More precisely, we construct sequences [Formula: see text] and [Formula: see text] that were difficult to deal with previously, which satisfy the conditions of our theorems.