Convergence of weighted ergodic averages

Let $(X, \mathcal{A},\mu)$ be a probability space and let $T$ be a contraction on $L^2(\mu)$. We provide suitable conditions over sequences $(w_k)$, $(u_k)$ and $(A_k)$ in such a way that the weighted ergodic limit $\lim\limits_{N\rightarrow\infty}\frac{1}{A_N}\sum_{k=0}^{N-1} w_k T^{u_k}(f)=0$ $\mu$-a.e. for any function $f$ in $L^2(\mu)$. As a consequence of our main theorems, we also deal with the so-called one-sided weighted ergodic Hilbert transforms.