On possible estimates of the rate of pointwise convergence in the Birkhoff ergodic theorem

We study the separation from zero of a sequence $\phi$ to obtain the estimates of the form ${\phi(n)/n}$ for the rate of pointwise convergence of ergodic averages. Each of these $\phi$ is shown to be separated from zero for mixings which is not always so for weak mixings. Moreover, for the characteristic function of a nontrivial set, it is shown that there exists a measure preserving transformation with arbitrarily slow decay of ergodic averages.