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Given a probability space $(X,\mu)$, a square integrable function $f$ on suchspace and a (unilateral or bilateral) shift operator $T$, we prove undersuitable assumptions that the ergodic means $N^{-1}\sum_{n=0}^{N-1} T^nf$converge pointwise almost everywhere to zero with a speed of convergence which,up to a small logarithmic transgression, is essentially of the order of$N^{-1/2}$. We also provide a few applications of our results, especially inthe case of shifts associated with toral endomorphisms.

On the speed of convergence in the ergodic theorem for shift operators