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Given a probability space 
$(X,\mu )$, a square integrable function f on such space and a (unilateral or bilateral) shift operator T, we prove under suitable 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 in the case of shifts associated with toral endomorphisms.
