Rates of convergence for the iterates of Cesàro operators

We obtain sharp rates of convergence in the usual sup-norm for the $n$th iterates $D^nf$ and $C^nf$ of continuous and discrete Cesàro operators, respectively. In both cases the best possible rate of convergence is $n^{-1/2}$, and such a rate is attained under appropriate integrability conditions on $f$. Otherwise, the rates of convergence could be extremely poor, depending on the behavior of $f$ near the boundary. We introduce probabilistic representations of $D^nf$ and $C^nf$ involving standardized sums of independent identically distributed random variables and binomial mixtures, respectively, which allow us to use the classical Berry-Esseen theorem.