We obtain sharp rates of convergence in the usual sup-norm for the
n
n
th iterates
D
n
f
D^nf
and
C
n
f
C^nf
of continuous and discrete Cesàro operators, respectively. In both cases the best possible rate of convergence is
n
−
1
/
2
n^{-1/2}
, and such a rate is attained under appropriate integrability conditions on
f
f
. Otherwise, the rates of convergence could be extremely poor, depending on the behavior of
f
f
near the boundary. We introduce probabilistic representations of
D
n
f
D^nf
and
C
n
f
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.