  We study the stochastic convergence of the Ces\`{a}ro mean of a sequence of
random variables. These arise naturally in statistical problems that have a
sequential component, where the sequence of random variables is typically
derived from a sequence of estimators computed on data. We show that
establishing a rate of convergence in probability for a sequence is not
sufficient in general to establish a rate in probability for its Ces\`{a}ro
mean. We also present several sets of conditions on the sequence of random
variables that are sufficient to guarantee a rate of convergence for its
Ces\`{a}ro mean. We identify common settings in which these sets of conditions
hold.
