  In 1973, Williams introduced two interesting discrete Markov processes,
namely $C$-processes and $A$-processes, which are related to record times in
statistics and Engel's series in number theory respectively. Moreover, he
showed that these two processes share the same classical limit theorems, such
as the law of large numbers, central limit theorem and law of the iterated
logarithm. In this paper, we consider the large deviations for these two Markov
processes, which indicate that there is a difference between $C$-processes and
$A$-processes in the context of large deviations.
