  This paper develops techniques to study the number of descents in random
permutations via martingales. We relax an assumption in the Berry-Esseen
theorem of Bolthausen (1982) to extend the theorem's scope to martingale
differences of time-dependent variances. This extension leads to a new proof of
the fact that the number of descents in random permutations is asymptotically
normal with an error bound of order $1/\sqrt{n}.$ The same techniques are shown
to be applicable to other descent and descent-related statistics as they
satisfy certain recurrence relation conditions. These statistics include
inversions, descents in signed permutations, descents in Stirling permutations,
the length of the longest alternating subsequences, descents in matchings and
two-sided Eulerian numbers.
