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Martingales and descent statistics
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abstract
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.
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Cited by 1 Pith paper
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A central limit theorem for the two-sided descent statistic on Coxeter groups
For well-behaved sequences of finite Coxeter groups, the descent-plus-inverse-descent statistic satisfies a central limit theorem exactly when its variance tends to infinity.
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