Asymptotics of a locally dependent statistic on finite reflection groups

Abstract

This paper discusses the asymptotic behaviour of the number of descents in a random signed permutation and its inverse, which was posed as an open problem by Chatterjee and Diaconis in a recent publication. For that purpose, we generalize their result for the asymptotic normality of the number of descents in a random permutation and its inverse to other finite reflection groups. This is achieved by applying their proof scheme on signed permutations, so elements of Coxeter groups of type $B_n$, which is also known as the hyperoctahedral group. Furthermore, a similar central limit theorem for elements of Coxeter groups of type $D_n$ is derived via Slutsky's Theorem and a bound on the Wasserstein distance of certain normalized statistics with local dependency structures and bounded local components is proven for both types of Coxeter groups. In addition, we show a two-dimensional central limit theorem via the Cramér-Wold device.

9 pages

References in corpus (1)

• Counting inversions and descents of random elements in finite Coxeter groups

Cited by in corpus (2)

• A central limit theorem for descents of a Mallows permutation and its inverse
• A central limit theorem for the two-sided descent statistic on Coxeter groups