On the central limit theorem for the two-sided descent statistics in Coxeter groups

TL;DR

This paper proves that the two-sided descent statistics in certain Coxeter groups follow an asymptotic normal distribution, removing previous regularity assumptions using Wasserstein distance techniques.

Contribution

It offers a shorter, more general proof that the two-sided descent statistics are asymptotically normal in Coxeter groups without regularity constraints.

Findings

Asymptotic normality of two-sided descent statistics established

New proof technique using Wasserstein distance introduced

Results apply broadly to sequences of Coxeter groups

Abstract

In 2018, Kahle and Stump raised the following problem: identify sequences of finite Coxeter groups $W_n$ for which the two-sided descent statistics on a uniform random element of $W_n$ is asymptotically normal. Recently, Br\"uck and R\"ottger provided an almost-complete answer, assuming some regularity condition on the sequence $W_n$. In this note, we provide a shorter proof of their result, which does not require any regularity condition. The main new proof ingredient is the use of the second Wasserstein distance on probability distributions, based on the work of Mallows (1972).