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

Valentin Féray

doi:10.1214/20-ecp309

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, Bruck and Rottger 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 (Ann. Math. Statist., 1972).

Highlights

In 2018, Kahle and Stump raised the following problem: identify sequences of finite Coxeter groups Wn for which the two-sided descent statistics on a uniform random element of Wn is asymptotically normal
The main new proof ingredient is the use of the second Wasserstein distance on probability distributions, based on the work of Mallows
Asymptotic normality of permutation statistics is a vast topic in discrete probability, dating back at least to Goncharov [Gon44] and Hoeffding [Hoe51]; we refer to [Vat[96], Ful[04], CD17, Özd19] for more recent works on the descent and two-sided descent statistics

Summary

Introduction

In 2018, Kahle and Stump raised the following problem: identify sequences of finite Coxeter groups Wn for which the two-sided descent statistics on a uniform random element of Wn is asymptotically normal. There has been some interest into generalizing such asymptotic normality results to statistics of Coxeter group elements[1]. Of descents) of a uniform random element in Wn is asymptotically normal.

Results

Conclusion