A central limit theorem for descents of a Mallows permutation and its inverse

TL;DR

This paper establishes a central limit theorem for the sum of descents in a Mallows permutation and its inverse, revealing their joint normal distribution and correlation structure, using Stein's method and regenerative processes.

Contribution

It provides the first joint CLT for descents of a Mallows permutation and its inverse, including a Berry-Esseen bound, advancing understanding of permutation statistics under non-uniform measures.

Findings

Asymptotic normality of descent sums and inverses

Explicit correlation structure depending on parameter q

Berry-Esseen bounds for convergence rate

Abstract

This paper studies the asymptotic distribution of descents $\des(w)$ in a permutation $w$, and its inverse, distributed according to the Mallows measure. The Mallows measure is a non-uniform probability measure on permutations introduced to study ranked data. Under this measure, permutations are weighted according to the number of inversions they contain, with the weighting controlled by a parameter $q$. The main results are a Berry-Esseen theorem for $\des(w)+\des(w^{-1})$ as well as a joint central limit theorem for $(\des(w),\des(w^{-1}))$ to a bivariate normal with a non-trivial correlation depending on $q$. The proof uses Stein's method with size-bias coupling along with a regenerative process associated to the Mallows measure.