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

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.

v2 some added references and minor changes to introduction. 35 pages, 1 figure, 1 table. Comments are welcome!

References in corpus (3)

• Interacting particle systems and random walks on Hecke algebras
• A central limit theorem for the two-sided descent statistic on Coxeter groups
• Central limit theorem for peaks of a random permutation in a fixed conjugacy class of $S_n$

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