Local central limit theorem for Mallows measure

TL;DR

This paper establishes a local central limit theorem for the height function under the Mallows measure on permutations, extending classical results and providing new multi-point versions.

Contribution

It introduces a local CLT for the Mallows measure's height function and derives related probabilistic limit theorems, including multi-point extensions.

Findings

Proves a local central limit theorem for the Mallows measure

Derives law of large numbers and large deviation principles

Establishes a multi-point local CLT

Abstract

We study the statistics of the Mallows measure on permutations in the limit pioneered by Starr (2009). Our main result is the local central limit theorem for its height function. We also re-derive versions of the law of large numbers and the large deviation principle, obtain the standard central limit theorem from the local one, and establish a multi-point version of the local central limit theorem.