A functional combinatorial central limit theorem

TL;DR

This paper develops a functional version of the Hoeffding combinatorial CLT, using Stein's method to show convergence of a Gaussian process approximation to the original process, with applications to permutation tableaux.

Contribution

It introduces a functional combinatorial CLT with a Gaussian process approximation and convergence analysis under weak conditions.

Findings

Gaussian process approximation is within Lyapounov ratio distance

Pre-limiting process converges to Gaussian limit under weak conditions

Application to the shape analysis of random permutation tableaux

Abstract

The paper establishes a functional version of the Hoeffding combinatorial central limit theorem. First, a pre-limiting Gaussian process approximation is defined, and is shown to be at a distance of the order of the Lyapounov ratio from the original random process. Distance is measured by comparison of expectations of smooth functionals of the processes, and the argument is by way of Stein's method. The pre-limiting process is then shown, under weak conditions, to converge to a Gaussian limit process. The theorem is used to describe the shape of random permutation tableaux.