A Functional Combinatorial Central Limit Theorem

Andrew Barbour,Svante Janson

doi:10.1214/ejp.v14-709

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.

Highlights

Let a0(n) := (a0(n)(i, j), 1 ≤ i, j ≤ n), n ≥ 1, be a sequence of real matrices
The paper establishes a functional version of the Hoeffding combinatorial central limit theorem
The theorem is used to describe the shape of random permutation tableaux

Summary

Introduction

Let a0(n) := (a0(n)(i, j), 1 ≤ i, j ≤ n), n ≥ 1, be a sequence of real matrices. Hoeffding’s (1951) combinatorial central limit theorem asserts that if π is a uniform random permutation of {1, 2, . . . , n}, under appropriate conditions, the distribution of the sum n. In Theorem 2.1, we approximate the random function Y by the Gaussian process n. The error in the approximation is expressed in terms of a probability metric defined in terms of comparison of expectations of certain smooth functionals of the processes, and it is bounded by a multiple of the Lyapounov ratio n i, j=1. We state our main results for an arbitrary normalization In most circumstances, such an approximation by Z = Z(n) depending on n is in itself not useful; one would prefer to have some fixed, and if possible well-known limiting approximation. This requires making additional assumptions about the sequence of matrices a(n) as n → ∞. The main tool in proving this is Theorem 3.3, applied to the matrices a0(n)(i, j) := 1{i≤j}

The pre-limiting approximation

A functional limit theorem

Rate of convergence

The shape of permutation tableaux