On the error bound in a combinatorial central limit theorem

TL;DR

This paper establishes an explicit bound on the error in approximating the distribution of a sum of randomly permuted array entries by a normal distribution, using Stein's method and concentration inequalities.

Contribution

It provides a new explicit error bound for the combinatorial central limit theorem using Stein's method and exchangeable pairs.

Findings

Bound on Kolmogorov distance involving third moments of array entries

Explicit constant 451 in the error bound

Method applicable to standardized sums of permuted array entries

Abstract

Let $\mathbb{X}=\{X_{ij}: 1\le i,j\le n\}$ be an $n\times n$ array of independent random variables where $n\ge2$. Let $\pi$ be a uniform random permutation of $\{1,2,\dots,n\}$, independent of $\mathbb{X}$, and let $W=\sum_{i=1}^nX_{i\pi(i)}$. Suppose $\mathbb{X}$ is standardized so that ${\mathbb{E}}W=0,\operatorname {Var}(W)=1$. We prove that the Kolmogorov distance between the distribution of $W$ and the standard normal distribution is bounded by $451\sum_{i,j=1}^n{\mathbb{E}}|X_{ij}|^3/n$. Our approach is by Stein's method of exchangeable pairs and the use of a concentration inequality.