On the Error Bound in a Combinatorial Central Limit Theorem

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