An $L_p$ Bound for the Remainder in a Combinatorial Central Limit Theorem

Abstract

For $n \geqq 2$ let $X_{nij}, i, j = 1, \cdots, n$, be a square array of independent random variables with finite variances and let $\pi_n = (\pi_n(1), \cdots, \pi_n(n))$ be a random permutation of $(1, \cdots, n)$ independent of the $X_{nij}$'s. By using Stein's method, a bound is obtained for the $L_p$ norm $(1 \leqq p \leqq \infty)$ with respect to the Lebesgue measure of the difference between the distribution function of $(W_n - EW_n)/(\operatorname{Var} W_n)^{\frac{1}{2}}$ and the standard normal distribution function where $W_n = \sum^n_{i=1} X_{ni\pi_n(i)}$. This result generalizes and improves a number of known results. In particular, it provides bounds for Motoo's combinatorial central limit theorem as well as the central limit theorem.