A Uniform Bound on a Combinatorial Central Limit Theorem

Abstract

This paper establishes a combinatorial central limit theorem for an array of independent random variables (X ij ), 1 ≤ i, j ≤ n, (n → ∞) with finite third moments. Let π = (π(1), π(2), …, π(n)) be a permutation of {1, 2, …, n}, and define W n = ∑ i X iπ(i). Then the authors prove the following uniform central limit property: , where F n is the distribution of is the strandrad normal distribution, and with [Xcirc] ij is a suitable normalization of X ij . The proof uses Stein's method and the result generalizes and improves a number of known results.