On the error bound in a combinatorial central limit theorem

Let x = {X ij: 1 ≤ i. j ≤ n} be an n x n array of independent random variables where n ≥ 2. Let π be a uniform random permutation of {1,2,...n}, independent of x, and let W =$\Sigma _{i = 1}^n{{\text{X}}_{i\pi \left( i \right)}}$. Suppose x is standardized so that EW = 0, Var(W) = 1. We prove that t...

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Bibliographic Details
Published in:Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability 2015-02, Vol.21 (1), p.335-359
Main Authors: CHEN, LOUIS H. Y., FANG, XIAO
Format: Article
Language:English
Subjects:
combinatorial central limit theorem
concentration inequality
exchangeable pairs
Stein’s method
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Summary:Let x = {X ij: 1 ≤ i. j ≤ n} be an n x n array of independent random variables where n ≥ 2. Let π be a uniform random permutation of {1,2,...n}, independent of x, and let W =$\Sigma _{i = 1}^n{{\text{X}}_{i\pi \left( i \right)}}$. Suppose x is standardized so that EW = 0, Var(W) = 1. We prove that the Kolmogorov distance between the distribution of W and the standard normal distribution is bounded by 451 $\Sigma _{i,j = 1}^n\mathbb{K}{\left| {{{\text{X}}_{ij}}} \right|^3}/n$. Our approach is by Stein's method of exchangeable pairs and the use of a concentration inequality.
ISSN:1350-7265
1573-9759
DOI:10.3150/13-bej569