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April, 1978 An $L_p$ Bound for the Remainder in a Combinatorial Central Limit Theorem
Soo-Thong Ho, Louis H. Y. Chen
Ann. Probab. 6(2): 231-249 (April, 1978). DOI: 10.1214/aop/1176995570


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.


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Soo-Thong Ho. Louis H. Y. Chen. "An $L_p$ Bound for the Remainder in a Combinatorial Central Limit Theorem." Ann. Probab. 6 (2) 231 - 249, April, 1978.


Published: April, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0375.60028
MathSciNet: MR478291
Digital Object Identifier: 10.1214/aop/1176995570

Primary: 60F05
Secondary: 62E20 , 62G99

Keywords: $L_p$ bound , Berry-Esseen bound , combinatorial central limit theorem , Normal approximation , permutation tests , Stein's method

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 2 • April, 1978
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