The Annals of Probability

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

Soo-Thong Ho and Louis H. Y. Chen

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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.

Article information

Ann. Probab., Volume 6, Number 2 (1978), 231-249.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory 62G99: None of the above, but in this section

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


Ho, Soo-Thong; Chen, Louis H. Y. An $L_p$ Bound for the Remainder in a Combinatorial Central Limit Theorem. Ann. Probab. 6 (1978), no. 2, 231--249. doi:10.1214/aop/1176995570.

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