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Journal Article

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

Soo-Thong Ho and Louis H. Y. Chen
The Annals of Probability
Vol. 6, No. 2 (Apr., 1978), pp. 231-249
Stable URL: http://www.jstor.org/stable/2243215
Page Count: 19

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## 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.

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