## Bernoulli

• Bernoulli
• Volume 21, Number 1 (2015), 335-359.

### On the error bound in a combinatorial central limit theorem

#### Abstract

Let $\mathbb{X}=\{X_{ij}\colon\ 1\le i,j\le n\}$ be an $n\times n$ array of independent random variables where $n\ge2$. Let $\pi$ be a uniform random permutation of $\{1,2,\dots,n\}$, independent of $\mathbb{X}$, and let $W=\sum_{i=1}^{n}X_{i\pi(i)}$. Suppose $\mathbb{X}$ is standardized so that $\mathbb{E}W=0$, $\operatorname{Var}(W)=1$. We prove that the Kolmogorov distance between the distribution of $W$ and the standard normal distribution is bounded by $451\sum_{i,j=1}^{n}\mathbb{E}|X_{ij}|^{3}/n$. Our approach is by Stein’s method of exchangeable pairs and the use of a concentration inequality.

#### Article information

Source
Bernoulli, Volume 21, Number 1 (2015), 335-359.

Dates
First available in Project Euclid: 17 March 2015

https://projecteuclid.org/euclid.bj/1426597072

Digital Object Identifier
doi:10.3150/13-BEJ569

Mathematical Reviews number (MathSciNet)
MR3322321

Zentralblatt MATH identifier
1354.60011

#### Citation

Chen, Louis H.Y.; Fang, Xiao. On the error bound in a combinatorial central limit theorem. Bernoulli 21 (2015), no. 1, 335--359. doi:10.3150/13-BEJ569. https://projecteuclid.org/euclid.bj/1426597072

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