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2009 Normal Approximation for Isolated Balls in an Urn Allocation Model
Mathew Penrose
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Electron. J. Probab. 14: 2155-2181 (2009). DOI: 10.1214/EJP.v14-699


Consider throwing $n$ balls at random into $m$ urns, each ball landing in urn $i$ with probability $p(i)$. Let $S$ be the resulting number of singletons, i.e., urns containing just one ball. We give an error bound for the Kolmogorov distance from the distribution of $S$ to the normal, and estimates on its variance. These show that if $n$, $m$ and $(p(i))$ vary in such a way that $n p(i)$ remains bounded uniformly in $n$ and $i$, then $S$ satisfies a CLT if and only if ($n$ squared) times the sum of the squares of the entries $p(i)$ tends to infinity, and demonstrate an optimal rate of convergence in the CLT in this case. In the uniform case with all $p(i)$ equal and with $m$ and $n$ growing proportionately, we provide bounds with better asymptotic constants. The proof of the error bounds is based on Stein's method via size-biased couplings.


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Mathew Penrose. "Normal Approximation for Isolated Balls in an Urn Allocation Model." Electron. J. Probab. 14 2155 - 2181, 2009.


Accepted: 6 October 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1191.60029
MathSciNet: MR2550296
Digital Object Identifier: 10.1214/EJP.v14-699

Primary: 60F05
Secondary: 60C05 , 62E17

Keywords: Berry-Esseen bound , central limit theorem , occupancy scheme , size biased coupling , Stein's method

Vol.14 • 2009
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