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Normal Approximation for Isolated Balls in an Urn Allocation Model


 
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1. Title Title of document Normal Approximation for Isolated Balls in an Urn Allocation Model
 
2. Creator Author's name, affiliation, country Mathew D. Penrose; University of Bath; United Kingdom
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Berry-Esseen bound, central limit theorem, occupancy scheme, size biased coupling, Stein's method.
 
3. Subject Subject classification Primary 60F05. Secondary 62E17, 60C05.
 
4. Description Abstract 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Alexander von Humboldt Foundation
 
7. Date (YYYY-MM-DD) 2009-10-06
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/699
 
10. Identifier Digital Object Identifier 10.1214/EJP.v14-699
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 14
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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