Normal approximation for isolated balls in an urn allocation model
| dc.creator | Penrose, Mathew D. | |
| dc.date | 2009-01-22 | |
| dc.date.accessioned | 2026-07-07T12:32:59Z | |
| dc.date.available | 2026-07-07T12:32:59Z | |
| dc.description | 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 $S$ to the normal, and estimates on its variance. These show that if $n$, $m$ and $(p_i, 1 \leq i \leq m)$ vary in such a way that $\sup_i p_i = O(n^{-1})$, then $S$ satisfies a CLT if and only if $n^2 \sum_i p_i^2$ tends to infinity, and demonstrate an optimal rate of convergence in the CLT in this case. In the uniform case $(p_i \equiv m^{-1}) with $m$ and $n$ growing proportionately, we provide bounds with better asymptotic constants. The proof of the error bounds are based on Stein's method via size-biased couplings. | |
| dc.description | 32 Pages | |
| dc.identifier | https://arxiv.org/abs/0901.3493 | |
| dc.identifier | http://arxiv.org/abs/0901.3493 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/216973 | |
| dc.subject | Probability | |
| dc.subject | 60F05; 62E17 | |
| dc.title | Normal approximation for isolated balls in an urn allocation model | |
| dc.type | text |