Normal approximation for isolated balls in an urn allocation model
Mathew D. Penrose, University of Bath
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.
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