Central limit theorems for infinite urn models. (English) Zbl 0685.60023

Dutko, Michael

Ann. Probab. 17, No. 3, 1255-1263 (1989).

Author’s summary: An urn model is defined as follows: n balls are independently placed in an infinite set of urns and each ball has probability \(p_ k>0\) of being assigned to the k th urn. We assume that \(p_ k\geq p_{k+1}\) for all k and that \(\sum^{\infty}_{k=1}p_ k=1\). A random variable \(Z_ n\) is defined to be the number of occupied urns after n balls have been thrown.
The main result is that \(Z_ n\), when normalized, converges in distribution to the standard normal distribution. Convergence to N(0,1) holds for all sequences \(\{p_ k\}\) such that \(\lim_{n\to \infty}Var Z_{N(n)}=\infty,\) where N(n) is a Poisson random variable with mean n. This generalizes a result of S. Karlin [J. Math. Mech. 17, 373-401 (1967; Zbl 0154.437)].

Reviewer: L.Mutafchiev

MSC:

60F05	Central limit and other weak theorems
60C05	Combinatorial probability

Keywords:

central limit theorems; urn model; infinite set of urns

Citations:

Zbl 0154.437