Central Limit Theorems for Infinite Urn Models

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 kth urn. We assume that pk≥ pk + 1for all k and that ∑∞ k = 1pk= 1. A random variable Znis defined to be the number of occupied urns after...

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Bibliographic Details
Published in:The Annals of probability 1989-07, Vol.17 (3), p.1255-1263
Main Author: Dutko, Michael
Format: Article
Language:English
Subjects:
60C05
60F05
Central limit theorem
Continuous variables
Exact sciences and technology
Gaussian distributions
Limit theorems
Mathematical theorems
Mathematics
Perceptron convergence procedure
Probability and statistics
Probability theory
Probability theory and stochastic processes
Random variables
Sciences and techniques of general use
urn model
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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 kth urn. We assume that pk≥ pk + 1for all k and that ∑∞ k = 1pk= 1. A random variable Znis defined to be the number of occupied urns after n balls have been thrown. The main result is that Zn, when normalized, converges in distribution to the standard normal distribution. Convergence to N(0, 1) holds for all sequences {pk} such that$\lim_{n \rightarrow \infty} \operatorname{Var}Z_{N(n)} = \infty$, where N(n) is a Poisson random variable with mean n. This generalizes a result of Karlin.
ISSN:0091-1798
2168-894X
DOI:10.1214/aop/1176991268