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A unified approach to limit theorems for urn models

Published online by Cambridge University Press:  14 July 2016

Lars Holst
Lars Holst*
Affiliation:
Uppsala University
*
∗Postal Address: Uppsala Universitet, Matematiska Institutionen, Thunbergsv. 3, S-75238 Uppsala, Sweden.
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Abstract

An urn contains A balls of each of N colours. At random n balls are drawn in succession without replacement, with replacement or with replacement together with S new balls of the same colour. Let Xk be the number of drawn balls having colour k, k = 1, …, N. For a given function f the characteristic function of the random variable ZM = f(X1)+ … + f(XM), M ≦ N, is derived. A limit theorem for ZM when M, N, n → ∞is proved by a general method. The theorem covers many special cases discussed separately in the literature. As applications of the theorem limit distributions are obtained for some occupancy problems and for dispersion statistics for the binomial, Poisson and negative-binomial distribution.

Keywords

URN MODELSMULTINOMIAL DISTRIBUTIONPÓLYA DISTRIBUTIONLIMIT THEOREMSOCCUPANCY PROBLEMSDISPERSION TESTS
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 1 , March 1979 , pp. 154 - 162
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

This research was originally supported in part by Mathematics Research Center, University of Wisconsin-Madison, under United States Army Contract No. DAAG 29–75-C-0024.

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