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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.

References

Bartlett, M. S. (1938) The characteristic function of a conditional statistic. J. London Math. Soc. 13, 62–67.Google Scholar
David, F. N. and Barton, D. E. (1962) Combinatorial Chance. Hafner, New York.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Holst, L. (1972) Asymptotic normality and efficiency for certain goodness-of-fit tests. Biometrika 59, 137–145.Google Scholar
Holst, L. (1979) Two conditional limit theorems with applications. Ann. Statist. To appear.Google Scholar
Johnson, N. L. and Kotz, S. (1969) Distributions in Statistics: Discrete Distributions. Houghton Mifflin, Boston.Google Scholar
Johnson, N. L. and Kotz, S. (1977) Urn Models and Their Applications: An Approach to Modern Discrete Probability Theory. Wiley, New York.Google Scholar
Kaplan, N. (1977) A generalization of a result of Erdös and Rényi. J. Appl. Prob. 14, 212–216.Google Scholar
Kolchin, V. F. and Chistyakov, V. P. (1974) Combinatorial problems of probability theory. Itogi Nauk. Tekhn. Teor. Veroyatnist. Mat. Statist. Teor. Kibernet. 11, 5–45. (English translation in J. Soviet Maths 4, 217–243).Google Scholar
Le Cam, L. (1958) Un théorème sur la division d'un intervalle par des points pris au hasard. Publ. Inst. Statist. Univ. Paris 7, 7–16.Google Scholar
Pothoff, R. F. and Whittinghill, M. (1966) Testing for homogeneity II: The Poisson distribution. Biometrika 53, 183–190.Google Scholar
Rao, C. R. (1973) Linear Statistical Inference and its Applications, 2nd edn. Wiley, New York.Google Scholar
Renyi, A. (1962) Three new proofs and a generalization of a theorem of Irving Weiss. Publ. Math. Inst. Hung. Acad. Sci. 7, 203–214.Google Scholar
Samuel-Cahn, E. (1974) Asymptotic distributions for occupancy and waiting time problems with positive probability of falling through the cells. Ann. Prob. 2, 515–521.Google Scholar
Vit, P. (1974) Testing for homogeneity: the geometric distribution. Biometrika 61, 565–568.CrossRefGoogle Scholar
Wisniewski, T. K. M. (1968) Testing for homogeneity of a binomial series. Biometrika 55, 426–428.Google Scholar