Hostname: page-component-84b7d79bbc-7nlkj Total loading time: 0 Render date: 2024-07-25T14:15:29.913Z Has data issue: false hasContentIssue false
  • English
  • Français

Sequential urn schemes and birth processes

Published online by Cambridge University Press:  01 July 2016

Lars Holst  and
Jürg Hüsler
Lars Holst*
Affiliation:
Uppsala University
Jürg Hüsler*
Affiliation:
University of Bern
*
∗Dept of Mathematics, Uppsala University, Thunbergsvägen 3, S-752 38 Uppsala, Sweden.
∗∗Dept of Mathematical Statistics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland.
Get access Rights & Permissions [Opens in a new window]

Abstract

An urn contains balls of different colours, which are randomly drawn one at a time. After each draw the number of balls in the urn with the same colour as the ball last drawn is changed. Special cases are sampling with and without replacement, and Pólya sampling. The drawing stops when a given number of colours has been drawn a given number of times. The number of times the different colours have been drawn is studied in this paper by imbedding the urn scheme in birth processes. Both exact and asymptotic results are obtained. In particular, waiting times for sampling with and without replacement and for Pólya sampling are considered.

Keywords

SAMPLING WITH AND WITHOUT REPLACEMENTPOLYA’S URN MODELSEQUENTIAL OCCUPANCYEXACT DISTRIBUTION THEORYASYMPTOTIC DISTRIBUTION THEORY
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 2 , June 1985 , pp. 257 - 279
Copyright
Copyright © Applied Probability Trust 1985 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, K., Sobel, M. and Uppuluri, V. R. R. (1982) Quota fulfilment times. Canad. J. Statist. 10, 73–88.CrossRefGoogle Scholar
Békéssy, A. (1964) On classical occupany problems II (sequential occupancy). Magy. Tud. Akad. Mat. Kutató. Int. Közl. 9, 133–141.Google Scholar
David, H. A. (1980) Order Statistics. Wiley, New York.Google Scholar
Feller, W. (1957) An Introduction to Probability Theory and its Applications , Vol. 1. Wiley, New York.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications , Vol. 2. Wiley, New York.Google Scholar
Holst, L. (1979) A unified approach to limit theorems for urn models. J. App. Prob. 16, 154–162.CrossRefGoogle Scholar
Holst, L. (1981) On sequential occupancy problems. J. Appl. Prob. 18, 435–442.CrossRefGoogle Scholar
Janson, S. (1983) Limit theorems for some sequential occupancy problems. J. Appl. Prob. 20, 545–553.CrossRefGoogle 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
Rao, C. R. (1973) Linear Statistical Inference and its Applications , 2nd edn. Wiley, New York.CrossRefGoogle 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.CrossRefGoogle Scholar