Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T07:22:44.365Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic probabilities in a sequential urn scheme related to the matchbox problem

Part of: Limit theorems Markov processes Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

Wolfgang Stadje
Wolfgang Stadje*
Affiliation:
Universität Osnabrück
*
∗ Postal address: Universität Osnabrück, Fachbereich Mathematik/Informatik, Albrechtstr 28, D-49076, Osnabrück, Germany. Email address: wolfgang@mathematik.uni-osnabruck.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Generalizing the classical Banach matchbox problem, we consider the process of removing two types of ‘items’ from a ‘pile’ with selection probabilities for the type of the next item to be removed depending on the current numbers of remaining items, and thus changing sequentially. Under various conditions on the probability pn1,n2 that the next removal will take away an item of type I, given that n1 and n2 are the current numbers of items of the two types, we derive asymptotic formulas (as the initial pile size tends to infinity) for the probability that the items of type I are completely removed first and for the number of items left. In some special cases we also obtain explicit results.

Keywords

Matchbox probleminteracting speciesextinction probabilityresidual population sizeasymptotic behaviorvariable removal probability

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 60F99: None of the above, but in this section 60C05: Combinatorial probability
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 831 - 849
Copyright
Copyright © Applied Probability Trust 1998 

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.Google Scholar
Ben-David, A. and Jagerman, D. L. (1993). A non-deterministic approach towards conflict resolution in inconsistent training sets. Preprint, Business School, The Hebrew University of Jerusalem.Google Scholar
Bender, E. A. (1973). Central and local limit theorems applied to asymptotic enumeration. J. Combinatorial Theory A 15, 91–111.Google Scholar
Berg, S. (1987). A variant of Banach's match box problem. Ann. Discr. Math. 33, 1–8.Google Scholar
Billard, L. (1974). Competition between two species. Stoch. Proc. Appl. 2, 391–398.CrossRefGoogle Scholar
Billard, L. (1977). On Lotka-Volterra predator prey models. J. Appl. Prob. 14, 375–381.CrossRefGoogle Scholar
Billard., L. (1981). Generalized two-dimensional birth and death processes and some applications. J. Appl. Prob. 18, 335–347.CrossRefGoogle Scholar
Billard, L. and Kryscio, R. J. (1977). The transition probabilities of a bounded bivariate pure death process. Math. Biosci. 37, 205–221.CrossRefGoogle Scholar
Chesson, J. (1976). A non-central multivariate hypergeometric distribution arising from biased sampling with application to selective predation. J. Appl. Prob. 13, 795–797.Google Scholar
Chung, K. L. (1968). A Course in Probability Theory. Harcourt, Brace & World, New York.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Goczyla, K. (1986). The generalized Banach match-box problem. Acta Appl. Math. 5, 27–36.Google Scholar
Groeneveld, R. A. and Arnold, B. C. (1984). Limit laws in the best of 2n-1 Bernoulli trials. Nav. Res. Logist. Quart. 31, 275–281.CrossRefGoogle Scholar
Harris, B. (1971). Generalized Banach match box problems and asymptotic distributions for inverse multinomial sampling. Bull. Inst. Internat. Statist. 44, 202–206.Google Scholar
Hitchcock, S. E. (1986). Extinction probabilities in predator-prey models. J. Appl. Prob. 23, 1–13.Google Scholar
Holst, L. (1989). A note on Banach's match box problem. Statist. Prob. Lett. 8, 441–443.Google Scholar
Knuth, D. (1984). The toilet paper problem. Amer. Math. Monthly 91, 465–470.Google Scholar
Maisel, H. (1966). Best k of 2k-1 comparisons. J. Amer. Statist. Assoc. 61, 329–344.Google Scholar
Mendelson, H., Pliskin, J. S. and Yechiali, U. (1980). A stochastic allocation problem. Operat. Res. 28, 687–693.Google Scholar
Menon, V. V. and Indira, N. K. (1983). On the asymptotic normality of the number of replications of a paired comparison. J. Appl. Prob. 20, 554–562.Google Scholar
Nagaraja, H. N. and Chan, W. T. (1989) On the number of games played in the best of (2n-1) series. Nav. Res. Logist. 36, 297–310.Google Scholar
Ridler-Rowe, C. J. (1978). On competition between two species. J. Appl. Prob. 15, 457–465.CrossRefGoogle Scholar
Severo, N. C. (1969). A recursion theorem on solving differential-difference equations and applications to some stochastic processes. J. Appl. Prob. 6, 673–681.Google Scholar
Sigrist, K. (1989). n-point, win-by-k games. J. Appl. Prob. 26, 807–814.Google Scholar
Stadje, W. (1986) A note on the Neyman–Pearson fundamental lemma. Methods Oper. Res. 53, 661–670.Google Scholar
Stirzaker, D. (1988). A generalization of the matchbox problem. Math. Sci. 13, 104–114.Google Scholar
Uppuluri, V. R. R. and Blot, W. J. (1974). Asymptotic properties of the number of replications of a paired comparison. J. Appl. Prob. 11, 43–52.Google Scholar