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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: [email protected]
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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 

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