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On the N-tower problem and related problems

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

F. Thomas Bruss ,
Guy Louchard  and
John W. Turner
F. Thomas Bruss*
Affiliation:
Université Libre de Bruxelles
Guy Louchard*
Affiliation:
Université Libre de Bruxelles
John W. Turner*
Affiliation:
Université Libre de Bruxelles
*
∗ Postal address: Université Libre de Bruxelles, Département de Mathématique, CP 210 Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
∗∗ Postal address: Université Libre de Bruxelles, Département d'Informatique, CP 212, Boulevard du Triomphe, B-1050 Bruxelles, Belgium. Email address: [email protected]
∗∗∗ Postal address: Université Libre de Bruxelles, Département de Physique, CP 231, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
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Abstract

Consider N towers each made up of a number of counters. At each step a tower is chosen at random, a counter removed which is then added to another tower also chosen at random. The probability distribution for the time needed to empty one of the towers is obtained in the case N = 3. Arguments are set forward as to why no simple formulae can be expected for N > 3. An asymptotic expression for the mean time before one of the towers becomes empty is derived in the case of four towers when they all initially contain a comparably large number of counters. We then study related problems, in particular the ruin problem for three players. Here we use simple martingale methodology as well as a solution proposed by T. S. Ferguson for a slightly modified problem. Throughout the paper it is our main objective to shed light on the reasons why the case N > 3 is so substantially different from the case N ≤ 3.

Keywords

Eigenfunctionsojourn probabilityhitting timeharmonic analysismartingaleruin probabilityFerguson's problemholomorphyconformal mappingLiouville's theorem

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60J65: Brownian motion
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 278 - 294
Copyright
Copyright © Applied Probability Trust 2003 

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