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MARTINGALES IN THE OK CORRAL

Published online by Cambridge University Press:  01 September 1999

J. F. C. KINGMAN
J. F. C. KINGMAN
Affiliation:
University of Bristol, Senate House, Tyndall Avenue, Bristol BS8 1TH
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Abstract

In the model of the OK Corral formulated by Williams and McIlroy [2]: ‘Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or there is no one left on the other side.’ They are interested in the number S of survivors when the shooting ceases, and the surprising result, for which they give both numerical and heuristic evidence, is that when m=n,

formula here

as m→∞ (where E denotes expectation).

It is the occurrence of this curious power of m, rather than any application to real gunfights, which makes the Williams–McIlroy process of interest. The purpose of this paper is to give an essentially elementary proof of the fact that if m and n are not too different, then S/m3/4 has a simple asymptotic distribution that leads at once to (1.1).

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 31 , Issue 5 , September 1999 , pp. 601 - 606
Copyright
© The London Mathematical Society 1999

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