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Martingales in the Ok Corral
Author(s) -
Kingman J. F. C.
Publication year - 1999
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609399006098
Subject(s) - mathematics , simple (philosophy) , combinatorics , subject (documents) , mathematics subject classification , distribution (mathematics) , calculus (dental) , mathematical economics , mathematical analysis , epistemology , philosophy , medicine , dentistry , library science , computer science
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] 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 / m 3/4 has a simple asymptotic distribution that leads at once to (1.1). 1991 Mathematics Subject Classification 60F05.