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The Martin boundary for Polya's urn scheme, and an application to stochastic population growth

Published online by Cambridge University Press:  14 July 2016

David Blackwell  and
David Kendall
David Blackwell
Affiliation:
University of California
David Kendall
Affiliation:
University of Cambridge
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Extract

1. In 1923 Eggenberger and Pólya introduced the following ‘urn scheme’ as a model for the development of a contagious phenomenon. A box contains b black and r red balls, and a ball is drawn from it at random with ‘double replacement’ (i.e. whatever ball is drawn, it is returned to the box together with a fresh ball of the same colour); the procedure is then continued indefinitely. A slightly more complicated version with m-fold replacement is sometimes discussed, but it will be sufficient for our purposes to keep m = 2 and it will be convenient further to simplify the scheme by taking b = r = 1 as the initial condition. We shall however generalise the scheme in another direction by allowing an arbitrary number k(≧2) of colours. Thus initially the box will contain k differently coloured balls and successive random drawings will be followed by double replacement as before. We write sn (a k-vector with jth component ) for the numerical composition of the box immediately after the nth replacement, so that and we observe that is a Markov process for which the state-space consists of all ordered k-ads of positive integers, the (constant) transition-probability matrix having elements determined by where Sn is the sum of the components of sn and (e(i))j = δij. We shall calculate the Martin boundary for this Markov process, and point out some applications to stochastic models for population growth.

Type
Research Papers
Information
Journal of Applied Probability , Volume 1 , Issue 2 , December 1964 , pp. 284 - 296
Copyright
Copyright © Applied Probability Trust 

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