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

Abstract

1. In 1923 Eggenberger and Pólya introduced the following ‘urn scheme’ as a model for the development of a contagious phenomenon. A box containsbblack andrred 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 ballof 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 keepm= 2 and it will be convenient further to simplify the scheme by takingb=r= 1 as the initial condition. We shall however generalise the scheme in another direction by allowing an arbitrary numberk(≧2) of colours. Thus initially the box will containkdifferently coloured balls and successive random drawings will be followed by double replacement as before. We writesn(ak-vector withjth component) for the numerical composition of the box immediately after the nth replacement, so thatand we observe thatis a Markov process for which the state-space consists of all orderedk-ads of positive integers, the (constant) transition-probability matrix having elements determined bywhere Snis the sum of the components of snand (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.