Celebratio Mathematica — Blackwell

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D. Blackwell and D. Kendall: “The Martin boundary for Pólya’s urn scheme, and an application to stochastic population growth,” J. Appl. Probability 1 : 2 (December 1964), pp. 284–296. MR 0176518 Zbl 0129.10702 article

In 1923 Eggenberger and Pólya [1923] 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 \( k \) (\( \geq 2 \)) 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 \( s^n \) (a \( k \)-vector with \( j \)-th component \( s_j^n \)) for the numerical composition of the box immediately after the \( n \)-th replacement, so that \( s^0 = (1 \), \( 1,\dots \), \( 1) \), and we observe that \( \{s^n: \) \( n=0 \), \( {}1 \), \( 2,\dots\} \) is a Markov proces for which the state-space consists of all ordered \( k \)-ads of positive integers, the (constant) transition-probability matrix having elements determined by \[ Pr\{s^{n+1} = s^n + e(i)\mid s^n\} = \frac{s_i^n}{S^n} ,\] where \( S^n \) is the sum of the components of \( s^n \) and \( (e(i))_j = \delta_{ij} \). We shall calculate the Martin boundary for this Markov process, and point out some applications to stochastic models for population growth.