A note on the rechargeable Polya urn scheme

A very simple specific case of a Polya urn scheme is as follows. At each trial one draws a ball from an urn with balls of two different colours. Then, one looks at the ball, and returns the ball to the urn together with another ball of the same colour. Then one makes another draw. Et cetera. At the first draw there is one ball of each colour. The rechargeable Polya urn scheme is essentially the same except that between each draw there is a fixed probability that the process starts over with two balls in the urn having different colours. Now, for n = 1, 2, ..., let B(n) and G(n) denote respectively the number of blue and yellow balls in the urn and let Y(n) denote the colour of the ball drawn at the nth draw. Further let Z(n) denote the probability distribution of (B(n),G(n)) given that we have observed Y(m), from m = 1 to m = n. In this note we prove that the sequence Z(1),Z(2),.... converges in distribution.