A time-dependent version of Pólya's urn (Q920465)

A Pólya urn is considered with the modification that the number of extra balls added of the color drawn is a function of time. An urn containing R red balls and B black balls at time \(n=1\) is considered. A ball is drawn at random from the urn and replaced along with another ball of the same color, so that at time \(n=2\) there are \(R+B+1\) balls; \(R+1\) of them are red with \(R/(R+B)\) and R of them are red otherwise. Draw and replacement are repeated ad infinitum, with the probability of drawing a red ball always equal to the proportion of balls in the urn that are red at that time. The proportions of red balls converge almost surely to a limit that is random and has beta distribution with parameters R and B. The author shows that the proportion of red balls in the urn converges to a random limit that may have a nonzero probability or being 0 or 1, but is nonatomic elsewhere.