Almost sure convergence of urn models in a random environment (Q1848051)

The classical Pólya urn model can be described as follows. An urn contains balls of two colors, black and white. In each step a ball is drawn from the urn. If it is black, we add \(a>0\) black balls to the urn and none white, and if it is white, we add \(a\) of the white ones. This replacement policy can be in general described through the replacement matrix \(R=(r_{ij})\), \(i,j=1,2\), giving the number \(r_{ij}\) of balls of color \(j\) to add if ball of color \(i\) is drawn. The authors present the same setting defined in an analogue fashion for \(L\geq 2\) colors. The Robbins-Monro scheme of stochastic approximation theory is used to obtain an a.s.\ convergence result for the initial model, as well as for the generalization (with the wider class of replacement matrices).