On generalized Pólya urn models
The authors consider a two-color urn model characterized by a \(2\times 2\) matrix of nonnegative integers \((\begin{smallmatrix} a & b\\ c & d \end{smallmatrix})\) as follows. The urn process evolves in stages. At each stage a ball is drawn and its color is observed. If a ball of color 1 is drawn at a stage, it is returned to the urn together with \(a\) balls of color 1 and \(b\) balls of color 2. If a ball of color 2 is drawn at a stage, the same manipulation with the matrix row \((c,d)\) is made. Exact distributions of the number of balls of a specific color that appear after \(n\) draws are derived. The variety of essentially different asymptotics that appear for various matrices as \(n\to \infty\) is illustrated by an example, where the authors find an asymptotic mean value.
scientific article; zbMATH DE number 1507149