On death processes and urn models

In this work we are concerned with so-called Pólya-Eggenberger urn models, which in the simplest case of two colors can be described as follows. At the beginning, the urn contains n white and m black balls. At every step, we choose a ball at random from the urn, examine its color and put it back into the urn and then add/remove balls according to its color by the following rules: if the ball is white, then we put α white and β black balls into the urn, while if the ball is black, then γ white balls and δ black balls are put into the urn. The values α, β, γ, δ ∈ Z are fixed integer values and the urn model is specified by the transition matrix M = ( α β γ δ ) . Models with r (≥ 2) types of colors can be described in an analogous way and are specified by an r × r transition matrix. Urn models are simple, useful mathematical tools for describing many evolutionary processes in diverse fields of application such as analysis of algorithms and data structures, and statistical genetics. Due to their importance in applications, there is a huge literature on the stochastic behavior of urn models; see for example [10, 11, 18]. Recently, a few different approaches have been proposed, which yield deep and far-reaching results [1, 4, 5, 8, 9, 19]. Most papers in the literature impose the so-called tenability condition on the transition matrix, so that the process of adding/removing balls can be continued ad infinitum. However, in some applications, examples given below, there are urn models with a very different nature, which we will refer to as “diminishing urn models.” Such models have recently received some attention, see for example [21, 22, 2, 20, 4, 6]. For simplicity of presentation, we describe diminishing urn models in the case of balls with two types of colors, black and white. We