A Study of the tenability of Pólya urn schemes

Pólya urn schemes are statistical models that use colored balls to represent a growth or decay process. A ball is drawn from the urn at random, and depending on the color drawn, certain rules of a replacement matrix are followed that may change the composition of the urn. If these rules can always be applied, the urn scheme is said to be tenable. The tenability of an urn scheme is of importance as it represents the sustainability of the process. It is therefore of interest to analyze the specific characteristics of tenable urn schemes to determine if a particular urn scheme is tenable, and if it is untenable, to uncover the specific reasons why. Characteristics of tenable urn schemes are defined mathematically through linear algebraic restrictions on the replacement matrix as well as the vector of initial conditions. Characteristics are also explored using graphical notation and tree structures. Additionally, properties such as the principal eigenvalues of the replacement matrix are analyzed as eigenvalues play a key role in the limiting distribution of urn composition over time