Characterization and Enumeration of Certain Classes of Tenable Pólya Urns Grown by Drawing Multisets of Balls

Abstract

We study necessary and sufficient conditions for the strong tenability of Polya urn schemes under the sampling of multisets of balls. We also investigate sufficient conditions for the tenability (not necessarily in the strong sense) of Polya urn schemes under the sampling of multisets of balls. We enumerate certain balanced classes and give algorithmic constructions for the replacement matrices for members in the class. We probabilistically analyze the zero-balanced tenable class, and find the asymptotic average proportion of each color, when the starting number of balls is large. We also give an algorithm to determine tenability and construct the Markov chain underlying the scheme, when it is tenable.