Mathematics Colloquia and Seminars

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Since their conception 90 years ago, Polya urn models and generalizations have been of great interest to mathematicians, statisticians, biologists, and more recently, computer scientists. The basic model is that an urn contains balls of different colors and at sequential steps a ball is randomly chosen from the urn, its color noted, and then the contents of the urn are altered based on this color; in the classical Polya urn the ball drawn is returned to the urn along with another of the same color. This talk will focus on the limiting distribution of the composition of the urn in a collection of these models, some of which (both the limits and the models) do not appear to have been studied previously. The limiting distributions are characterized as unique fixed points of certain distributional transformations and are explicitly written as products of powers of independent beta and gamma variables. Our methods suggest some conjectures for limiting distributions in further models and these conjectures suggest a bigger picture that we don't yet have. Joint work with Erol Pekoz and Adrian Roellin.