On Some Classes of Urn Models with Multiple-drawing
Open AccessWe study three classes of multiple-drawing urn models: Self-equilibrium Friedman-like urns, affine diminishing urns and dynamic Friedman urns with opposite-reinforcement.Chapter 1 starts with providing a general introduction on urn models, which is followed by a detailed introduction on the classes we investigate: multiple-drawing urns, diminishing urns and dynamic urns. Chapter 2 presents the main analytic tools we use for proving the theorems.Chapter 3 opens our discussion on multiple-drawing urns by formulating a system of recurrences of the moments. The system reveals useful properties of the class of multiple-drawing urns.In Chapter 4, we introduce a class of self-equilibrium Friedman-like urns. The random replacement matrix of this class has the symmetry property of a Friedman type urn. We assume opposite-reinforcement so that the color has lower appearance in the drawn sample gets reinforced stronger. We prove almost-sure convergence and find a central limit theorem for the proportion of white balls by the method of stochastic approximation. An application in stock trading is discussed in the end.In Chapter 5, we propose a class of affine diminishing urns following the research on affine multiple-drawing urns in [48]. In the tenability range, we investigate the composition of the urn at different stages of drawing. Through asymptotic analysis, we find a phase transition from a sublinear number of draws to a linear number of draws. In both phases, we get central limit theorems for the number of white balls through martingale convergence theorem. A generalized OK Corral urn model and a model to measure the dynamics of market depth are discussed as applications.In Chapter 6, we investigate a class of Dynamic Friedman urns with opposite-reinforcement. We have a classification into small-increment schemes and large-increment schemes based on the rate of a dynamic function. For small-increment urns, we prove almost-sure convergence and get a central limit theorem for the proportion of white balls by the convergence of stochastic approximation algorithms. For large-increment urns, by imposing the affinity condition, we prove almost-sure convergence and show a way to identify the limit distribution of the proportion of white balls via martingale theory. Chapter 7 is where some potential future developments of the above mentioned three classes are discussed.
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