## Doctoral Theses

### Infinite Color Urn Models.

4-28-2015

4-28-2016

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Abstract (Summary of the Work)

In recent years, there has been a wide variety of work on random reinforcement models of various kinds. Urn models form an important class of random reinforcement models, with numerous applications in engineering and informatics and bioscience. In recent years there have been several works on different kinds of urn models and their generalizations. For occupancy urn models, where one considers recursive addition of balls into finite or infinite number of boxes, there are some works which introduce models with infinitely many colors, typically represented by the boxes.As observed in [51], the earliest mentions of urn models are in the post-Renaissance period in the works of Huygen, de Moivre, Laplace and other noted mathematicians and scientists. The rigorous study of urn models began with the seminal work of Polya [57, 56], where he introduced the model to study the spread of infectious diseases. We will refer to this model as the classical Polya urn model. Since then, various types of urn schemes with finitely many colors have been widely studied in literature. See [54] for an extensive survey of the known results. However, other than the classical work by Blackwell and MacQueen [13], there has not been much development of infinite color generalization of the Polya urn scheme. In this thesis, we introduce and analyze a new Polya type urn scheme with countably infinite number of colors.1.1 Model description A generalized Polya urn model with finitely many colors can be described as follows: Consider an urn containing finitely many balls of different colors. At any time n â‰¥ 1, a ball is selected uniformly at random from the urn, the color of the selected ball is noted, the selected ball is returned to the urn along with a set of balls of various colors which may depend on the color of the selected ball.The goal is to study the asymptotic properties of the configuration of the urn. Suppose there are K â‰¥ 1, different colors and we denote the configuration of the urn at time n by Un = (Un,1, Un,2 . . . , Un,K), where Un,j denotes the number of balls of color j, 1 â‰¤ j â‰¤ K. The dynamics of the urn model depend on the replacement policy. The replacement policy can be described by a K Ã— K matrix, say R with non negative entries. The (i, j)-th entry of R is the number of balls of color j which are to be added to the urn if the selected color is i. In literature, R is termed as the replacement matrix. Let Zn denote the random color of the ball selected at the (n + 1)-th draw. w. The dynamics of the model can then be written asUn+1 = Un + RZnwhere RZn is the Zn-th row of the replacement matrix R.

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