On Pólya Urn Schemes with Infinitely Many Colors.

In this talk, we extend the mutlicolor P/'olya urn schemes to countably infinitely many colors. We index the colors by \mathbb{Z}. Throughout the talk, we discuss mainly replacement matrices arising out of random walks. We show that the proportion of colors with suitable centering and scaling show central tendencies. Also the centering and scaling are fairly general. This behavior is in sharp contrast with the finite color case, where the asypmtotic behavior of the proportion of colors are determined by the qualitative properties (transience or recurrence) of the Markov chain underlying the replacement matrix. We also extend the infinite color case to fairly general graphs on \mathbb{R}^{d} and show that the proportion of colors show central tendencies similar to that in the case for \mathbb{Z}. Even the centering and scaling remains same.