Seminar at SMU Delhi

In this talk we will present a new urn model consisting of balls of infinite but countably many colors which we index by the integers. We will consider two special replacement matrices, one arriving from the right shift operator, and the other arriving from the simple symmetric random walk on the one dimensional integer lattice. We show using martingale techniques that in both the cases the expected proportion of colors converges to a standard normal distribution after an appropriate centering and scaling by $\sqrt{\log n}$. This shows that even though the associated Markov chain has different qualitative properties, namely one is transient and the other is null recurrent, the infinite color urn models have same asymptotic behavior. This is in sharp contrast to what is generally observed for finite color urn models.