Theoretical Statistics and Mathematics Unit, ISI Delhi

June 3, 2015 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Debleena Thacker,
Indian Statistical Institute, Delhi

Title:
Infinite Color Urn Models

Abstract of Talk

In this work, we study the asymptotic properties of certain infinite color balanced urn model. P\'{o}lya urn scheme with infinitely many colors was first introduced by Blackwell and MacQueen in 1973, with only self-reinforcement. We extend the study of infinite color urn models by considering replacement schemes with possible reinforcement of other colors, resulting to possible non-zero off-diagonal entries. For a balanced replacement matrix, one can easily associate a Markov chain on the set of colors as the state space and the normalized replacement matrix as the transition probability matrix. It is also well known that for a balanced urn with finitely many colors, the asymptotic properties of the urn are often determined by the qualitative properties of this Markov chain. We present an explicit coupling of the marginal distribution of the randomly selected color with the associated Markov chain sampled at independent, but random times. We then use this coupling to derive central and local limit theorems, as well as, rate of convergence and the large deviation principles for the randomly selected color, when the associated Markov chain is a random walk on $d$-dimensional integer lattice. We further study general replacement schemes with infinite but countably many colors and show that if the chain is irreducible, aperiodic and positive recurrent with stationary distribution $\pi$, then the distribution of the randomly selected color converges to $\pi$. This is a generalization of the earlier known result for finitely many colors.