Almost Sure Convergence of Randomized Urn Models with Finite Mean

Ujan Gangopadhyay, Krishanu Maulik

Published 2019-09-24Version 1

We consider a randomized urn model containing objects of finitely many colors in this article. The replacement matrices are allowed to be random, subject to the minimum conditions that the mechanism to choose color and the replacement matrix at each step are conditionally independent given the past, as well as, the conditional expectations of the replacement matrices are close to a (possibly random) irreducible (and hence positive recurrent) matrix. We obtain almost sure convergence of the configuration vector, the proportion vector and the count vector under finite first moment condition alone. The convergence is shown to be $L^1$ as well. We show that first moment assumption is sufficient when the replacement matrix sequence is i.i.d. and independent of the past choices of the color. This significantly improves the similar results for urn models obtained by Athreya and Ney (1972), by weakening the moment assumptions on replacement matrices from $L \log_+ L$ to $L^1$. For more general adaptive sequence of replacement matrix, a little more than $L \log_+ L$ condition is required.