Almost Sure Convergence of Randomized Urn Models with Application to Elephant Random Walk

We consider a randomized urn model with objects of finitely many colors. The replacement matrices are random, and are conditionally independent of the color chosen given the past. Further, the conditional expectations of the replacement matrices are close to an almost surely irreducible matrix. We obtain almost sure and $L^1$ convergence of the configuration vector, the proportion vector and the count vector. We show that first moment is sufficient for i.i.d.
replacement matrices independent of past color choices. This significantly improves the similar results for urn models obtained in Athreya and Ney (1972) requiring $L{\log }_{+}L$ moments. For more general adaptive sequence of replacement matrices, a little more than $L{\log }_{+}L$ condition is required. Similar results based on $L^1$ moment assumption alone has been considered independently and in parallel in Zhang (2018). Finally, using the result, we study a delayed elephant random walk on the nonnegative orthant in $d$ dimension with random memory.