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