We consider an urn model where the replacement matrices are random and may depend on past information. Such models are useful in several situations including clinical trials. Using stochastic approximation techniques, we study convergence of scaled color count and configuration vectors, when the replacement matrices have only first moment and the expected replacement matrix "converges to" an irreducible, but not necessarily balanced, matrix. We show the convergence to be in probability. Under some additional L log L -type assumptions, we extend the convergence to be almost sure. We shall compare the assumptions and the stochastic approximation results in these two setups and give a brief sketch of proof.
This is a joint work with Ujan Gangopadhyay from University of Southern California.