Asymptotic theorems for urn models with nonhomogeneous generating matrices

The generalized Friedman’s urn (GFU) model has been extensively applied to biostatistics. However, in the literature, all the asymptotic results concerning the GFU are established under the assumption of a homogeneous generating matrix, whereas, in practical applications, the generating matrices are often nonhomogeneous. On the other hand, even for the homogeneous case, the generating matrix is assumed in the literature to have a diagonal Jordan form and satisfies λ>2 Re(λ1), where λ and λ1 are the largest eigenvalue and the eigenvalue of the second largest real part of the generating matrix (see Smythe, 1996, Stochastic Process. Appl. 65, 115–137). In this paper, we study the asymptotic properties of the GFU model associated with nonhomogeneous generating matrices. The results are applicable to a variety of settings, such as the adaptive allocation rules with time trends in clinical trials and those with covariates. These results also apply to the case of a homogeneous generating matrix with a general Jordan form as well as the case where λ = 2 Re(λ1).