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 conce rning the GFU are established under the assumption of a homogeneous generat ing matrix, whereas, in practical applications, the generating matrices are often nonhomogeneous. On the other hand, even for the homogeneous case, th e generating matrix is assumed in the literature to have a diagonal Jordan form and satisfies lambda>2 Re(lambda(1)), where lambda and lambda(1) are t he 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 a ssociated with nonhomogeneous generating matrices. The results are applicab le to a variety of settings, such as the adaptive allocation rules with tim e trends in clinical trials and those with covariates. These results also a pply to the case of a homogeneous generating matrix with a general Jordan f orm as well as the case where lambda = 2 Re(lambda(1)). (C) 1999 Elsevier S cience B.V. All rights reserved.