The Gaussian approximation for multi-color generalized Friedman’s urn model

Abstract

The generalized Friedman’s urn model is a popular urn model which is widely used in many disciplines. In particular, it is extensively used in treatment allocation schemes in clinical trials. In this paper, we show that both the urn composition process and the allocation proportion process can be approximated by a multi-dimensional Gaussian process almost surely for a multi-color generalized Friedman’s urn model with both homogeneous and non-homogeneous generating matrices. The Gaussian process is a solution of a stochastic differential equation. This Gaussian approximation is important for the understanding of the behavior of the urn process and is also useful for statistical inferences. As an application, we obtain the asymptotic properties including the asymptotic normality and the law of the iterated logarithm for a multi-color generalized Friedman’s urn model as well as the randomized-play-the-winner rule as a special case.