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

作　　者：ZHANG LiXin HU FeiFang 

机构地区：[1]Department of Mathematics,Zhejiang University,Hangzhou 310027,China [2]Department of Statistics,University of Virginia,Halsey Hall,Charlottesville,Virginia 22904-4135,USA

出　　处：《Science China Mathematics》2009年第6期1305-1326,共22页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No. 10771192);National Science Foundation of USA (Grant No. DMS-0349048)

摘　　要：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.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.

关 键 词：strong invariance Gaussian approximation the law of iterated logarithm asymptotic normality urn model randomized play-the-winner rule 60F15 62E20 62L05 60F05 62F12 

分 类 号：O212[理学—概率论与数理统计]