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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    LiXin Zhang

  2. Department of Statistics, University of Virginia, Halsey Hall, Charlottesville, Virginia, 22904-4135, USA

    FeiFang Hu

Authors
  1. LiXin Zhang
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  2. FeiFang Hu
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Correspondence to LiXin Zhang.

Additional information

Dedicated to Professor Zhidong Bai on the occasion of his 65th birthday

This work was supported by National Natural Science Foundation of China (Grant No. 10771192) and National Science Foundation of USA (Grant No. DMS-0349048)

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Zhang, L., Hu, F. The Gaussian approximation for multi-color generalized Friedman’s urn model. Sci. China Ser. A-Math. 52, 1305–1326 (2009). https://doi.org/10.1007/s11425-009-0092-9

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  • DOI: https://doi.org/10.1007/s11425-009-0092-9

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