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

TL;DR

This paper establishes a Gaussian approximation for the composition and allocation processes in a multi-color generalized Friedman's urn model, enhancing understanding and statistical inference capabilities for this widely used stochastic process.

Contribution

It proves a Gaussian approximation for the urn process with non-homogeneous matrices, providing new theoretical insights and tools for analysis.

Findings

Gaussian process approximation for urn composition and allocation

Asymptotic normality of the urn model

Law of the iterated logarithm for the model

Abstract

The 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 prove 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 non-homogeneous generating matrices. The Gaussian process is a solution of a stochastic differential equation. This Gaussian approximation together with the properties of the Gaussian process 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.