Negatively Reinforced Balanced Urn Schemes

TL;DR

This paper studies negatively reinforced urn schemes with finitely many colours, proving almost sure convergence and asymptotic normality under certain conditions, especially when the replacement matrix is doubly stochastic.

Contribution

It introduces a class of negatively reinforced urn models with specific convergence properties and asymptotic behavior analysis.

Findings

Almost sure convergence of the urn configuration.

Convergence to the uniform distribution when R is doubly stochastic.

Asymptotic normality for large number of colours.

Abstract

We consider weighted negatively reinforced urn schemes with finitely many colours. An urn scheme is called negatively reinforced, if the selection probability for a colour is proportional to the weight $w$ of the colour proportion, where $w$ is a non-increasing function. Under certain assumptions on the replacement matrix $R$ and weight function $w$, such as, $w$ is differentiable and $w(0) < \infty$, we obtain almost sure convergence of the random configuration of the urn model. In particular, we show that if $R$ is doubly stochastic the random configuration of the urn converges to the uniform vector, and asymptotic normality holds, if the number of colours in the urn are sufficiently large.