Generalized P\'olya Urn Schemes with Negative but Linear Reinforcements

TL;DR

This paper introduces a negatively reinforced urn scheme with linear decreasing weights, establishing its almost sure limits, conditions for uniform distribution, and central limit theorems for configurations and counts.

Contribution

It develops a new negatively reinforced urn model with linear weights, providing limit theorems and conditions for uniformity, extending classical Polya urn results.

Findings

Limit of the configuration is uniform iff the replacement matrix is doubly stochastic.

Almost sure convergence of the colour counts and configurations.

Central limit theorems for the configuration and colour counts.

Abstract

In this paper, we consider a new type of urn scheme, where the selection probabilities are proportional to a weight function, which is linear but decreasing in the proportion of existing colours. We refer to it as the \emph{negatively reinforced} urn scheme. We establish almost sure limit of the random configuration for any \emph{balanced} replacement matrix $R$. In particular, we show that the limiting configuration is uniform on the set of colours, if and only if, $R$ is a \emph{doubly stochastic} matrix. We further establish almost sure limit of the vector of colour counts and prove central limit theorems for the random configuration, as well as, for the colour counts.