Multicolor urn models with reducible replacement matrices

TL;DR

This paper studies multicolor urn models with specific reducible replacement matrices, revealing their asymptotic behavior and limiting distributions, including variance mixtures of normals and almost sure limits, contrasting with irreducible cases.

Contribution

It introduces analysis of non-irreducible replacement matrices in multicolor urns, deriving new asymptotic results and limiting distributions for these models.

Findings

Certain linear combinations have variance mixture normal limits

Almost sure limits are obtained in some cases

Contrasts with irreducible case behaviors

Abstract

Consider the multicolored urn model where, after every draw, balls of the different colors are added to the urn in a proportion determined by a given stochastic replacement matrix. We consider some special replacement matrices which are not irreducible. For three- and four-color urns, we derive the asymptotic behavior of linear combinations of the number of balls. In particular, we show that certain linear combinations of the balls of different colors have limiting distributions which are variance mixtures of normal distributions. We also obtain almost sure limits in certain cases in contrast to the corresponding irreducible cases, where only weak limits are known.