Describing the asymptotic behaviour of multicolour P\'olya urns via smoothing systems analysis

TL;DR

This paper analyzes the long-term behavior of multicolour Pólya urns by decomposing their composition vectors and characterizing the asymptotic distributions of their projections, revealing new insights into their complex stochastic dynamics.

Contribution

It provides a detailed description of the asymptotic composition of balanced, tenable, and irreducible multicolour Pólya urns using Jordan decomposition and smoothing systems analysis.

Findings

Projections onto small Jordan spaces are asymptotically Gaussian.

Projections onto large Jordan spaces converge to a limiting random variable W.

The asymptotic behavior depends on the urn's parameters and the Jordan decomposition.

Abstract

The present paper aims at describing in details the asymptotic composition of a class of d-colour P\'olya urn: namely balanced, tenable and irreducible urns. We decompose the composition vector of such urns according to the Jordan decomposition of their replacement matrix. The projections of the composition vector onto the so-called small Jordan spaces are known to be asymptotically gaussian, but the asymptotic behaviour of the projections onto the large Jordan spaces are not known in full details up to now and are discribed by a limiting random variables called W, depending on the parameters of the urn.