Balanced multicolour Pólya urns via smoothing systems analysis

The present paper aims at describing in details the asymptotic composition of a class of d-colour Pólya urns: 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 described by a limit random variable called W, depending on the parameters of the urn. We prove, via the study of smoothing systems, that the variable W has a density and that it is moment-determined.