Birth-and-death Polya urns and stationary random partitions

TL;DR

This paper introduces birth-and-death Polya urns that model sampling and removal processes, revealing a phase transition in the structure of induced partitions and identifying their stationary measures.

Contribution

It presents a new class of urn models with birth-and-death dynamics and characterizes their asymptotic partition regimes and invariant measures.

Findings

Phase transition between infinite and stationary partitions

Identification of invariant and reversible measures

Stationary measure as a mixture of Ewens sampling formulas

Abstract

We introduce a class of birth-and-death Polya urns, which allow for both sampling and removal of observations governed by an auxiliary inhomogeneous Bernoulli process, and investigate the asymptotic behaviour of the induced allelic partitions. By exploiting some embedded models, we show that the asymptotic regimes exhibit a phase transition from partitions with almost surely infinitely many blocks and independent counts, to stationary partitions with a random number of blocks. The first regime corresponds to limits of Ewens-type partitions and includes a result of Arratia, Barbour and Tavar\'e (1992) as a special case. We identify the invariant and reversible measure in the second regime, which preserves asymptotically the dependence between counts, and is shown to be a mixture of Ewens sampling formulas, with a tilted Negative Binomial mixing distribution on the sample size.