P\'olya urns with immigration at random times

TL;DR

This paper analyzes a modified Pólya urn model with random black ball additions at unpredictable times, characterizing the limiting distribution of white balls and connecting it to fixed points of certain distributional transformations.

Contribution

It introduces a new Pólya urn model with random immigration times and characterizes the complex limiting distributions through probabilistic fixed points.

Findings

Limiting distributions depend on inter-arrival times and are fixed points of distributional transformations.

The model's results apply to degree distributions in preferential attachment graphs.

Convergence to these limits is established under certain moment conditions.

Abstract

We study the number of white balls in a classical P\'olya urn model with the additional feature that, at random times, a black ball is added to the urn. The number of draws between these random times are i.i.d. and, under certain moment conditions on the inter-arrival distribution, we characterize the limiting distribution of the (properly scaled) number of white balls as the number of draws goes to infinity. The possible limiting distributions obtained in this way vary considerably depending on the inter-arrival distribution and are difficult to describe explicitly. However, we show that the limits are fixed points of certain probabilistic distributional transformations, and this fact provides a proof of convergence and leads to properties of the limits. The model can alternatively be viewed as a preferential attachment random graph model where added vertices initially have a random number of edges, and from this perspective, our results describe the limit of the degree of a fixed vertex.