• Bernoulli
  • Volume 25, Number 1 (2019), 189-220.

Pólya urns with immigration at random times

Erol Peköz, Adrian Röllin, and Nathan Ross

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We study the number of white balls in a classical Pólya 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.

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Bernoulli, Volume 25, Number 1 (2019), 189-220.

Received: February 2017
Revised: July 2017
First available in Project Euclid: 12 December 2018

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distributional convergence distributional fixed point equation Pólya urns preferential attachment random graph


Peköz, Erol; Röllin, Adrian; Ross, Nathan. Pólya urns with immigration at random times. Bernoulli 25 (2019), no. 1, 189--220. doi:10.3150/17-BEJ983.

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