A.s. convergence for infinite colour Pólya urns associated with random walks

We consider Pólya urns with infinitely many colours that are of a random walk type, in two related versions. We show that the colour distribution a.s., after rescaling, converges to a normal distribution, assuming only second moments on the offset distribution. This improves results by Bandyopadhyay and Thacker (2014–2017; convergence in probability), and Mailler and Marckert (2017; a.s. convergence assuming exponential moment).

2010 Mathematics Subject Classification

60C05

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Partly supported by the Knut and Alice Wallenberg Foundation.

Received 19 June 2019

Received revised 18 October 2020

Accepted 27 October 2020

Published 4 May 2021