Arkiv för Matematik

Volume 59 (2021)

Number 1

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

Pages: 87 – 123



Svante Janson (Department of Mathematics, Uppsala University, Uppsala, Sweden)


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


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