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