  We consider P\'olya urns with infinitely many colours that are of a random
walk type, in two related version. 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).
