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).

中文翻译：

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作为无限颜色的融合，与随机游走相关的Pólya骨灰盒

我们认为Pólya骨灰盒具有两个随机版本的无限多种随机行走的颜色。我们显示，重新缩放后的颜色分布会收敛到正态分布，假设偏移分布上只有第二个矩。Bandyopadhyay和Thacker（2014-2017；概率收敛），以及Mailler和Marckert（2017；收敛假设指数矩）改善了结果。