论文标题
Hammond-Sheffield urn中通过合并概率的渐近高斯性
Asymptotic Gaussianity via coalescence probabilities in the Hammond-Sheffield urn
论文作者
论文摘要
对于随机$ \ pm 1 $ 1 $颜色的重新规定，$ \ mathbb z $的连接组件是由Hammond和Sheffield的“ Power Law law pollya's urn''共同续签流程产生的，我们证明了与棕色动作的功能融合，在紧密的论文中弥补了差距的差异。 此外，在强大的更新定理方面，我们可以深入了解Hammond-Sheffield urn的合并更新过程（例如，最近最新共同祖先的渐近深度）能够控制两个，三个和四个人的合并概率，这些概率是从$ [n] $中随机采样的。这使我们能够获得更一般色彩的重质量总和的渐近高斯性（包括功能融合）的新的概念证明，这可以看作是Hammond和Sheffield的主要结果以外的不变性原理。 在此证明中，独立利益的关键要素是基于Stein方法的$ [n] $的随机随机分区中重新颜色的随机分区中重质量总和的渐近高斯性的足够标准。 一路走来，我们还证明了关于Blath，GonzálezCasanova，Kurt和Spanò的远程种子银行模型中合并概率的渐近学的陈述。
For the renormalised sums of the random $\pm 1$-colouring of the connected components of $\mathbb Z$ generated by the coalescing renewal processes in the "power law Pólya's urn" of Hammond and Sheffield we prove functional convergence towards fractional Brownian motion, closing a gap in the tightness argument of their paper. In addition, in the regime of the strong renewal theorem we gain insights into the coalescing renewal processes in the Hammond-Sheffield urn (such as the asymptotic depth of most recent common ancestors) and are able to control the coalescence probabilities of two, three and four individuals that are randomly sampled from $[n]$. This allows us to obtain a new, conceptual proof of the asymptotic Gaussianity (including the functional convergence) of the renormalised sums of more general colourings, which can be seen as an invariance principle beyond the main result of Hammond and Sheffield. In this proof, a key ingredient of independent interest is a sufficient criterion for the asymptotic Gaussianity of the renormalised sums in randomly coloured random partitions of $[n]$, based on Stein's method. Along the way we also prove a statement on the asymptotics of the coalescence probabilities in the long-range seedbank model of Blath, González Casanova, Kurt, and Spanò.
