发表机构
The Catholic University of America; J.W. Goethe University; The George Washington University（美国天主教大学; 法兰克福歌德大学; 乔治华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入$m$叉递归树并分析其距离，推导深度分布，通过耦合得到高度极限，并用加权范数收缩法建立维纳指数与内部路径长度的二元极限定律。
AI 中文摘要
我们引入了 $m$ 叉递归树，这是一种从单个节点开始的树结构。在生长过程的每一步中，选择一个随机节点（称为招募者），并将 $m$ 个新节点连接到该招募者。主要目标是对该结构中的距离进行渐近分析。我们首先推导出此类树中节点深度的分布，作为独立（但非同分布）伯努利随机变量的卷积。该表示为深度提供了具有量化误差的正态近似和泊松近似。我们还通过耦合到具有适当选择权重的加权随机递归树来研究这些树的高度，即最长的根到叶距离。该耦合将加权递归树上的强大数定律和紧致性结果转移到 $m$ 叉递归树。此外，我们使用收缩方法建立了维纳指数（即所有成对距离之和）和内部路径长度（即所有节点深度之和）的二元极限定律。作为我们方法的一个新颖技术特点，我们在收缩方法中使用加权范数而不是欧几里得范数。
英文摘要
We introduce the $m$-ary recursive tree, a tree structure that starts with a single node. At each step of the growth process, a random node, called the recruiter, is selected, and $m$ new nodes are attached to this recruiter. The main objective is an asymptotic analysis of distances in this structure. We first derive the distribution of the depth of nodes in such a tree as a convolution of independent (though not identically distributed) Bernoulli random variables. This representation yields normal and Poisson approximations with quantified errors for the depth. We also investigate the height of these trees, that is, the longest root-to-leaf distance, via a coupling to a weighted random recursive tree with suitably chosen weights. The coupling transfers the results on a strong law and tightness from the weighted recursive tree to the $m$-ary recursive tree. Furthermore, we establish a bivariate limit law for the Wiener index, that is, the sum of all pairwise distances, and the internal path length, that is, the sum of the depths of all nodes, using the contraction method. As a novel technical feature of our approach, we employ weighted norms instead of Euclidean norms within the contraction method.
