AI 中文总结
本文针对具有独立边权重的随机几何图，通过分割与迭代路径构造法得到最大生成树权重增长的偏差界，利用鞅差方法确定最大权重L²收敛的充分条件。
AI 中文摘要
本文研究由n个顶点构成的随机几何图（RGG）G的最大权重生成树，其中每条边以特定概率独立地为“开放”或“闭合”，且配备独立的随机正权重。我们采用分割和迭代路径构造方法，得到生成树最大权重增长阶的偏差界，该偏差界以边权重互补累积分布函数（ccdf）的倒数形式呈现，并针对幂律和指数衰减的特殊情形说明所得结果。随后，我们利用鞅差方法分别估计顶点位置随机性、边状态及边权重带来的方差贡献，确定经适当缩放和中心化后的最大权重L²收敛的充分条件。
英文摘要
In this paper, we study maximum weight spanning trees of the random geometric graph (RGG)~\(G\) formed by~\(n\) vertices where each edge is independently either open or closed with a certain probability and is also equipped with an independent random positive weight. We use segmentation and iterative path construction to obtain deviation bounds for the order of growth of the maximum weight of a spanning tree in terms of an inverse of the edge weight complementary cumulative distribution function (ccdf) and also illustrate our results for the special cases of power law and exponential decay. We then use martingale difference methods to individually estimate the variance contribution due to randomness in vertex locations and edge states/weights and determine sufficient conditions for~\(L^2-\)convergence of the maximum weight, appropriately scaled and centred.
