AI 中文总结
研究临界 Bienaymé-Galton-Watson 树中稀有子树模式，结合多种方法克服大小条件依赖性，建立泊松近似，应用于最大叶高、完全$r$元树高度等，给出不同情况下的渐近结果及行为特点。
AI 中文摘要
我们为临界 Bienaymé-Galton-Watson 树中具有在稳定律吸引域内的后代分布$\mu$且条件为有大量顶点的稀有局部模式建立了一般泊松近似。模式由边缘子树上的序列相关标记指定。若标记边缘子树保持微观且附近标记出现的聚类可忽略，则其计数在均值有界时在全变差中渐近泊松分布；均值发散时满足大数定律。主要困难在于大小条件产生的全局依赖性，通过结合循环移位表示、Chen-Stein 界的精细形式和控制局部标记与条件随机游走其余部分相互作用的桥移除估计来克服。对于非边缘模式，重叠出现可能形成聚类且原始计数不一定渐近泊松分布，我们引入解聚指标并证明其计数的一般泊松近似。作为应用，我们得到了最大叶高（等价于最大保护数）以及作为非边缘子树出现的最大完全$r$元树高度的精确渐近。一元链最大值以及$\mu_1>0$时的最大叶高表现出格点调制的 Gumbel 行为。$r\ge2$时的完全$r$元模式以及$\mu_1=0$时的最大叶高定位在一或两个连续整数上。结果不需要指数矩且包括具有无限方差的后代分布。
英文摘要
We establish a general Poisson approximation for rare local patterns in critical Bienaymé--Galton--Watson trees with offspring distribution $μ$ in the domain of attraction of a stable law, conditioned to have a large number of vertices. A pattern is specified by a sequence-dependent mark on fringe subtrees. If marked fringe subtrees remain microscopic and nearby marked occurrences have negligible clustering, then their count is asymptotically Poisson in total variation whenever its mean remains bounded; when the mean diverges, the count satisfies a law of large numbers. The main difficulty is the global dependence created by size conditioning. We overcome it by combining the cyclic-shift representation with a refined form of the Chen--Stein bound and a bridge-removal estimate controlling the interaction between a local mark and the remainder of the conditioned random walk. For non-fringe patterns, overlapping occurrences may form clusters and the raw count need not be asymptotically Poisson. We introduce declumped indicators which select boundary witnesses of these clusters and prove a general Poisson approximation for their count. As applications, we obtain sharp asymptotics for the maximum leaf-height, equivalently the maximum protection number, and for the height of the largest complete $r$-ary tree appearing as a non-fringe subtree. Unary-chain maxima, and the maximum leaf-height when $μ_1>0$, exhibit lattice-modulated Gumbel behavior. Complete $r$-ary patterns for $r\ge2$, and the maximum leaf-height when $μ_1=0$, are localized on one or two consecutive integers. The results require no exponential moment and include offspring distributions with infinite variance.
