Motivated by applications to the study of depth functions for tree-indexed random variables generated by point processes, we describe functional limit theorems for the intensity measure of point processes. Specifically, we establish uniform laws of large numbers and uniform central limit theorems over a class of bounded measurable functions for estimates of the intensity measure. Using these results, we derive the uniform asymptotic properties of half-space depth and, as corollaries, obtain the asymptotic behavior of medians and other quantiles of the standardized intensity measure. Additionally, we obtain uniform concentration upper bound for the estimator of half-space depth. As a consequence of our results, we also derive uniform consistency and uniform asymptotic normality of Lotka-Nagaev and Harris-type estimators for the Laplace transform of the point processes in a branching random walk.

翻译：受树指标随机变量深度函数研究（由点过程生成）的动机驱动，我们描述了点过程强度测度的泛函极限定理。具体地，在一类有界可测函数上，我们建立了强度测度估计的一致大数定律和一致中心极限定理。利用这些结果，我们导出了半空间深度的一致渐近性质，并作为推论得到了标准化强度测度的中位数及其他分位数的渐近行为。此外，我们获得了半空间深度估计量的一致浓度上界。作为我们结果的推论，我们还推导了分支随机游走中点过程拉普拉斯变换的Lotka-Nagaev型与Harris型估计量的一致相合性与一致渐近正态性。