  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.
