2024/02/07 by Giacomo Francisci, Francisci, Giacomo, Anand N. Vidyashankar +1
Mathematics · #60F17 #60G50 #60G55 (Primary) 60J80 #62G05 #62M99 (Secondary) #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2402.05087
openalex publication_date 2024/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.