Central limit theorems and asymptotic independence for local $U$-statistics on diverging halfspaces

TL;DR

This paper establishes limit theorems for local U-statistics of Poisson processes in diverging halfspaces, revealing asymptotic independence under light tails and quantifying convergence rates using advanced probabilistic methods.

Contribution

It provides the first finite-dimensional CLTs for local U-statistics in diverging halfspaces, including tail-specific results and convergence rate bounds.

Findings

Asymptotic independence for U-statistics with diverging halfspaces at different angles.

No asymptotic independence for heavy-tailed densities.

Faster convergence rates for lighter tails in Kolmogorov distance.

Abstract

We consider the stochastic behavior of a class of local $U$-statistics of Poisson processes$-$which include subgraph and simplex counts as special cases, and amounts to quantifying clustering behavior$-$for point clouds lying in diverging halfspaces. We provide limit theorems for distributions with light and heavy tails. In particular, we prove finite-dimensional central limit theorems. In the light tail case we investigate tails that decay at least as slow as exponential and at least as fast as Gaussian. These results also furnish as a corollary that $U$-statistics for halfspaces diverging at different angles are asymptotically independent, and that there is no asymptotic independence for heavy-tailed densities. Using state-of-the-art bounds derived from recent breakthroughs combining Stein's method and Malliavin calculus, we quantify the rate of this convergence in terms of Kolmogorov distance. We also investigate the behavior of local $U$-statistics of a Poisson Process conditioned to lie in diverging halfspace and show how the rate of convergence in the Kolmogorov distance is faster the lighter the tail of the density is.