Concentration inequalities for Poisson $U$-statistics

TL;DR

This paper derives new concentration inequalities for Poisson U-statistics, providing bounds with optimal order in the tail probability, applicable to geometric structures like Gilbert graphs and hyperplane processes.

Contribution

The paper introduces the first concentration bounds for Poisson U-statistics with explicit, optimal-order tail decay, under general kernel and measure assumptions.

Findings

Derived concentration bounds with optimal order in tail probability

Explicit formulas for the concentration function $I(\gamma,t)$

Applications to Gilbert graphs and Poisson hyperplane processes

Abstract

In this article we obtain concentration inequalities for Poisson $U$-statistics $F_m(f,\eta)$ of order $m\ge 1$ with kernels $f$ under general assumptions on $f$ and the intensity measure $\gamma \Lambda$ of underlying Poisson point process $\eta$. The main result are new concentration bounds of the form \[ \mathbb{P}(|F_m ( f , \eta) -\mathbb{E} F_m ( f , \eta)| \ge t)\leq 2\exp(-I(\gamma,t)), \] where $I(\gamma,t)$ is of optimal order in $t$, namely it satisfies $I(\gamma,t)=\Theta(t^{1\over m}\log t)$ as $t\to\infty$ and $\gamma$ is fixed. The function $I(\gamma,t)$ is given explicitly in terms of parameters of the assumptions satisfied by $f$ and $\Lambda$. One of the key ingredients of the proof is bounding the centred moments of $F_m(f,\eta)$. We discuss the optimality of obtained concentration bounds and consider a number of applications related to Gilbert graphs and Poisson hyperplane processes in constant curvature spaces.