Concentration inequalities for Poisson $U$-statistics

Abstract

In this article we obtain concentration inequalities for Poisson $U$-statistics $F_m(f,\hat{I}\cdot )$ of order $m\ge 1$ with kernels $f$ under general assumptions on $f$ and the intensity measure ${\hat{I}}^3\hat{I}›$ of underlying Poisson point process $\hat{I}\cdot$. The main result are new concentration bounds of the form \[ \mathbb{P}(|F_m ( f , η) -\mathbb{E} F_m ( f , η)| \ge t)\leq 2\exp(-I(γ,t)), \] where $I({\hat{I}}^3,t)$ is of optimal order in $t$, namely it satisfies $I({\hat{I}}^3,t)=\hat{I}˜(t^{\frac{1}{m}}\log t)$ as $t\to \infty$ and ${\hat{I}}^3$ is fixed. The function $I({\hat{I}}^3,t)$ is given explicitly in terms of parameters of the assumptions satisfied by $f$ and $\hat{I}›$. One of the key ingredients of the proof is bounding the centred moments of $F_m(f,\hat{I}\cdot )$. 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.