Concentration inequalities for Poisson $U$-statistics

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 $\gamma \mathrm{\Lambda }$ of underlying Poisson point process $\eta$. The main result are new concentration bounds of the form $$\mathbb{P}(\mathrm{\mid }{F}_m(f,\eta )-\mathbb{E}{F}_m(f,\eta )\mathrm{\mid }\ge t)\le 2\exp (-I(\gamma ,t)),$$ where $I(\gamma ,t)$ is of optimal order in $t$, namely it satisfies $I(\gamma ,t)=\mathrm{\Theta }(t^{\frac{1}{m}}\log t)$ as $t\to \mathrm{\infty }$ and $\gamma$ is fixed. The function $I(\gamma ,t)$ is given explicitly in terms of parameters of the assumptions satisfied by $f$ and $\mathrm{\Lambda }$. 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.