Concentration inequalities for Poisson U-statistics

In this article we obtain concentration inequalities for Poisson U-statistics Fm(f,η) of order m≥1 with kernels f under general assumptions on f and the intensity measure γΛ of underlying Poisson point process η. The main result consists of new concentration bounds of the form P(|Fm(f,η)−EFm(f,η)|≥t)≤2exp(−I(γ,t)), where I(γ,t) satisfies I(γ,t)=Θ(t1 mlogt) as t→∞ and γ is fixed. This improves the existing bounds by a logarithmic factor log(t) in the exponent and provides a bound of optimal order in t. The function I(γ,t) is given explicitly in terms of parameters of the assumptions satisfied by f and Λ. One of the key ingredients of the proof is bounding the centred moments of Fm(f,η). We establish the optimality of our concentration inequalities and demonstrate their broad applicability across various settings, including classical statistics of Gilbert graphs and Poisson hyperplane processes in constant curvature spaces. In several cases, our bounds outperform existing results, and, notably, in hyperbolic spaces, they appear to be the first of their kind.