2024/04/25 by Bonnet, Gilles, Gusakova, Anna
#05C80 #60D05 #60F10 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2404.16756
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 are new concentration bounds of the form ℙ(|Fm ( f , η) -𝔼 Fm ( f , η)| ≥ t)≤ 2exp(-I(γ,t)), where I(γ,t) is of optimal order in t, namely it satisfies I(γ,t)=Θ(t1\over mlog t) as t→∞ and γ is fixed. 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 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.