Gaussian Approximation for Sums of Region-Stabilizing Scores

Chinmoy Bhattacharjee, Ilya Molchanov

Published 2021-01-13Version 1

We consider the Gaussian approximation for functionals of a Poisson process that are expressible as a sum of stabilizing score functions and provide a bound on the rate of convergence in the Kolmogorov metric. Such results have previously been shown in \cite{LSY19}, but we relax some conditions assumed there and provide further insights into the results. This is achieved by working with stabilization regions that may differ from balls of random radii commonly used in the literature concerning stabilizing functionals. As an application, we consider the Gaussian approximation of number of minimal points in a homogeneous Poisson process in $[0,1]^d$ and provide presumably optimal rate of convergence.