A Flexible Approach for Normal Approximation of Geometric and Topological Statistics

TL;DR

This paper introduces a flexible method for normal approximation of geometric and topological statistics derived from point processes, broadening the scope beyond traditional sum-based approaches.

Contribution

It develops a novel, adaptable framework using an enhanced add-one cost operator combined with stabilization theory to derive normal approximations for complex statistics.

Findings

Applicable to a wide class of stabilizing functionals

Unifies and extends existing normal approximation results

Demonstrates effectiveness on practical geometric and topological statistics

Abstract

We derive normal approximation results for a class of stabilizing functionals of binomial or Poisson point process, that are not necessarily expressible as sums of certain score functions. Our approach is based on a flexible notion of the add-one cost operator, which helps one to deal with the second-order cost operator via suitably appropriate first-order operators. We combine this flexible notion with the theory of strong stabilization to establish our results. We illustrate the applicability of our results by establishing normal approximation results for certain geometric and topological statistics arising frequently in practice. Several existing results also emerge as special cases of our approach.