Normal approximation of stabilizing Poisson pair functionals with column-type dependence

TL;DR

This paper establishes conditions under which certain stabilizing Poisson pair functionals with column-type dependence can be approximated by a normal distribution, with applications to graph crossings and topological data analysis summaries.

Contribution

It provides the first sufficient conditions for normal approximation of these complex Poisson functionals with mixed local and non-local dependence structures.

Findings

Upper bounds on Wasserstein distance to normal for the functionals.

Normal approximation for rectilinear crossing number in random graphs.

Normal approximation for barcode summaries in topological data analysis.

Abstract

In this paper, we study two specific types of $d$-dimensional Poisson functionals: a double-sum type and a sum-log-sum type, both over pairs of Poisson points. On these functionals, we impose column-type dependence, i.e., local behavior in the first $k$ directions and allow non-local, yet stabilizing behavior in the remaining $d-k$ directions. The main contribution of the paper is to establish sufficient conditions for Normal approximation for sequences of such functionals over growing regions. Specifically, for any fixed region, we provide an upper bound on the Wasserstein distance between each functional and the standard Normal distribution. We then apply these results to several examples. Inspired by problems in computer science, we prove a Normal approximation for the rectilinear crossing number, arising from projections of certain random graphs onto a 2-dimensional plane. From the field of topological data analysis, we examine two types of barcode summaries, the inversion count and the tree realization number, and establish Normal approximations for both summaries under suitable models of the topological lifetimes.