Covariance matrices of length power functionals of random geometric graphs -- an asymptotic analysis

TL;DR

This paper investigates the asymptotic behavior of covariance matrices of length power functionals in random geometric graphs, revealing regime-dependent properties and stochastic implications as the underlying Poisson process intensity grows.

Contribution

It provides a systematic analysis of the covariance matrix properties across different regimes, offering new insights into the asymptotic structure of functionals in random geometric graphs.

Findings

Covariance matrices exhibit distinct properties in different regimes.

Asymptotic covariance matrices are characterized by their rank, definiteness, and eigenstructure.

Stochastic consequences for random geometric graphs are derived from the asymptotic analysis.

Abstract

Asymptotic properties of a vector of length power functionals of random geometric graphs are investigated. More precisely, its asymptotic covariance matrix is studied as the intensity of the underlying homogeneous Poisson point process increases. This includes a systematic discussion of matrix properties like rank, definiteness, determinant, eigenspaces or decompositions of interest. For the formulation of the results a case distinction is necessary. Indeed, in the three possible regimes the respective covariance matrix is of quite different nature which leads to different statements. Finally, stochastic consequences for random geometric graphs are derived.