2022/07/12 by Matthias Reitzner, Reitzner, Matthias, Tim Römer +3 · 2 citations
Mathematics · #15B52 #60D05 (Primary) 05C80 #60G55 (Secondary) #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications #Rings and Algebras (math.RA) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2207.05450
openalex publication_date 2022/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.