Central Limit Theorems for Persistent Betti Numbers

Abstract

Various persistent invariants have been developed in recent years. In this paper, we develop a method based on homological algebra for deriving central limit theorems for persistent Betti numbers associated with $\mathbb{R}$-indexed chain complexes constructed from a homogeneous Poisson point process of unit intensity on $\mathbb{R}^d$. Our method also applies to Gibbs point processes. This algebraic method reduces the verification of the required conditions to checking properties of the kernels and cokernels of add one maps between chain complexes. As an application, we recover the known central limit theorem for persistent Betti numbers arising from simplicial complex filtrations. In addition, we prove a central limit theorem for persistent Betti numbers of $\ell_p$-Vietoris-Rips simplicial homology, including blurred magnitude homology of random geometric graphs. We also establish central limit theorems for relative $\ell_p$-Vietoris-Rips homology and, in particular, for magnitude Betti numbers of random geometric graphs. These results suggest that our algebraic method can be applied more broadly to derive central limit theorems for persistent Betti numbers.