Functional limit theorems for the euler characteristic process in the critical regime

Abstract

This study presents functional limit theorems for the Euler characteristic of Vietoris–Rips complexes. The points are drawn from a nonhomogeneous Poisson process on $\mathbb{R}^d$ , and the connectivity radius governing the formation of simplices is taken as a function of the time parameter t, which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have nontrivial topology. We establish two ‘functional-level’ limit theorems, a strong law of large numbers and a central limit theorem, for the appropriately normalized Euler characteristic process.

On approximation theorems for the Euler characteristic with applications to the bootstrap

“…Moreover, using the continuity properties of theČech filtration, we now extend the findings of [39] who provide a functional central limit theorem for the Vietoris-Rips complex and a Poisson sampling scheme. We remark that a functional central limit theorem for the binomial sampling scheme has not been established yet for either filtration type and follows from a Poissonization argument covered in the technical details of Section 4.…”

“…To the best of our knowledge the present contribution is the first to obtain rates of convergence for this topological invariant. Moreover, the presented FCLT is the first that also covers the case forČech filtration, completing the investigation of Thomas and Owada [39] who observed the additional technical difficulties that arise when proving the tightness of the processes in (1.1) for theČech complex. In addition, we extend the FCLT to a binomial sampling scheme by a Poissonisation argument, this setting has previously been not considered.…”

“…Note that the choice of ρ in the above result is arbitrary and in particular independent of the density κ. It is known that the EC tends to a Gaussian process for a Poisson sampling scheme and the Vietoris-Rips filtration, see [39]. We generalize this statement in the following to theČech filtration and the binomial sampling scheme and quantify the convergence.…”

“…The following properties are crucial for the desired tightness results of the EC which we will prove below. We begin with the Vietoris-Rips complex, see also [39] for a similar result for this filtration.…”

Random Simplicial Complexes: Models and Phenomena

“…• Functional limit theorems: One can examine functionals such as the Betti numbers and the Euler characteristic in a dynamic setting, and seek a limit in the form of a stochastic (Gaussian) process. In [103,108], geometric complexes were studied, and the dynamics was the growing connectivity radius in the complex. The results show that the limiting process is indeed Gaussian.…”