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 non-homogeneous Poisson process on R d , and the connectivity radius governing the formation of simplices is taken as a function of 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 non-trivial topology. We establish two "functionallevel" 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 the Čech filtration, we extend the findings of Thomas and Owada (2019) 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.…”

“…It is known that the Euler characteristic tends to a Gaussian process for a Poisson sampling scheme and the Vietoris-Rips filtration, see Thomas and Owada (2019). We generalize this statement in the following to the Čech filtration and the binomial sampling scheme and quantify the convergence.…”

“…In the following, we will point out the dependencies of g on the different parameters. We begin with the Vietoris-Rips complex, see also Thomas and Owada (2019) for a similar result for this filtration.…”

On the law of the iterated logarithm and strong invariance principles in stochastic geometry

“…For recent applications involving the Euler characteristic, see Adler (2008), Decreusefond et al (2014), Crawford et al (2016). Multivariate or functional central limit theorems for the Euler characteristic where proved in Hug et al (2016), Thomas and Owada (2019). Ergodic theorems for the Euler characteristic were established in Schneider and Weil (2008).…”