Abstract:

To investigate the functional limit properties of $f$-vectors in $d$-dimensional Euclidean space, this paper introduces random Vietoris-Rips complexes generated by point sets of inhomogeneous Poisson processes. By projecting the distance threshold parameter onto the time axis $t$, the $f$-vector in the complex is constructed as a stochastic process $F(t)$. The asymptotic behavior of the expectation of $F(t)$ and the convergence limit of its covariance are clarified. Furthermore, employing Prokhorov’s theorem and weak convergence theory, a functional central limit theorem for $F(t)$ in the Skorokhod space is established. This result extends the limit characteristics of $F(t)$ in function spaces, imposing only basic topological well-defined conditions on the point process.

Key words: $f$-vector, functional central limit theorem, random simplicial complexes, nonhomogeneous Poisson process