  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 $\mathbb{R}^d$, 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
"functional-level" limit theorems, a strong law of large numbers and a central
limit theorem for the appropriately normalized Euler characteristic process.
