  The objective of this study is to examine the asymptotic behavior of Betti
numbers of \v{C}ech complexes treated as stochastic processes and formed from
random points in the $d$-dimensional Euclidean space $\mathbb{R}^d$. We
consider the case where the points of the \v{C}ech complex are generated by a
Poisson process with intensity $nf$ for a probability density $f$. We look at
the cases where the behavior of the connectivity radius of \v{C}ech complex
causes simplices of dimension greater than $k+1$ to vanish in probability, the
so-called sparse and Poisson regimes, as well when the connectivity radius is
on the order of $n^{-1/d}$, the critical regime. We establish limit theorems in
all of the aforementioned regimes, a central limit theorem for the sparse and
critical regimes, and a Poisson limit theorem for the Poisson regime. When the
connectivity radius of the \v{C}ech complex is $o(n^{-1/d})$, i.e., the sparse
and Poisson regimes, we can decompose the limiting processes into a
time-changed Brownian motion and a time-changed homogeneous Poisson process
respectively. In the critical regime, the limiting process is a centered
Gaussian process but has much more complicated representation, because the
\v{C}ech complex becomes highly connected with many topological holes of any
dimension.
