  Consider a Poisson point process within a convex set in a Euclidean space.
The Vietoris-Rips complex is the clique complex over the graph connecting all
pairs of points with distance at most $\delta$. Summing powers of the volume of
all $k$-dimensional faces defines the volume-power functionals of these random
simplicial complexes. The asymptotic behavior of the volume-power functionals
of the Vietoris-Rips complex is investigated as the intensity of the underlying
Poisson point process tends to infinity and the distance parameter goes to
zero. Univariate and multivariate central limit theorems are proven. Analogous
results for the \v{C}ech complex are given.
