  Motivated by open problems in applied and computational algebraic topology,
we establish multivariate normal approximation theorems for three random
vectors which arise organically in the study of random clique complexes. These
are: (1) the vector of critical simplex counts attained by a lexicographical
Morse matching, (2) the vector of simplex counts in the link of a fixed
simplex, and (3) the vector of total simplex counts. The first of these random
vectors forms a cornerstone of modern homology algorithms, while the second one
provides a natural generalisation for the notion of vertex degree, and the
third one may be viewed from the perspective of U-statistics. To obtain
distributional approximations for these random vectors, we extend the notion of
dissociated sums to a multivariate setting and prove a new central limit
theorem for such sums using Stein's method.
