  We establish central and local limit theorems for the number of vertices in
the largest component of a random $d$-uniform hypergraph $\hnp$ with edge
probability $p=c/\binnd$, where $(d-1)^{-1}+\eps<c<\infty$. The proof relies on
a new, purely probabilistic approach, and is based on Stein's method as well as
exposing the edges of $H_d(n,p)$ in several rounds.
