  Consider a measure $\mu_\lambda = \sum_x \xi_x \delta_x$ where the sum is
over points $x$ of a Poisson point process of intensity $\lambda$ on a bounded
region in $d$-space, and $\xi_x$ is a functional determined by the Poisson
points near to $x$, i.e. satisfying an exponential stabilization condition,
along with a moments condition (examples include statistics for proximity
graphs, germ-grain models and random sequential deposition models). A known
general result says the $\mu_\lambda$-measures (suitably scaled and centred) of
disjoint sets in $R^d$ are asymptotically independent normals as $\lambda \to
\infty$; here we give an $O(\lambda^{-1/(2d + \epsilon)})$ bound on the rate of
convergence. We illustrate our result with an explicit multivariate central
limit theorem for the nearest-neighbour graph on Poisson points on a finite
collection of disjoint intervals.
