Suppose $X = (X_x)_{x\in\mathbb{Z}^d}$ is a white noise process, and
$H (B)$, defined for finite subsets $B$ of $\math{Z}^d$, is determined in a
stationary way by the restriction of $X$ to $B$. Using a martingale approach,
we prove a central limit theorem (CLT) for $H$ as $B$ becomes large, subject to
$H$ satisfying a “stabilization” condition (the effect of
changing $X _x$ at a single site needs to be local). This CLT is then applied
to component counts for percolation and Boolean models, to the size of the big
cluster for percolation on a box, and to the final size of a spatial
epidemic.