  In this paper we give a central limit theorem for the weighted quadratic
variations process of a two-parameter Brownian motion. As an application, we
show that the discretized quadratic variations $\sum_{i=1}^{[n s]}
\sum_{j=1}^{[n t]} | \Delta_{i,j} Y |^2$ of a two-parameter diffusion
$Y=(Y_{(s,t)})_{(s,t)\in[0,1]^2}$ observed on a regular grid $G_n$ is an
asymptotically normal estimator of the quadratic variation of $Y$ as $n$ goes
to infinity.
