  We consider vectors of random variables, obtained by restricting the length
of the nodal set of Berry's random wave model to a finite collection of
(possibly overlapping) smooth compact subsets of $\mathbb{R}^2$. Our main
result shows that, as the energy diverges to infinity and after an adequate
normalisation, these random elements converge in distribution to a Gaussian
vector, whose covariance structure reproduces that of a homogeneous
independently scattered random measure. A by-product of our analysis is that,
when restricted to rectangles, the dominant chaotic projection of the nodal
length field weakly converges to a standard Wiener sheet, in the Banach space
of real-valued continuous mappings over a fixed compact set. An analogous study
is performed for complex-valued random waves, in which case the nodal set is a
locally finite collection of random points.
