Functional limit theorems for measures of level surfaces of a gaussian random field

Functional limit theorems for measures of level surfaces of a gaussian random field
We consider the measures of level sets of a smooth Gaussian random field observed in some bounded window. These measures define a random process indexed by the levels. Assuming some general condition on covariance function we prove a functional limit theorem establishing the convergence of these process to a Gaussian one in the space of continuous functions, when the observation windows grow to infinity. The convergence of such process has been proven before only in the sense of finite-dimennsional distributions or in the Hilbert space sense.