A functional central limit theorem for integrals of stationary mixing random fields

We prove a functional central limit theorem for the integrals $\int_W f(X(t))\, dt$, where $(X(t))_{t\in\mathbf{R}^d}$ is a stationary mixing random field and the stochastic process is indexed by the function $f$, as the integration domain $W$ grows unboundedly in the Van Hove sense. We also discuss properties of the covariance function of the limiting Gaussian process.