On statistical properties of non-linear functionals of random fields

This thesis studies the asymptotic behaviour of non-linear transformations of Gaussian vector random fields via integral functionals and solutions of stochastic partial differential equations. The components of vector fields considered in this thesis admit different types of dependence properties. These random fields and their transformations play a central role in the asymptotic theory and various statistical applications. They can produce non-Gaussian limit theorems. The thesis also investigates stochastic hyperbolic diffusion equations with random initial conditions given by random~fields. First, the reduction principle for non-linear functionals of Gaussian vector random fields is derived. The components of such fields can exhibit different behaviour under long-range dependence. It is shown that the asymptotic distributions of integral functionals of Gaussian vector random fields are not necessarily determined by the leading components at the Hermite rank level. The concept of dominant components which determine the asymptotic distribution is discussed. The non-central limit theorem for the first Minkowski functionals of Fisher-Snedecor random fields is derived. Next, the asymptotics of non-linear functionals of vector random fields with strongly and weakly dependent components are studied. A version of the reduction theorem is provided which proves that the asymptotic distributions of such functionals are not determined by their Hermite ranks. These functionals converge in distribution to random variables which are determined by sums of the multiple Wiener-It\^{o} stochastic integrals. Applications to the first Minkowski functionals of Student random fields are provided. Last, the solutions of hyperbolic diffusion equations with random initial conditions given by random fields are investigated. The restriction of solutions to the unit sphere is considered. The dependence structures of spherical hyperbolic diffusion random fields are studied. The approximations of the exact solution random fields are investigated. The smoothness properties of the exact solution and its approximation are obtained.