Asymptotic Behaviour of Discretised Functionals of Long-Range Dependent Functional Data

Abstract: The paper studies the asymptotic behaviour of weighted functionals of long-range dependent data over increasing observation windows. Various important statistics, including sample means, high order moments, occupation measures can be given by these functionals. It is shown that in the discrete sampling case additive functionals have the same asymptotic distribution as the corresponding integral functionals for the continuous functional data case. These results are applied to obtain non-central limit theorems for weighted additive functionals of random fields. As the majority of known results concern the discrete sampling case the developed methodology helps in translating these results to functional data without deriving them again. Numerical studies suggest that the theoretical findings are valid for wider classes of long-range dependent data.