Limit theorems for filtered long-range dependent random fields

Abstract

This article investigates general scaling settings and limit distributions of functionals of filtered random fields. The filters are defined by the convolution of non-random kernels with functions of Gaussian random fields. The case of long-range dependent fields and increasing observation windows is studied. The obtained limit random processes are non-Gaussian. Most known results on this topic give asymptotic processes that always exhibit non-negative auto-correlation structures and have the self-similar parameter H ∈ ( 1 2 , 1). In this work we also obtain convergence for the case H ∈ (0, 1 2 ) and show how the Hurst parameter H can depend on the shape of the observation windows. Various examples are presented.Keywords filtered random fields · long-range dependence · self-similar processes · non-central limit theorem · Hurst parameter Mathematics Subject Classification (2010) 60G60 · 60F17 · 60F05 · 60G35

On Asymptotics of Functionals of Random Fields With Long-Range Dependence

“…Most published results on this topic give asymptotic random processes that always exhibit nonnegative autocorrelation structures and have the self-similar parameter H ∈ ( 1 2 , 1). The limit processes obtained in this thesis can have the self-similar parameter H ∈ (0, 1 2 ). These results extend the theory for one-dimensional processes and sequences given in [5,10,11] to multidimensional settings.…”