A limit theorem for random fields with a singularity in the spectrum

Abstract

Abstract. Homogeneous isotropic random fields with singularities in spectra at nonzero frequencies are studied. This class of fields generalizes the case of random fields with long range dependence where the spectrum has a singularity at the origin. We obtain a limit theorem for integral weight functionals of the field. We also discuss the difference between this class and the long range dependence.

Weak convergence of weighted additive functionals of long-range dependent fields

“…There are numerous papers on non-central limit theorems either for integrals or additive functionals of random fields, see, for example, [1,2,[7][8][9][10][11][12][13][14][15][16][20][21][22][23][24]. The presented results below give a rigorous proof that under rather general assumptions, limits coincide for the above functionals.…”

Limit theorems for filtered long-range dependent random fields

“…These three cases correspond to the three types of behaviour called 1 f noise, ultraviolet and infrared catastrophes by Mandelbrot and Taqqu [39]. The literature shows a variety of limit theorems with asymptotics given by non-Gaussian self-similar processes that exhibit non-negative auto-correlation structures with parameter H ∈ (0.5, 1), see [14,20,29,30,36,38] and references therein. However, there are only few results where asymptotic processes have H ∈ (0, 0.5).…”

“…These functionals play an important role in various fields, such as physics, cosmology, telecommunications, just to name a few. In particular, asymptotic results were obtained either for integrals or additives functionals of random fields under long-range dependence, see [2,11,22,24,[28][29][30] and the references therein.…”