Limit theorems for weighted functionals of cyclical long-range dependent random fields

The paper investigates isotropic random fields for which the spectral density is unbounded at some frequencies. Limit theorems for weighted functionals of these random fields are established. It is shown that for a wide class of functionals, which includes the Donsker scheme, the limit is not affected by singularities at non-zero frequencies. For general schemes, in contrast to the Donsker line, we demonstrate that the singularities at non-zero frequencies play a role even for linear functionals.Comment: 19 pages, 2 figures. This is an Author's Accepted Manuscript of an article in the Stochastic Analysis and Applications, Vol. 31, No. 2. (2013), 199--213. [copyright Taylor \& Francis], available online at: http://www.tandfonline.com/ [DOI:10.1080/07362994.2013.741410

Sojourn measures of Student and Fisher-Snedecor random fields

Limit theorems for the volumes of excursion sets of weakly and strongly dependent heavy-tailed random fields are proved. Some generalizations to sojourn measures above moving levels and for cross-correlated scenarios are presented. Special attention is paid to Student and Fisher-Snedecor random fields. Some simulation results are also presented.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ529 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Tauberian and Abelian theorems for long-range dependent random fields

This paper surveys Abelian and Tauberian theorems for long-range dependent random fields. We describe a framework for asymptotic behaviour of covariance functions or variances of averaged functionals of random fields at infinity and spectral densities at zero. The use of the theorems and their limitations are demonstrated through applications to some new and less-known examples of covariance functions of long-range dependent random fields.Comment: Will appear in Methodology and Computing in Applied Probability. 26 pages, 10 figures. The final publication is available at link.springer.com. DOI: 10.1007/s11009-012-9276-