  Long-range dependent random fields with spectral densities which are
unbounded at some frequencies are investigated. We demonstrate new examples of
covariance functions which do not exhibit regular varying asymptotic behaviour
at infinity. However, variances of averaged functionals of these fields are
regularly varying. Limit theorems for weighted functionals of cyclical
long-range dependent fields are obtained. The order of normalizing constants
and relations between the weight functions and singularities in
non-degenerative asymptotics are discussed.
