  This paper investigates asymptotic properties of multifractal products of
random fields. The obtained limit theorems provide sufficient conditions for
the convergence of cumulative fields in the spaces $L_q.$ New results on the
rate of convergence of cumulative fields are presented. Simple unified
conditions for the limit theorems and the calculation of the R\'enyi function
are given. They are less restrictive than those in the known one-dimensional
results. The developed methodology is also applied to multidimensional
multifractal measures. Finally, a new class of examples of geometric
$\varphi$-sub-Gaussian random fields is presented. In this case, the general
assumptions have a simple form and can be expressed in terms of covariance
functions only.
