We give a theorem of Vijayaraghavan type for summability methods for double sequences, which allows a conclusion from boundedness
in a mean and a one-sided Tauberian condition to the boundedness of the sequence itself. We apply the result to certain power
series methods for double sequences improving a recent Tauberian result by S. Baron and the author .
A paper by Chow  contains (i.a.) a strong law for delayed sums, such that the length of the edge of the nth window equals nα for 0 < α < 1. In this paper we consider the kind of intermediate case when edges grow like n=L(n), where L is slowly varying at infinity, thus at a higher rate than any power less than one, but not quite at a linear rate. The typical
example one should have in mind is L(n) = log n. The main focus of the present paper is on random field versions of such strong laws.
We discuss the relations between weighted mean methods and ordinary convergence for double sequences. In particular, we study
Tauberian theorems also for methods not being products of the related one-dimensional summability methods. For the special
case of theC1,1-method, the results contain a classical Tauberian theorem by Knopp  as special case and generalize theorems given by Móricz
 thereby showing that one of his Tauberian conditions can be weakened.