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 .
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.
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.