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.