  In some earlier work we have considered extensions of Lai's (1974) law of the
single logarithm for delayed sums to a multiindex setting with the same as well
as different expansion rates in the various dimensions. A further
generalization concerns window sizes that are regularly varying with index 1
(on the line). In the present paper we establish multiindex versions of the
latter as well as for some mixtures of expansion rates. In order to keep things
within reasonable size we confine ourselves to some special cases for the index
set $\mathbb{Z}_+^2$.
