  We consider delayed sums of the type S_{n+an}-Sn where a_n is possibly a
positive integer valued random variable satisfying certain conditions and S_n
is the sum of independent random variables X_n with distribution functions F_n
in {G_1, G_2} . We study the limiting behavior of delayed sums and prove laws
of the iterated logarithm of Chover- type. These results extend the results in
Vasudeva and Divanji (1992) and Chen (2008).
