  We prove limit theorems for row sums of a rowwise independent infinitesimal
array of random variables with values in a locally compact Abelian group. First
we give a proof of Gaiser's theorem, since it does not have an easy access and
it is not complete. This theorem gives sufficient conditions for convergence of
the row sums, but the limit measure can not have a nondegenerate idempotent
factor. Then we prove necessary and sufficient conditions for convergence of
the row sums, where the limit measure can be also a nondegenerate Haar measure
on a compact subgroup. Finally, we investigate special cases: the torus group,
the group of p-adic integers and the p-adic solenoid.
