The talk is concerned with the limiting behavior of row sums of infinitesimal arrays of independent random variables taking values in a locally compact Abelian group. By a theorem of K.R. Parthasarathy, any possible limit of such row sums is weakly infinite divisible and as such a convolution product of an idempotent measure, a Dirac measure, a Gaussian measure and a generalized Poisson measure. Following the classical lines sufficient conditions in terms of the characteristics of the above factors are established in order to obtain convergence of the row sums. Specialization to symmetric arrays and to the torus group illustrates the slightly technical results.