60E15, 60F05, 60G40.

We consider random numbers of independent, identically distributed (i.i.d.) random variables and their sums . Whereas Blum, Hanson and Rosenblatt [3] proved a central limit theorem for such sums and Landers and Rogge [8] derived the corresponding approximation order, a Berry-Esseen type result seems to be missing. Using an inequality for the asymmetry of distributions, which seems to be of its own interest, we prove, under the assumption (in an appropriate sense), a Berry-Esseen theorem for random summation. ;