Poisson approximations for sequences of random variables

Let [xi]1, [xi]2, ... be a sequence of independent identically distributed lattice variables, E[xi]12+[delta] x [infinity], Sn = [xi]1 + [xi]2 + ··· + [xi]n. The classical approach is to approximate the distribution of Sn by the normal law. We show that for a suitably centered Sn the Poisson approximation also can be applied. Moreover, this approximation holds for all Borel sets which is impossible for the normal distribution. The rate of accuracy is determined by Ibragimov's necessary and sufficient conditions.