Deviation of discrete distributions-positive and negative results

The paper is devoted to the problem of estimating the deviation of two discrete probability distributions in terms of the supremum distance between their generating functions over the interval [0, 1]. The deviation can be measured by the difference of the kth terms, or by total variation distance. In addition to upper bounds we illustrate the limitations of such estimations by counterexamples. Such problems arise e.g. when Poisson limit results are proved by sieve methods.