On a method of expansion of the probabilities of lattice random variables

Asymptotic expansions are widely used in studying properties of probabilities. One can mention in this connection papers by Cramйr, Richter, Petrov, Statulyavichus, Saulis, Franken, Kalinin, Bikyalis, Zhemaitis, and many others. Expansions of such a kind are constructed, as a rule, by means of semi-invariants in the case of continuous random variables (r.v.'s) and by means of factorial semi-invariants in the case of lattice r.v.'s. In this paper different expansions for the probabilities of sums of independent identically distributed lattice r.v.'s are considered. These expansions use other numerical characteristics, namely, the values of the derivatives of the logarithm of a generating function at point zero. The advantages of the method are discussed in detail.