Edgeworth expansions for independent bounded integer valued random variables

We obtain asymptotic expansions for local probabilities of partial sums uniformly bounded independent but not necessarily identically distributed integer-valued random variables. The involve products polynomials and trigonometric polynomials. These results also have counterparts triangular arrays. Our do require any additional assumptions. As an application our we find necessary sufficient conditions the classical Edgeworth expansion. It turns out that there are two possible obstructions validity expansion order r. First, distance between distribution underlying modulo some h∈N uniform could fail to be o(σN1−r), where σN is standard deviation sum. Second, this required closeness unstable, in sense it destroyed by removing finitely many terms. In first case, r fails. second case may or hold depending on behavior derivatives characteristic functions summands whose removal causes break-up distribution. show a quantitative version Prokhorov condition (for strong central limit theorem) expansions, moreover is, sense, optimal.