  We obtain asymptotic expansions for local probabilities of partial sums for
uniformly bounded independent but not necessarily identically distributed
integer-valued random variables. The expansions involve products of polynomials
and trigonometric polynomials. Our results do not require any additional
assumptions. As an application of our expansions we find necessary and
sufficient conditions for the classical Edgeworth expansion. It turns out that
there are two possible obstructions for the validity of the Edgeworth expansion
of order $r$. First, the distance between the distribution of the underlying
partial sums modulo some $h\in \bbN$ and the uniform distribution could fail to
be $o(\sigma_N^{1-r})$, where $\sigma_N$ is the standard deviation of the
partial sum. Second, this distribution could have the required closeness but
this closeness is unstable, in the sense that it could be destroyed by removing
finitely many terms. In the first case, the expansion of order $r$ fails. In
the second case it may or may not hold depending on the behavior of the
derivatives of the characteristic functions of the summands whose removal
causes the break-up of the uniform distribution. We also show that a
quantitative version of the classical Prokhorov condition (for the strong local
central limit theorem) is sufficient for Edgeworth expansions, and moreover
this condition is, in some sense, optimal.
