In this paper, we consider approximating expansions for the distribution of
integer valued random variables, in circumstances in which convergence in law
cannot be expected. The setting is one in which the simplest approximation to
the n'th random variable Xn is by a particular member Rn of a given
family of distributions, whose variance increases with n. The basic
assumption is that the ratio of the characteristic function of Xn and that
of R_n$ converges to a limit in a prescribed fashion. Our results cover a
number of classical examples in probability theory, combinatorics and number
theory
