Mod-discrete expansions

Abstract

In this paper, we consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law (without normalization) 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 to that of Rn converges to a limit in a prescribed fashion. Our results cover and extend a number of classical examples in probability, combinatorics and number theory. © 2013 Springer-Verlag Berlin Heidelberg.