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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 \(X_n\) is by a particular member \(R_n\) of a given family of distributions, whose variance increases with \(n\). The basic assumption is that the ratio of the characteristic function of \(X_n\) to that of \(R_n\) converges to a limit in a prescribed fashion. Our results cover and extend a number of classical examples in probability, combinatorics and number theory.
KeywordsMod–Poisson convergence Characteristic function Poisson–Charlier expansion Erdős–Kac theorem
Mathematics Subject Classification (2000)62E17 60F05 60C05 60E10 11N60
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