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Probability Theory and Related Fields

, Volume 158, Issue 3–4, pp 859–893 | Cite as

Mod-discrete expansions

  • A. D. Barbour
  • E. Kowalski
  • A. Nikeghbali
Article

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 \(X_n\) 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.

Keywords

Mod–Poisson convergence Characteristic function Poisson–Charlier expansion Erdős–Kac theorem 

Mathematics Subject Classification (2000)

62E17 60F05 60C05 60E10 11N60 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut für Mathematik, Universität ZürichZürichSwitzerland
  2. 2.ETH Zurich, D-MATHZürichSwitzerland

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