Journal of Theoretical Probability

, Volume 26, Issue 2, pp 360–385 | Cite as

Moderate Deviations via Cumulants

  • Hanna Döring
  • Peter EichelsbacherEmail author


The purpose of the present paper is to establish moderate deviation principles for a rather general class of random variables fulfilling certain bounds of the cumulants. We apply a celebrated lemma of the theory of large deviations probabilities due to Rudzkis, Saulis, and Statulevičius. The examples of random objects we treat include dependency graphs, subgraph-counting statistics in Erdös–Rényi random graphs and U-statistics. Moreover, we prove moderate deviation principles for certain statistics appearing in random matrix theory, namely characteristic polynomials of random unitary matrices and the number of particles in a growing box of random determinantal point processes such as the number of eigenvalues in the GUE or the number of points in Airy, Bessel, and sine random point fields.


Moderate deviations Cumulants Large deviation probabilities Dependency graphs Random graphs U-statistics Characteristic polynomials Random matrix ensembles Determinantal point processes 

Mathematics Subject Classification

60F10 05C80 62G20 60B20 



Both authors have been supported by Deutsche Forschungsgemeinschaft via SFB/TR 12. The first author was supported by the international research training group 1339 of the DFG.


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institut für Mathematik, MA 767Technische Universität BerlinBerlinGermany
  2. 2.Fakultät für Mathematik, NA 3/68Ruhr-Universität BochumBochumGermany

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