Journal of Theoretical Probability

, Volume 32, Issue 1, pp 1–46 | Cite as

Singularity Analysis for Heavy-Tailed Random Variables

  • Nicholas M. Ercolani
  • Sabine JansenEmail author
  • Daniel Ueltschi


We propose a novel complex-analytic method for sums of i.i.d. random variables that are heavy-tailed and integer-valued. The method combines singularity analysis, Lindelöf integrals, and bivariate saddle points. As an application, we prove three theorems on precise large and moderate deviations which provide a local variant of a result by Nagaev (Transactions of the sixth Prague conference on information theory, statistical decision functions, random processes, Academia, Prague, 1973). The theorems generalize five theorems by Nagaev (Litov Mat Sb 8:553–579, 1968) on stretched exponential laws \(p(k) = c\exp ( -k^\alpha )\) and apply to logarithmic hazard functions \(c\exp ( - (\log k)^\beta )\), \(\beta >2\); they cover the big-jump domain as well as the small steps domain. The analytic proof is complemented by clear probabilistic heuristics. Critical sequences are determined with a non-convex variational problem.


Local limit laws Large deviations Heavy-tailed random variables Asymptotic analysis Lindelöf integral Singularity analysis Bivariate steepest descent 

Mathematics Subject Classification (2010)

05A15 30E20 44A15 60F05 60F10 



The authors wish to thank the Laboratoire Jean Dieudonné of the University of Nice and the Institut Henri Poincaré (during the program of the spring 2013 organized by M. Esteban and M. Lewin) for their kind hospitality and for the opportunity to discuss this project. S. J. thanks V. Wachtel for pointing out Nagaev’s article [14].


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Authors and Affiliations

  1. 1.Department of MathematicsThe University of ArizonaTucsonUSA
  2. 2.Mathematisches InstitutLudwigs-Maximilians Universität MünchenMunichGermany
  3. 3.Department of MathematicsUniversity of WarwickCoventryUK

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