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Singularity Analysis for Heavy-Tailed Random Variables

Author

Listed:
  • Nicholas M. Ercolani

    (The University of Arizona)

  • Sabine Jansen

    (Ludwigs-Maximilians Universität München)

  • Daniel Ueltschi

    (University of Warwick)

Abstract

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 $$p(k) = c\exp ( -k^\alpha )$$ p ( k ) = c exp ( - k α ) and apply to logarithmic hazard functions $$c\exp ( - (\log k)^\beta )$$ c exp ( - ( log k ) β ) , $$\beta >2$$ β > 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.

Suggested Citation

  • Nicholas M. Ercolani & Sabine Jansen & Daniel Ueltschi, 2019. "Singularity Analysis for Heavy-Tailed Random Variables," Journal of Theoretical Probability, Springer, vol. 32(1), pages 1-46, March.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:1:d:10.1007_s10959-018-0832-2
    DOI: 10.1007/s10959-018-0832-2
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    References listed on IDEAS

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    1. Armendáriz, Inés & Grosskinsky, Stefan & Loulakis, Michail, 2013. "Zero-range condensation at criticality," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3466-3496.
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