Singularity Analysis for Heavy-Tailed Random Variables

Nicholas M. Ercolani (), Sabine Jansen () and Daniel Ueltschi ()
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Nicholas M. Ercolani: The University of Arizona
Sabine Jansen: Ludwigs-Maximilians Universität München
Daniel Ueltschi: University of Warwick

Journal of Theoretical Probability, 2019, vol. 32, issue 1, 1-46

Abstract: 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.

Keywords: Local limit laws; Large deviations; Heavy-tailed random variables; Asymptotic analysis; Lindelöf integral; Singularity analysis; Bivariate steepest descent; 05A15; 30E20; 44A15; 60F05; 60F10 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-018-0832-2

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