Singularity Analysis for Heavy-Tailed Random Variables

Nicholas M. Ercolani, Sabine Jansen, Daniel Ueltschi

Research output: Contribution to journalArticlepeer-review

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) = cexp (- k α ) and apply to logarithmic hazard functions cexp (- (log k) β ) , 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.

Original languageEnglish (US)
JournalJournal of Theoretical Probability
Volume32
Issue number1
DOIs
StatePublished - Mar 15 2019

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

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