Singularity Analysis for Heavy-Tailed Random Variables

Nicholas M Ercolani, Sabine Jansen, Daniel Ueltschi

Research output: Contribution to journalArticle

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 (Formula presented.) and apply to logarithmic hazard functions (Formula presented.), (Formula presented.); 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)
Pages (from-to)1-46
Number of pages46
JournalJournal of Theoretical Probability
DOIs
StateAccepted/In press - May 26 2018

Fingerprint

Singularity Analysis
Random variable
Nonconvex Variational Problems
Theorem
Sums of I.i.d. Random Variables
Moderate Deviations
Hazard Function
Random process
Information Theory
Saddlepoint
Large Deviations
Transactions
Logarithmic
Jump
Heuristics
Cover
Generalise
Integer
Random variables
Singularity

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

Cite this

Singularity Analysis for Heavy-Tailed Random Variables. / Ercolani, Nicholas M; Jansen, Sabine; Ueltschi, Daniel.

In: Journal of Theoretical Probability, 26.05.2018, p. 1-46.

Research output: Contribution to journalArticle

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