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Singularity analysis for heavy-tailed random variables

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Ercolani, Nicholas M., Jansen, Sabine and Ueltschi, Daniel (2019) Singularity analysis for heavy-tailed random variables. Journal of Theoretical Probability, 321 . pp. 1-46. doi:10.1007/s10959-018-0832-2

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Official URL: https://doi.org/10.1007/s10959-018-0832-2

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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 S. V. Nagaev (1973). The theorems generalize five theorems by A. V. Nagaev (1968) on stretched exponential laws p (k) = c exp(− k α) and apply to logarithmic hazard functions c exp(− (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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Random variables, Integrals
Journal or Publication Title: Journal of Theoretical Probability
Publisher: Springer New York LLC
ISSN: 0894-9840
Official Date: 15 March 2019
Dates:
DateEvent
15 March 2019Published
26 May 2018Available
17 May 2018Accepted
Volume: 321
Page Range: pp. 1-46
DOI: 10.1007/s10959-018-0832-2
Status: Peer Reviewed
Publication Status: Published
Publisher Statement: This is a post-peer-review, pre-copyedit version of an article published in Journal of Theoretical Probability. The final authenticated version is available online at: https://doi.org/10.1007/s10959-018-0832-2
Access rights to Published version: Restricted or Subscription Access
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
DMS-1212167National Science Foundationhttp://dx.doi.org/10.13039/100000001
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