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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
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 | ||||||||
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| 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: |
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| 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 | ||||||||
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