Singularity analysis for heavy-tailed random variables
| Autor: | Ercolani, Nicholas M., Jansen, Sabine, Ueltschi, Daniel |
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| Rok vydání: | 2015 |
| Předmět: | |
| Zdroj: | J. Theor. Probab. 32, 1-46 (2019) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s10959-018-0832-2 |
| Popis: | 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, Lindel\"of 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 $p(k) = c\exp( -k^\alpha)$ and apply to logarithmic hazard functions $c\exp( - (\log k)^\beta)$, $\beta>2$; 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. Comment: 32 pages, 3 figures |
| Databáze: | arXiv |
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