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, Lindelof 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) and apply to logarithmic hazard functions cexp(-(logk)), >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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Faculties: | Mathematics, Computer Science and Statistics > Mathematics |
Subjects: | 500 Science > 510 Mathematics |
ISSN: | 0894-9840 |
Language: | English |
Item ID: | 82387 |
Date Deposited: | 15. Dec 2021, 15:01 |
Last Modified: | 13. Aug 2024, 12:43 |