Singularity Analysis for Heavy-Tailed Random Variables
| Autor: | Nicholas M. Ercolani, Daniel Ueltschi, Sabine Jansen |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Statistics and Probability
Asymptotic analysis General Mathematics 01 natural sciences Domain (mathematical analysis) 010104 statistics & probability Saddle point FOS: Mathematics Mathematics - Combinatorics Complex Variables (math.CV) 0101 mathematics QA Mathematics Discrete mathematics Mathematics - Complex Variables Stochastic process Probability (math.PR) 010102 general mathematics 05A15 30E20 44A15 60F05 60F10 Cover (topology) Large deviations theory Combinatorics (math.CO) Statistics Probability and Uncertainty Random variable Mathematics - Probability Analytic proof |
| Zdroj: | Journal of Theoretical Probability. 32:1-46 |
| ISSN: | 1572-9230 0894-9840 |
| 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: | OpenAIRE |
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