### 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 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 (Formula presented.) and apply to logarithmic hazard functions (Formula presented.), (Formula presented.); 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1-46 |

Number of pages | 46 |

Journal | Journal of Theoretical Probability |

DOIs | |

State | Accepted/In press - May 26 2018 |

### Fingerprint

### Keywords

- Asymptotic analysis
- Bivariate steepest descent
- Heavy-tailed random variables
- Large deviations
- Lindelöf integral
- Local limit laws
- Singularity analysis

### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Theoretical Probability*, 1-46. https://doi.org/10.1007/s10959-018-0832-2

**Singularity Analysis for Heavy-Tailed Random Variables.** / Ercolani, Nicholas M; Jansen, Sabine; Ueltschi, Daniel.

Research output: Contribution to journal › Article

*Journal of Theoretical Probability*, pp. 1-46. https://doi.org/10.1007/s10959-018-0832-2

}

TY - JOUR

T1 - Singularity Analysis for Heavy-Tailed Random Variables

AU - Ercolani, Nicholas M

AU - Jansen, Sabine

AU - Ueltschi, Daniel

PY - 2018/5/26

Y1 - 2018/5/26

N2 - 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 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 (Formula presented.) and apply to logarithmic hazard functions (Formula presented.), (Formula presented.); 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.

AB - 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 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 (Formula presented.) and apply to logarithmic hazard functions (Formula presented.), (Formula presented.); 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.

KW - Asymptotic analysis

KW - Bivariate steepest descent

KW - Heavy-tailed random variables

KW - Large deviations

KW - Lindelöf integral

KW - Local limit laws

KW - Singularity analysis

UR - http://www.scopus.com/inward/record.url?scp=85047416183&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047416183&partnerID=8YFLogxK

U2 - 10.1007/s10959-018-0832-2

DO - 10.1007/s10959-018-0832-2

M3 - Article

AN - SCOPUS:85047416183

SP - 1

EP - 46

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

ER -