Singularity analysis for heavy-tailed random variables

TL;DR

This paper introduces a new complex-analytic approach to analyze sums of heavy-tailed, integer-valued i.i.d. random variables, extending classical deviation results with precise asymptotics and probabilistic insights.

Contribution

It develops a novel method combining singularity analysis, Lindelöf integrals, and saddle points to derive precise large and moderate deviation theorems for heavy-tailed distributions.

Findings

Proves three theorems on large and moderate deviations for heavy-tailed variables.

Generalizes classical results to stretched exponential and logarithmic hazard functions.

Provides a probabilistic heuristic and identifies critical sequences via a variational problem.

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, 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.