Singularity analysis for heavy-tailed random variables

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 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.

32 pages, 3 figures

References in corpus (2)

• Large deviations for random walks under subexponentiality: the big-jump domain
• Zero Range Condensation at Criticality

Cited by in corpus (1)

• Ground state representation for the fractional Laplacian with Hardy potential in angular momentum channels