Precise large deviations through a uniform Tauberian theorem

TL;DR

This paper develops a uniform Tauberian theorem approach to derive large deviation principles for random variables in the domain of attraction of stable laws, applicable to complex cases beyond traditional methods.

Contribution

It introduces a versatile method using a uniform Tauberian theorem for Laplace-Stieltjes transforms, enabling analysis of large deviations in challenging scenarios.

Findings

Applicable to random walks with long-range memory kernels

Handles randomly stopped sums with infinite mean or non-concentrated stopping times

Reveals the role of characteristic functions when Cramér's condition fails

Abstract

We derive a large deviation principle for families of random variables in the basin of attraction of spectrally positive stable distributions by proving a uniform version of the Tauberian theorem for Laplace-Stieltjes transforms. The main advantage of this method is that it can be easily applied to cases that are beyond the reach of the techniques currently used in the literature. Notable examples include large deviations for random walks with long-ranged memory kernels, as well as for randomly stopped sums where the random time $\mathrm N$ is either not concentrated around its expectation or has an infinite mean. The method reveals the role of the characteristic function when Cram\'er's condition is violated and provides a unified approach within regular variation.