Precise large deviations through a uniform Tauberian theorem

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 Cramér's condition is violated and provides a unified approach within regular variation.