Tail Probability and Singularity of Laplace-Stieltjes Transform of a Heavy Tailed Random Variable

TL;DR

This paper establishes a sufficient condition linking the singularity of the Laplace-Stieltjes transform at the convergence boundary to the heavy tail behavior of a non-negative random variable, providing a Tauberian theorem for tail analysis.

Contribution

It introduces a new Tauberian theorem connecting the singularity of the Laplace-Stieltjes transform to heavy tail properties of distributions.

Findings

Identifies a sufficient condition for heavy tails based on transform singularity.

Provides a theoretical framework for tail probability asymptotics.

Links the decay rate of tail probabilities to the transform's singularity.

Abstract

In this paper, we will give a sufficient condition for a non-negative random variable $X$ to be heavy tailed by investigating the Laplace-Stieltjes transform of the probability distribution function. We focus on the relation between the singularity at the real point of the axis of convergence and the asymptotic decay of the tail probability. Our theorem is a kind of Tauberian theorems.