On the singularity of Laplace-Stieltjes transform of a heavy tailed random variable

In this paper, we give a sufficient condition for a non-negative random variable X being heavy tailed. A light tailed case is studied in. The sufficient condition is an analytic property of Laplace-Stieltjes transform of the probability distribution function of X. The proof is based on the complex Tauberian theorem by Graham-Vaaler.