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In this paper, we will give a sufficient condition for a non-negative randomvariable $X$ to be heavy tailed by investigating the Laplace-Stieltjestransform of the probability distribution function. We focus on the relationbetween the singularity at the real point of the axis of convergence and theasymptotic decay of the tail probability. Our theorem is a kind of Tauberiantheorems.

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