We give a sufficient condition for a non-negative random variable $X$ to be of Pareto type by investigating the Laplace-Stieltjes transform of the cumulative 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. For the proof of our theorems, we apply Graham-Vaaler’s complex Tauberian theorem. As an application of our theorems, we consider the asymptotic decay of the stationary distribution...

We say that an entire function $f\left(z\right)={\sum }_{k=0}{a}_k{z}^{n_k}(0=n_0\<n_1\<n_2\<...)$ has Fejér gaps if ${\sum }_{k=1}^{\infty }1/n_k\<\infty .$ The main result of this paper is as follows: An entire function with Fejér gaps has no finite deficient value.