The Real Powers of the Convolution of a Gamma Distribution and a Bernoulli Distribution

Abstract

In this paper, we essentially compute the set of $x,y>0$ such that the mapping $z \longmapsto \Big{(}1-r+r e^z\Big{)}^x \Big{(}\dis\fracλ{λ-z}\Big{)}^{y}$ is a Laplace transform. If $X$ and $Y$ are two independent random variables which have respectively Bernoulli and Gamma distributions, we denote by $μ$ the distribution of $X+Y.$ The above problem is equivalent to finding the set of $x>0$ such that $μ^{{\ast}x}$ exists.