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

TL;DR

This paper characterizes the conditions under which the convolution powers of a distribution, formed by a Bernoulli and a Gamma distribution, are valid Laplace transforms, revealing new insights into their combined behavior.

Contribution

It provides a detailed analysis of the set of parameters for which the convolution powers of a Bernoulli-Gamma distribution exist and are Laplace transforms, extending understanding of their algebraic properties.

Findings

Identifies the set of parameters where the convolution power exists

Characterizes the Laplace transform of the convolution of Bernoulli and Gamma distributions

Provides conditions for the existence of convolution powers

Abstract

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