New Explicit Examples of Fixed Points of Poisson Shot Noise Transforms

The main objective of this paper is to establish a close relation between the fixed points of Poisson shot noise transforms and perpetuities of a special type. With this relation it is shown that the gamma distributions, the generalized positive Linnik distributions, and the S2 distributions are fixed points of Poisson shot noise transforms. The paper also proves that log‐convexity of the response functions is not needed for non‐negative Poisson shot noise distributions to be self‐decomposable. Finally, the problems of existence and uniqueness of the above mentioned perpetuities are investigated.