The compound product distribution; a solution to the distributional equation X=AX+1

Abstract

The solution of $X=AX+1$ is analyzed for a discrete variable $A$ with $\mathbb{P}\left[A=0\right]>0$. Accordingly, a fast algorithm is presented to calculate the obtained heavy tail density. To exemplify, the compound product distribution is studied in detail for some particular families of distributions.

10 pages, 5 figures, one appendix