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

## TL;DR

This paper analyzes the solution of the distributional equation X=AX+1 for discrete A, presenting a fast algorithm to compute the heavy tail density, with detailed study of the compound product distribution for specific cases.

## Contribution

It introduces a fast algorithm for calculating the heavy tail density of solutions to the distributional equation X=AX+1, with detailed analysis of the compound product distribution.

## Key findings

- Developed a computationally efficient method for heavy tail density calculation.
- Provided detailed analysis for specific distribution families.
- Demonstrated the applicability to compound product distributions.

## 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.