# Joseph Mecke's last fragmentary manuscripts - a compilation

## TL;DR

This paper analyzes Joseph Mecke's last manuscripts, introducing the concept of thickening of random variables, and explores new relations between exponential distributions and Poisson processes using generating functions and Laplace transforms.

## Contribution

It introduces the novel concept of thickening of random variables and characterizes thickable variables, expanding the understanding of distribution transformations.

## Key findings

- Thickening is introduced as an inverse to thinning.
- Characterization of thickable random variables is provided.
- New relations between exponential distributions and Poisson processes are derived.

## Abstract

Summarizing results from Joseph Mecke's last fragmentary manuscripts, the generating function and the Laplace transform for nonnegative random variables are considered. The concept of thickening of a random variable, as an inverse operation to thinning (which is usually applied to point processes) is introduced, based on generating functions, and a characterization of thickable random variables is given. Further, some new relations between exponential distributions and their interpretation in terms of Poisson point processes are derived with the help of the Laplace transform.