# Short proof of the conditioning property for multi-dimensional Poisson point processes

## TL;DR

This paper provides a concise proof of the conditioning property for multi-dimensional Poisson point processes, addressing the challenge posed by the lack of independence in higher dimensions.

## Contribution

It introduces a novel, simplified proof of the conditioning property specifically for multi-dimensional Poisson point processes.

## Key findings

- The proof extends the conditioning property to higher dimensions.
- It clarifies the role of independence and exponential distribution in the proof.
- The approach simplifies understanding of multi-dimensional Poisson processes.

## Abstract

Poisson processes and one-dimensional Poisson point processes satisfy three main properties: superposition, thinning, and conditioning. The proof of the first two relies on basic estimates involving the Poisson distribution that are also true for multi-dimensional Poisson point processes. In contrast, the proof of conditioning uses that the distances between consecutive occurrences in time or entities in space are independent and exponentially distributed, which is nonsensical in higher dimensions. This paper gives a short proof of the conditioning property for multi-dimensional Poisson point processes.