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

2025/08/29 by Nicolas Lanchier, Lanchier, Nicolas
Biochemistry, Genetics and Molecular Biology · Mathematics · #60G55 #Diffusion and Search Dynamics #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2508.21303

openalex publication_date 2025/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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