Regenerative processes for Poisson zero polytopes

• Published in: Advances in applied probability Vol. 50; no. 4; pp. 1217 - 1226
• Main Authors: Martínez, Servet, Nagel, Werner
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.12.2018; Applied Probability Trust
• Subjects: Hyperplanes; Markov processes; Mathematical analysis; Original Article; Poisson distribution; Polytopes; Probability; Tessellation; Stochastic geometry; Primary 60D05; 60J75; zero polytope; Bernoulli flow; 37A35; random tessellation; STIT tessellation; 37A25; Poisson hyperplane tessellation; regenerative process; 60J25
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1017/apr.2018.57

Let (Mt:t>0) be a Markov process of tessellations of ℝℓ, and let ( t:t>0) be the process of their zero cells (zero polytopes), which has the same distribution as the corresponding process for Poisson hyperplane tessellations. In the present paper we describe the stationary zero cell process (at at:t∈ℝ),a>1, in terms of some regenerative structure and we show that it is a Bernoulli flow. An important application is to STIT tessellation processes.