Regenerative processes for Poisson zero polytopes

TL;DR

We study the stationary renormalized zero-cell process for tessellations of $\mathbb{R}^\ell$ with $a>1$, showing a regenerative structure and Bernoulli-flow property. By analyzing the associated 0-1 indicators $V^K_n={\bf 1}_{\{\Gamma_n\supset K\}}$, we construct a stationary renewal set $\mathcal{V}^*$ and derive the interarrival law via transition probabilities, revealing the dependence structure between scales $\mathcal{V}^{aK*}$ and $\mathcal{V}^{K*}$. The framework unifies Poisson hyperplane tessellations and the STIT tessellations under renormalization, establishing that the renormalized zero-cell processes are Bernoulli flows of infinite entropy through explicit factor maps. These results provide a rigorous ergodic-theoretic characterization of regenerative phenomena in random tessellations, with implications for the asymptotic behavior and complexity of STIT-like models.

Abstract

Let $(M_t: t > 0)$ be a Markov process of tessellations of ${\mathbb R}^\ell$ and $({\cal C}_t:\, t > 0)$ the process of their zero cells (zero polytopes) which has the same distribution as the corresponding process for Poisson hyperplane tessellations. Let $a>1$. Here we describe the stationary zero cell process $(a^t {\cal C}_{a^t}:\, t\in {\mathbb R})$ in terms of some regenerative structure and we prove that it is a Bernoulli flow. An important application are the STIT tessellation processes.

Theorems & Definitions (17)

• Lemma 1
• proof
• Proposition 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Proposition 2
• proof