Strong mixing property for STIT tessellations

TL;DR

This paper proves the strong mixing property for STIT tessellations, demonstrating how the dependence between events in disjoint regions diminishes with distance, and determines the optimal decay rate of this dependence.

Contribution

The paper establishes the strong mixing property for STIT tessellations and derives the optimal decay rate of dependence between events separated by a translation.

Findings

Proves strong mixing property for STIT tessellations

Derives the optimal decay rate of dependence between separated events

Quantifies the rate at which correlations diminish with distance

Abstract

The so-called STIT tessellations form the class of homogeneous (spatially stationary) tessellations of $\mathbb{R}^d$ which are stable under the nesting/iteration operation. In this paper, we establish the strong mixing property for these tessellations and give the optimal form of the rate of decay for the quantity $|\mathbb{P}({A}\cap Y=\emptyset,T_h B \cap Y=\emptyset)-\mathbb{P}({A}\cap Y=\emptyset)\mathbb{P}({B}\cap Y=\emptyset)|$ when $A$ and $B$ are two compact sets, $h$ a vector of $\mathbb{R}^d$, $T_{h}$ the corresponding translation operator and $Y$ a STIT Tessellation.