# The largest order statistics for the inradius in an isotropic STIT tessellation

## TL;DR

This paper analyzes the asymptotic distribution of the largest inradius among cells in a planar isotropic STIT tessellation, providing new limit theorems using the Chen-Stein method.

## Contribution

It introduces the first limit distribution results for the largest inradius in an isotropic STIT tessellation as the observation window grows.

## Key findings

- Limit distributions of the largest inradius are derived.
- Asymptotic behavior of maximum inradius is characterized.
- Results apply to large-scale tessellation observations.

## Abstract

A planar stationary and isotropic STIT tessellation at time $t>0$ is observed in the window $W_\rho={t^{-1}}\sqrt{\pi \ \rho}\cdot [-\frac{1}{2},\frac{1}{2}]^2$, for $\rho>0$. With each cell of the tessellation, we associate the inradius, which is the radius of the largest disk contained in the cell. Using the Chen-Stein method, we compute the limit distributions of the largest order statistics for the inradii of all cells whose nuclei are contained in $W_\rho$ as $\rho$ goes to infinity.